Entry Trajectory Optimization Research Articles

Entry trajectory optimization of lifting-body vehicle by successive difference-of-convex programming

The complexity of the three-dimensional entry trajectory optimization problem has escalated due to the need to liberalize the angle of attack and bank angle as control variables, thereby enhancing the inherent maneuverability and control capabilities of lifting-body vehicles. The difference-of-convex (DC) properties inherent in the constraints of the problem are exploited in this paper. A DC decomposition approach is utilized to address the nonlinear auxiliary control equations, and the DC relaxation technique is applied to resolve iteration infeasibilities arising from Taylor expansion. The dependence on the initial trajectory is diminished by the implementation of an exact penalty method, thus improving the applicability of the methods. Furthermore, a control variable oscillation suppression mechanism has been constructed to tackle the control variable oscillation issues arising from the relaxation of the angle of attack and bank angle. This mechanism effectively suppresses large jumps in the angle of attack and high-frequency oscillations in the bank angle. Two novel successive DC programming methods are proposed: the successive concave-convex procedure and the successive proximal bundle method, functioning independently of trust-region constraints. Numerical experiments have demonstrated that the two proposed successive DC optimization methods exhibit exceptional performance in accuracy, feasibility, adaptability, and low sensitivity to initial values when applied to solving the three-dimensional entry trajectory optimization problem.

Improved sequential convex programming based on pseudospectral discretization for entry trajectory optimization

Sequential convex programming (SCP) is widely used to solve entry trajectory optimization problems. However, challenges persist in scenarios with strict constraints, such as unsolvable convex subproblems, iterative solution oscillations, and slow convergence rates. This study introduces an enhanced SCP algorithm designed to address these limitations. First, hp Radau pseudospectral discretization is used instead of trapezoidal discretization to improve the efficiency in solving subproblems while maintaining discretization accuracy. Second, the trust region is adaptively updated on the basis of trajectory information during iterations. Additionally, constraint relaxation and virtual control are introduced to facilitate smooth iteration at the initial stage. The regularization technique is also utilized to improve the convergence rates. Finally, the proposed algorithm is validated through two examples: maximum-terminal-velocity entry and maximum-terminal-longitude entry. The results show that these two problems cannot be effectively solved using the basic SCP algorithm. However, the proposed algorithm, along with the trust-region SCP algorithm and GPOPS, can solve them efficiently. With comparable accuracy in the obtained solutions, the algorithm proposed in this paper requires only half the CPU time of the trust-region SCP algorithm and just 5 % of the computing time of the general-purpose open-source solver GPOPS.

Reduced-Order Desensitized Optimal Control for Mars Entry Trajectory Optimization

Reduced-Order Desensitized Optimal Control for Mars Entry Trajectory Optimization

Entry Trajectory Optimization With General Polygonal No-Fly Zone Constraints

Entry Trajectory Optimization With General Polygonal No-Fly Zone Constraints

A Hybrid ADMM for Six-Degree-of-Freedom Entry Trajectory Optimization Based on Dual Quaternions

This paper investigates the six-degree-of-freedom (6-DoF) entry trajectory optimization problem in a Human-Mars entry, powered descent, and landing mission. During the entry phase, aerodynamic forces are employed to decelerate the vehicle. Instead of being treated as a point mass, both translational and rotational motions of the entry vehicle are considered. Specifically, the 6-DoF rigid body motion of the entry vehicle is modeled using the unit dual quaternion representations to avoid highly non-linear terms in the flight dynamics expression originally based on the flight-path coordinates. Then, the entry trajectory optimization problem is to minimize the terminal speed subject to dynamical, operational, and mission constraints modeled by the new representation scheme. By applying the discretization technique and polynomial approximation, the entry trajectory optimization problem is reformulated as a nonconvex quadratically constrained quadratic programming (QCQP) problem, which is solved via a hybrid alternating direction method of multipliers (ADMM). The accuracy of the dual quaternion based model and the computational efficiency of the hybrid ADMM are validated via numerical simulations.

Entry trajectory optimization for hypersonic vehicles based on convex programming and neural network

Trajectory optimization is important in achieving long-range atmospheric entry hypersonic vehicles. However, the trajectory optimization problem for atmospheric entry of hypersonic vehicles is characterized by strong nonlinearity, parameter uncertainties and multiple constraints. This study proposes a novel online trajectory optimization method for hypersonic vehicles based on convex programming and a feedforward neural network. A sequential second-order cone programming (SOCP) method is obtained to describe the trajectory optimization problem after the Gauss pseudo-spectral discretization. Subsequently, multiple optimal trajectories under aerodynamic uncertainties are generated offline and classified as the training and validation datasets. Then, a multilayer feedforward neural network is trained using these datasets and to output the optimal control command online. This method yields approximately 95% shorter computation time compared with the offline SOCP method. Considering the existence of the aerodynamic uncertainties, three terminal states calculated by this method are all smaller than 4.1%. In conclusion, the proposed trajectory optimization method can provide a high-precision, robust entry trajectory for hypersonic vehicles efficiently.

A fast matrix generation method for solving entry trajectory optimization problems under the pseudospectral method

A fast matrix generation method based on correlations is proposed for the pseudospectral method (PSM) to solve the entry trajectory optimization problem. First, the optimization problem is analyzed to obtain some assumptions of performance indicators and constraints. Then, the chain rule and correlations among the variables are used to simplify the gradient calculations, resulting in a fast generation method for the gradient vector. Meanwhile, based on the previous assumptions, a fast generation method for the Jacobi matrix is derived semi-analytically using the correlation information among the variables. Finally, the trajectory optimization problem is studied using a typical aircraft. The results are verified via comparisons with the well-known GPOPS II software. Furthermore, the superiority of the proposed method in terms of the computational efficiency is demonstrated by comparing the relevant time indexes. The performance of the proposed method in calculating the Jacobian matrix and gradient matches that of the GPOPS, combined with a relatively short solution time.

End-to-End Performance Optimization for High-Speed Ascent–Entry Missions

The performance optimization for a combined ascent–entry mission subject to constraints on heating rate and heating load is studied. The ascent vehicle is modeled as a three-stage rocket that places the entry vehicle onto a suborbital exo-atmospheric trajectory after which the vehicle undergoes an unpowered entry and descent to a vertically downward terminal condition. The entry vehicle is modeled as a high-lift-to-drag-ratio vehicle that is capable of withstanding high levels of thermal and structural loads. A performance index is designed to improve control margin while attenuating phugoid oscillations during atmospheric entry. Furthermore, a mission corresponding to a prototype launch and target point is used in this study. The trajectory optimization problem is formulated as a multiple-phase optimal control problem, and the optimal control problem is solved using an adaptive Gaussian quadrature collocation method. The key features of the optimized trajectories and controls are identified, and the approach developed in this paper provides a systematic method for end-to-end ascent–entry trajectory optimization.

Solving Constrained Trajectory Planning Problems Using Biased Particle Swarm Optimization

Constrained trajectory optimization has been a critical component in the development of advanced guidance and control systems. An improperly planned reference trajectory can be a main cause of poor online control performance. Due to the existence of various mission-related constraints, the feasible solution space of a trajectory optimization model may be restricted to a relatively narrow corridor, thereby easily resulting in local minimum or infeasible solution detection. In this article, we are interested in making an attempt to handle the constrained trajectory design problem using a biased particle swarm optimization approach. The proposed approach reformulates the original problem to an unconstrained multicriterion version by introducing an additional normalized objective reflecting the total amount of constraint violation. Besides, to enhance the progress during the evolutionary process, the algorithm is equipped with a local exploration operation, a novel ε-bias selection method, and an evolution RS. Numerical simulation experiments, obtained from a constrained atmospheric entry trajectory optimization example, are provided to verify the effectiveness of the proposed optimization strategy. Main advantages associated with the proposed method are also highlighted by executing a number of comparative case studies.

Improved Sequential Convex Programming Algorithms for Entry Trajectory Optimization

Entry trajectory optimization for hypersonic vehicles has been formulated as constrained optimal control problems, which are difficult to solve because of the existing high nonlinearity and nonconvexity. In recent years, convex optimization has shown promise for real-time onboard applications with the state-of-the-art interior-point methods. In this paper, line-search and trust-region techniques are introduced to fundamentally improve the performance of the sequential convex programming method for entry trajectory optimization. Two improved algorithms are developed: line-search sequential convex programming and trust-region sequential convex programming. In addition, a new trajectory generation method is proposed by taking advantage of the predictor–corrector method to find an initial 3-D trajectory for the developed successive algorithms. As such, the convergence of the solution process is improved. To demonstrate the effectiveness and performance of the newly proposed algorithms, numerical simulations are presented for maximum-terminal-velocity and minimum-heat-load entry problems. Results show that these two problems are not well solved using the basic sequential convex programming algorithm. However, they can be efficiently solved by the two improved algorithms in a few seconds in MATLAB, showing a significant improvement over general-purpose solvers, such as GPOPS.

Entry Trajectory Optimization With Virtual Motion Camouflage Principle

This article proposes a bio-inspired algorithm called virtual motion camouflage (VMC) to reduce the dimension and computational cost of entry trajectory optimization problem for the reusable launch vehicle. In the VMC framework, the trajectory to be planned can be obtained by predefining the virtual prey motion according to the boundary conditions, then optimizing the reference point and path control parameters using sequential quadratic programming algorithm. Compared with Gauss pseudospectral method, due to the proposed approach reduce the number of discrete points and the equality constraints of the optimization model, it can improve the convergence speed and reduce computational cost effectively. Two minimum time entry examples, without and with no-fly-zone constraints, are presented to demonstrate the effectiveness of the proposed approach.

Rapid trajectory optimization for hypersonic entry using convex optimization and pseudospectral method

Purpose This paper aims to develop a novel trajectory optimization algorithm which is capable of producing high accuracy optimal solution with superior computational efficiency for the hypersonic entry problem. Design/methodology/approach A two-stage trajectory optimization framework is constructed by combining a convex-optimization-based algorithm and the pseudospectral-nonlinear programming (NLP) method. With a warm-start strategy, the initial-guess-sensitive issue of the general NLP method is significantly alleviated, and an accurate optimal solution can be obtained rapidly. Specifically, a successive convexification algorithm is developed, and it serves as an initial trajectory generator in the first stage. This algorithm is initial-guess-insensitive and efficient. However, approximation error would be brought by the convexification procedure as the hypersonic entry problem is highly nonlinear. Then, the classic pseudospectral-NLP solver is adopted in the second stage to obtain an accurate solution. Provided with high-quality initial guesses, the NLP solver would converge efficiently. Findings Numerical experiments show that the overall computation time of the two-stage algorithm is much less than that of the single pseudospectral-NLP algorithm; meanwhile, the solution accuracy is satisfactory. Practical implications Due to its high computational efficiency and solution accuracy, the algorithm developed in this paper provides an option for rapid trajectory designing, and it has the potential to evolve into an online algorithm. Originality/value The paper provides a novel strategy for rapid hypersonic entry trajectory optimization applications.

Rapid trajectory optimization for hypersonic entry using a pseudospectral-convex algorithm

Aiming at improving the autonomy of hypersonic entry vehicles, a rapid trajectory optimization algorithm, which has the potential to be implemented online and onboard, is proposed in this paper. The nonlinear and nonconvex hypersonic entry trajectory optimization problem is transformed into a series of convex subproblems through a proper combination of the pseudospectral method and an improved successive convexification method; thus, the high discretization accuracy of the pseudospectral method and the fast and deterministic convergence properties of the convex-optimization-based algorithm can be simultaneously exploited. The resulting subproblems can be solved efficiently by matured interior-point methods, and the solution converges rapidly by adopting a novel dynamic trust-region updating approach. The optimality of the solution is verified by the optimal control theory. The effectiveness of the algorithm is demonstrated by numerical experiments.

Mars atmospheric entry trajectory optimization with maximum parachute deployment altitude using adaptive mesh refinement

In this paper, we present an adaptive mesh refinement method that is used in conjunction with local direct collocation methods for solving Mars entry trajectory optimization problems with a maximum parachute deployment altitude. The study focuses on the capability of the mesh refinement method to solve Mars entry trajectory optimization problems and the potential benefits of enabling angle-of-attack control. The numerical results show that Mars entry trajectory optimization problems can be solved accurately and efficiently using the proposed method. The accuracy is found to be comparable to that of indirect methods, and the computational time is on the order of several seconds, depending on the complexity of the problem. Furthermore, it is found that parachute deployment altitude gains of 2.2–3.1 km over pure bank control are possible if angle-of-attack control is enabled for a Mars Science Laboratory-type vehicle. Efforts are also made to investigate the influences of bank angle limits on entry trajectories, and it is found that the structure of the optimal bank angle profiles are generally similar for different bank angle limits, but a larger bank angle range allows an higher maximum parachute deployment altitude.

Mars entry trajectory robust optimization based on evidence under epistemic uncertainty

The epistemic uncertainties caused by insufficient knowledge of the atmosphere, aerodynamic coefficient, and entry state render the entry process challenging, as they not only result in the deviation of the preplanned trajectory, but also may lead to the nonsatisfaction of path constraints. Herein, a robust epistemic uncertainty optimization (REUO) method based on evidence is proposed to solve the Mars entry trajectory optimization problem under epistemic uncertainty. A two-loop nested robust optimization (RO) model is formulated, in which the outer-loop optimization searches the optimal control while the inner-loop optimization calculates the extremal trajectory performances within each focal element (FE) to evaluate the evidence level. To solve the path constraint violation problem under uncertainties, the constraint design based on limitation bounds is considered in the RO model. The polynomial chaos expansion (PCE) is employed to obtain the approximate analytic function of the trajectory performance under uncertainties. Thereafter, the optimization based on ordinary stochastic entry dynamics in the inner loop is replaced by a simple parameter optimization of the analytic functions, which can be readily and rapidly solved. The REUO method is tested in a specific Mars entry mission. The simulation results show that the proposed method can identify the most robust solutions with the optimal trajectory performance under epistemic uncertainties.

Time-optimal memetic whale optimization algorithm for hypersonic vehicle reentry trajectory optimization with no-fly zones

A novel time-optimal memetic whale optimization algorithm (WOA) integrating the Gauss pseudo-spectral methods (GPM), is proposed in this paper for the hypersonic vehicle entry trajectory optimization problem with no-fly zones. The WOA is featured with the strong global search ability and non-sensitive to the initial values, but also shows poor searching convergence speed around the global optimum. Conversely, GPM may be sensitive to the initial solution and easily trapped in a local optimum, but it also possesses more rapid convergence speed around the optimum and higher searching accuracy. Thus, a memetic optimization algorithm which contains a two-stage approach mechanism is proposed for searching the global optimum. The first searching stage, which is driven by an improved WOA (IWOA), works as an initializer of the entire searching due to its strong global search ability and non-sensitive to the initial values. The local optimum reservation and adaptive amplitude factor updating strategy are established to improve the convergent speed and the global search ability of the WOA. Once the changing of fitness value satisfies the predefined criterion, the next searching stage driven by GPM will take the place of the IWOA to expedite the search process around optimum and to obtain a precise global optimal solution. By this hybrid way, the proposed optimization algorithm may find an optimum more quickly and accurately. Simulation results show the proposed algorithm possesses faster convergence speed, higher accuracy, and stronger robustness for the hypersonic vehicle entry trajectory optimization.

Re-entry trajectory optimization using pigeon inspired optimization based control profiles

In this paper, the entry trajectory optimization problem of lifting type re-entry vehicle with path constraints is solved using Pigeon Inspired Optimization (PIO). Entry trajectory optimization problem involves finding the control profiles, bank angle, and angle of attack to guide the vehicle safely to the destination. The proposed approach parametrizes the bank angle to be a linear function of energy while the angle of attack is considered to be a monotonic function of Mach number. Thus, the problem of finding control profiles is transformed into three parameter search problem. The PIO algorithm is used to find the values of these parameters that minimizes the objective function. The terminal heading angle offset is minimized using traditional bank reversal logic. A new approach is proposed in which the bank angle is modulated to eliminate the oscillations observed in the altitude profile of an entry vehicle with a high lift to drag ratio (L/D). A methodology to satisfy the given load factor constraint is also proposed, as an alternative to traditional penalty factor approach used for incorporating path constraints in PIO algorithm. The proposed approach is further validated by considering sub-cases with different load factor limits and bank angle as the only control variable. The angle of attack profile obtained from the previous case is considered as the nominal profile. The proposed trajectory optimization strategy using PIO algorithm is simulated for Common Aero Vehicle with high L/D ratio (CAV-H) with different load factor constraint limits. The results show that the obtained angle of attack profile minimizes the peak heat rate experienced by the vehicle and bank angle modulation eliminates the oscillations in the altitude profile as well as makes the entry trajectory satisfy the load factor constraint.

Autonomous Entry Guidance for Hypersonic Vehicles by Convex Optimization

Advanced entry guidance systems could potentially enable future vehicles to generate feasible/optimal flight trajectories onboard for the latest mission requirements and track the new trajectories under uncertainties. To this end, an autonomous entry guidance algorithm is proposed for hypersonic vehicles using convex optimization methods. Based on previous sequential convex programming methods for entry trajectory optimization, a different successive procedure is developed for the online generation and tracking of reference trajectories from the current location to a desired target. In each guidance cycle, a new reference trajectory is generated by solving a second-order cone programming problem, and an optimal feedback guidance law is designed using a quadratically constrained quadratic programming method to track this reference trajectory. Because these two problems are convex, they are expected to be solved by state-of-the-art interior-point methods very efficiently. To demonstrate the effectiveness and performance of the proposed entry guidance algorithm, an offline convex entry guidance method is also designed for comparison based on a predetermined reference trajectory. Numerical simulations are presented for both offline and online approaches, which show that the proposed autonomous entry guidance algorithm is robust and efficient for potential future onboard guidance applications.

Reliability-based trajectory optimization using nonintrusive polynomial chaos for Mars entry mission

This paper presents the reliability-based sequential optimization (RBSO) method to settle the trajectory optimization problem with parametric uncertainties in entry dynamics for Mars entry mission. First, the deterministic entry trajectory optimization model is reviewed, and then the reliability-based optimization model is formulated. In addition, the modified sequential optimization method, in which the nonintrusive polynomial chaos expansion (PCE) method and the most probable point (MPP) searching method are employed, is proposed to solve the reliability-based optimization problem efficiently. The nonintrusive PCE method contributes to the transformation between the stochastic optimization (SO) and the deterministic optimization (DO) and to the approximation of trajectory solution efficiently. The MPP method, which is used for assessing the reliability of constraints satisfaction only up to the necessary level, is employed to further improve the computational efficiency. The cycle including SO, reliability assessment and constraints update is repeated in the RBSO until the reliability requirements of constraints satisfaction are satisfied. Finally, the RBSO is compared with the traditional DO and the traditional sequential optimization based on Monte Carlo (MC) simulation in a specific Mars entry mission to demonstrate the effectiveness and the efficiency of the proposed method.

Stochastic gradient particle swarm optimization based entry trajectory rapid planning for hypersonic glide vehicles

In this paper, a novel stochastic gradient particle swarm optimization (SGPSO) algorithm is proposed, which combines the high-efficiency of gradient search with the randomness of particle swarm search. By adjusting the stochastic gradient obtained by the best historical positions of adjacent two generations, the proposed algorithm can effectively overcome the problems of premature convergence and poor accuracy of standard particle swarm optimization (PSO). Due to the capabilities of rapidity, optimality and adaptability, the proposed algorithm is applied as a global optimization approach to rapidly generate feasible and smooth entry trajectories for hypersonic glide vehicles with highly constraints. Under the constraints of Earth's rotation and oblateness, the entry trajectory planning model is established. By parameterizing the control variables including angle of attack (AOA) and bank angle, the entry trajectory optimization problem is then converted to a multi-parameter optimization problem, which can be solved by the proposed SGPSO algorithm. Considering Common Aero Vehicle (CAV) model, the simulations show that the proposed algorithm has better performance on optimization speed, stability and solution optimality than those of the classical methods, and it can realize the rapid optimization of entry trajectory for hypersonic glide vehicles.