Aircraft online trajectory optimization warm start method based on symmetric neighborhood centering

Abstract

The application discloses a method for online trajectory optimization warm start of an aircraft based on symmetric neighborhood centralization, and comprises the following steps: S1, obtaining an optimization result of a sequence convex algorithm in solving a second-order cone programming sub-problem in the kth outer iteration, and taking the optimization result as an initial solution of the second-order cone programming sub-problem in the k+1th outer iteration; S2, introducing a symmetric neighborhood; S3, judging whether the deviation of the complementary product of each pair of relaxation variables and dual variables in the initial solution with respect to the average complementary gap exceeds the symmetric neighborhood; S4, constructing a joint correction system; S5, solving the joint correction system to obtain a warm start initial point; and S6, solving the second-order cone programming sub-problem in the current k+1th outer iteration to obtain an optimal trajectory of the current outer iteration. The application can improve the centrality and feasibility of the warm start initial point, thereby ensuring numerical stability and improving the calculation efficiency of the original dual interior point method for solving the second-order cone programming sub-problem.

Description

Technical Field

[0001] This invention relates to the field of online trajectory optimization technology for aircraft, and specifically to a hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization. Background Technology

[0002] In recent years, with the continuous advancement of aerospace technology and profound changes in industrial models, commercial aerospace has gradually moved from the conceptual exploration stage to engineering and large-scale application, becoming an inevitable development direction that balances technical feasibility, cost control, and operational efficiency. Among these advancements, new aerospace technologies, represented by reusable launch vehicles, have matured, significantly lowering the overall cost threshold for space launches and missions. Existing technologies have achieved vertical landing and recovery of the first-stage booster of launch vehicles, enabling multiple reuses and significantly improving the frequency of launches and the economics of commercial operation. These technological breakthroughs have driven the transformation of space launch models from traditional one-off, long-cycle missions to high-frequency, large-scale deployments and high-efficiency, low-cost operations. Simultaneously, commercial aerospace has been explicitly included in the strategic emerging industries system, with multiple policy documents encouraging social capital participation in the development of launch vehicles, satellite manufacturing, and aerospace application services. Some commercial aerospace companies are actively promoting the engineering application of liquid-fueled launch vehicles and reusable technologies. High-frequency launch missions and autonomous recovery processes place higher demands on the real-time trajectory planning and online optimization capabilities of spacecraft under complex dynamic constraints, various uncertainties, and strict safety constraints.

[0003] In the aforementioned application scenarios, aircraft trajectory optimization is typically performed online or in a rolling time domain. Its core lies in repeatedly and rapidly solving a series of structurally similar optimization problems under limited computational resources and strict real-time constraints. These optimization problems are often based on aircraft dynamics models and state and control constraints, and are transformed into large-scale numerical optimization problems through discretization and approximation. Since the mission environment and flight state change relatively smoothly in adjacent time intervals, two consecutive optimization problems exhibit strong consistency in variable size, constraint structure, and objective function form. However, the complete solution process still needs to be completed within each control cycle, placing high demands on the computational efficiency and numerical stability of the optimization algorithm.

[0004] Extensive research has been conducted in academia and engineering on the online trajectory optimization problem for aircraft. Theoretically, the aircraft trajectory optimization problem is typically modeled as an optimal control problem with dynamic constraints, state constraints, and control constraints. Methods for solving optimal control problems are mainly divided into indirect and direct methods. Indirect methods, based on Pontryagin's maximum principle, first derive the necessary analytical conditions and then solve the two-point boundary value problem. Direct methods discretize the continuous-time optimal control problem, representing the trajectory as finite-dimensional optimization variables, thus transforming the problem into a finite-dimensional nonlinear programming problem. With the improvement of computing power and the development of convex optimization theory, more and more research in recent years has focused on transforming and approximating the problem structure to convert the original nonlinear, non-convex trajectory optimization problem into a series of efficiently solvable convex optimization problems.

[0005] Sequential Convex Programming (SCP) has been widely applied in spacecraft guidance and online trajectory planning. Existing techniques typically transform the original problem into a series of convex optimization subproblems for iterative solution by linearizing or approximating the nonlinear dynamic equations and nonconvex constraints near the current reference trajectory, and by introducing a trust region or penalty function mechanism. In engineering implementation, these convex subproblems often take the form of second-order cone programming (SOCP) or quadratic programming (QP), and are solved using efficient numerical algorithms such as the primal-dual interior-point method. This has been validated in scenarios such as rocket recovery and landing, spacecraft rendezvous and docking, and reentry guidance. At the numerical solution level, the primal-dual interior-point method is widely used for solving convex optimization problems such as SOCP due to its fast convergence speed and good numerical stability. To improve the efficiency of online applications, some research has begun to explore the introduction of a hot-start strategy, which uses the solution results of the previous or adjacent problems as the initial point of the current optimization problem to reduce the number of iterations in the interior-point method. However, existing research has mostly focused on initialization at the level of the original variables, and systematic research on the dual variables and their relationship with the central path is still relatively limited.

[0006] Although existing online trajectory optimization methods for aircraft have made some progress in theory and engineering practice, they still have several shortcomings in high-frequency, real-time application scenarios. First, while sequential convexification methods can effectively handle nonlinear and nonconvex constraints, they essentially still require repeatedly solving multiple convex optimization subproblems within each control cycle. When the problem size is large or the constraints are complex, the computational burden remains heavy, making it difficult to operate stably under strict time constraints. Second, for the solution process based on second-order cone programming, most existing hot-start techniques only use the trajectory solution from the previous time step to initialize the original variables, neglecting the important role of dual variables in the primal-dual interior-point method. The primal-dual interior-point method requires initialization of both the original and dual variables. Using the previous iteration result of the sequential convexification algorithm directly is often not feasible. Specifically, the slack variables or dual multipliers of the optimal solution in the previous iteration are close to the constraint boundary and far from the central path. Using these directly as initial values ​​will result in a very poor condition number of the KKT matrix of the original dual interior point method, leading to serious numerical instability. At the same time, the initial boundary point is often located at the edge of the central path, resulting in a very small step size of the interior point method in the initial iteration stage. This forces multiple expensive matrix decomposition iterations, thereby weakening the performance advantage of the hot-start strategy. Summary of the Invention

[0007] To overcome the shortcomings of existing technologies, the present invention aims to provide a hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization, so as to improve the centrality and feasibility of the initial point of hot start, thereby ensuring numerical stability and improving the computational efficiency of the original dual interior point method for solving second-order cone programming subproblems.

[0008] To achieve the objective of this invention, the following solution is adopted: A hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization includes the following steps: S1. Obtain the optimization result of the sequential convexification algorithm in solving the second-order cone programming subproblem in the kth outer layer iteration, and use the optimization result as the initial solution of the second-order cone programming subproblem in the (k+1)th outer layer iteration; the initial solution includes the original variables, slack variables and dual variables; S2. For the initial solution, a symmetric neighborhood is introduced, which is defined as a preset range of the deviation of the complementary product of the slack variable and the dual variable from the average complementary gap. S3. Determine whether the deviation of the complementary product of each pair of slack variables and dual variables in the initial solution from the average complementary gap exceeds the symmetric neighborhood. If it does, then perform a centering correction on the complementary product that deviates from the symmetric neighborhood so that it is located on the boundary of the symmetric neighborhood, and obtain the slack variables and dual variables after centering correction. S4. Construct a joint correction system, which simultaneously includes the complementary product constraints on the residuals of the original problem, the residuals of the dual problem, and the slack variables and dual variables after the centering correction. S5. Solve the joint correction system to obtain the centralized correction search direction, and use the centralized correction search direction to update the initial solution to obtain the hot start initial point; S6. Using the hot start initial point as the initial value for the original dual interior point method, solve the second-order cone programming subproblem of the current (k+1)th outer iteration to obtain the optimal trajectory of the current outer iteration.

[0009] Further, in step S2, the symmetric neighborhood is defined as: for any pair of slack variables With dual variables Their complementary product satisfies Where μ is the average complementary gap, The parameter controls the size of the symmetric neighborhood, and 0 < <1.

[0010] Furthermore, in step S3, the complementary products that deviate from the symmetric neighborhood are centered and corrected, specifically including: when the complementary products... Less than When, correct it to When complementary products Greater than When, correct it to When complementary products are located in a symmetric neighborhood [ , Within [the specified range], it remains unchanged.

[0011] Furthermore, in step S4, the joint correction system also introduces an adjustment parameter to coordinate and control the correction intensity of the residuals of the original problem and the dual problem, so as to achieve a balance between feasibility recovery and step size preservation.

[0012] Furthermore, in step S4, the joint correction system is constructed based on the Nesterov-Todd scaling matrix, which is used to maintain the coupling relationship and symmetry between the original variable, the dual variable and the relaxed variable during the correction process.

[0013] Furthermore, in step S5, after solving the joint correction system to obtain the centered correction search direction, a line search is performed inside the cone defined by the slack variables and dual variables to obtain the maximum correction search step size, and the initial solution is updated using this step size.

[0014] Furthermore, the second-order cone programming subproblem is transformed from a continuous optimal control problem using a first-order preserved discretization method, wherein the control variables are parameterized as piecewise linear functions.

[0015] Furthermore, in step S4, the joint correction system is constructed based on a self-dual embedding model, which introduces auxiliary variables to uniformly determine the feasibility, optimality, and infeasibility of the original problem and the dual problem.

[0016] Furthermore, in step S5, the hot start initial point satisfies the initial conditions of the self-dual embedding model, wherein the auxiliary variables in the self-dual embedding model are all set to preset initial values ​​so that the original problem residual, the dual problem residual, and the dual gap are less than a preset threshold.

[0017] Furthermore, by iteratively executing steps S1 to S6, all outer iterations of the sequential convexity algorithm are completed, and the optimal trajectory of the aircraft is obtained; wherein, in the first outer iteration of the sequential convexity algorithm, a cold start method is used to generate the initial point, and the warm start method is enabled from the second outer iteration.

[0018] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention significantly improves the centrality and feasibility of the initial hot-start point, thereby increasing the convergence speed. Addressing the structural similarity of adjacent subproblems in the sequential convexity algorithm, this invention utilizes the optimization result of the (k-1)th outer iteration as the initial solution for the kth outer iteration and introduces a symmetric neighborhood. By judging whether the deviation of the complementary product of each pair of relaxed and dual variables in the initial solution from the average complementary gap exceeds a preset range, the deviated complementary products are centered and corrected to lie on the boundary of the symmetric neighborhood. Simultaneously, by constructing a joint correction system including the residuals of the original problem, the residuals of the dual problem, and the corrected complementary product constraints, the initial solution is systematically updated, ensuring that the corrected initial hot-start point simultaneously satisfies the approximate feasibility of the original problem, the approximate feasibility of the dual problem, and lies within the symmetric neighborhood of the central path. Therefore, this invention effectively improves the centrality and feasibility of the initial hot-start point, enabling the original-dual interior-point method to obtain a larger effective step size in the initial iteration stage, significantly improving the convergence speed.

[0019] 2. This invention reduces the number of iterations in the interior-point method, thus lowering computational time. Most existing warm-start techniques only initialize the original variables, neglecting the centrality of the dual variables. This leads to the initial solution deviating from the central path, and the interior-point method is limited in step size during the initial iterations, forcing multiple matrix decomposition iterations. This invention, through a symmetric neighborhood centering correction system, coordinates and corrects the initial solution while maintaining the coupling relationship between the original, dual, and relaxed variables. This makes the corrected warm-start initial point closer to the central path, thereby reducing the condition number of the KKT matrix and improving the quality of the Newton direction. Experimental results show that in typical online trajectory optimization scenarios for aircraft, after adopting the method of this invention, the total number of iterations of the original-dual interior-point method in each outer iteration of the sequential convexification algorithm is reduced by approximately 35.5%, and the total computation time is reduced by approximately 31.9%, achieving a significant computational acceleration effect.

[0020] 3. This invention balances numerical stability and computational efficiency, making it suitable for real-time online applications. It can be integrated into existing solution processes as a front-end initial solution correction step without altering the core solution logic of the original dual interior-point method or second-order cone programming solver, offering advantages such as low implementation cost and minimal modification to existing systems. This invention is particularly suitable for high-frequency, high-real-time online trajectory optimization scenarios such as reusable launch vehicle recovery and reentry vehicle guidance, effectively supporting autonomous trajectory planning and real-time guidance of spacecraft under limited computational resources. Attached Figure Description

[0021] Figure 1 This is a flowchart of the hot start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization in an embodiment of the present invention; Figure 2 This is a comparison diagram of the dual gap, original problem residual, and dual problem residual at the initial moment of each SCP iteration for cold start and hot start in embodiments of the present invention. Detailed Implementation

[0022] The present invention will now be further described in conjunction with the accompanying drawings and specific embodiments. It should be noted that, without conflict, the various embodiments or technical features described below can be arbitrarily combined to form new embodiments.

[0023] This invention provides a hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization. Designed for online trajectory planning applications, it fully leverages the structural similarity between adjacent convex subproblems in the sequential convexization algorithm to construct high-quality initial points, achieving efficient hot-start for solving second-order cone programming (SOCP) convex subproblems using the primal-dual interior-point method. By reducing the number of iterations in the primal-dual interior-point method, it significantly improves overall computational efficiency while ensuring numerical stability and solution accuracy, thereby enabling real-time and rapid solution of the optimal aircraft trajectory.

[0024] Existing hot-start methods typically select the optimal solution from the previous iteration as the initial point for the current SOCP subproblem. However, this initial solution often fails to simultaneously satisfy the feasibility requirements of both the primal and dual problems within the current subproblem. More critically, constraints active in the previous iteration cause the slack variables or dual variables of the primal problem to approach zero, resulting in the initial point being located at the boundary of the feasible region, significantly deviating from the central path of the primal-dual interior point method. These problems directly lead to the numerical ill-conditioning of the KKT system, significantly limiting the feasible step size in the search direction, thereby increasing the number of iterations, reducing the convergence speed, and weakening the acceleration effect of the hot-start strategy in online trajectory optimization.

[0025] This invention introduces the concept of a symmetric neighborhood, which refers to the complementary relaxation conditions defined by the central path. The strict equality is further relaxed to a neighborhood constraint that meets certain limits, thus defining a feasible region containing the central path. Based on this symmetric neighborhood, this invention performs consistency correction on the relaxed variable s and dual variable z in the hot-start initial solution: when the product of any pair of complementary variables deviates from the range of the symmetric neighborhood, a correction strategy is constructed to adjust it back into the symmetric neighborhood. Simultaneously, for cases where the initial solution may not satisfy the feasibility conditions of the primal or dual problem of the current SOCP subproblem, the residuals of the primal and dual problems are coordinated and corrected together.

[0026] To address this, this invention constructs a primal-dual centralization correction system based on symmetric neighborhoods. While maintaining the coupling relationship between the original, dual, and relaxed variables, it solves for the corresponding correction search direction and updates the initial solution accordingly. This correction process not only avoids the asymmetric perturbations caused by simply pushing away from the boundary but also effectively maintains the centrality and balance between the original and dual variables.

[0027] Furthermore, this invention designs a set of adjustable parameters to control key factors in the initial value construction process, including: the size of the symmetric neighborhood and the correction magnitude of the residuals of the primal and dual problems. Through the coordinated adjustment of these parameters, it can be ensured that the corrected initial point remains consistent in terms of feasibility, centrality, and numerical stability, thereby enabling the primal-dual interior-point method to obtain a larger effective step size in the initial iteration stage.

[0028] The initial solution obtained based on the similarity of adjacent SOCP subproblems during the sequence convexification process, after being processed by the symmetric neighborhood centering correction method proposed in this invention, yields a warm-start initial solution with smaller residuals in both the original and dual problems, a lower duality gap, and an overall position closer to the central path of the original-dual interior point method. Compared to the cold-start initial solution that does not utilize any prior information, this warm-start initial solution has significant advantages in both numerical quality and convergence performance. Experimental and engineering application results show that the warm-start method of this invention can significantly reduce the number of iterations in the original-dual interior point method, achieving a computational speedup of over 30% compared to cold-start in typical online trajectory optimization scenarios.

[0029] like Figure 1 As shown, this embodiment of the invention provides a hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization, including the following steps: S1. Obtain the optimization result of the sequential convexity algorithm in solving the second-order cone programming subproblem in the kth outer iteration, and use the optimization result as the initial solution of the second-order cone programming subproblem in the (k+1)th outer iteration; the initial solution includes the original variables, slack variables and dual variables.

[0030] S2. For the initial solution, a symmetric neighborhood is introduced, which is defined as a preset range of the deviation of the complementary product of the slack variable and the dual variable from the average complementary gap.

[0031] S3. Determine whether the deviation of the complementary product of each pair of slack variables and dual variables in the initial solution from the average complementary gap exceeds the symmetric neighborhood. If it does, then center the complementary products that deviate from the symmetric neighborhood so that they are located on the boundary of the symmetric neighborhood, thus obtaining the slack variables and dual variables after centering correction.

[0032] S4. Construct a joint correction system, which simultaneously includes the complementary product constraints on the residuals of the original problem, the residuals of the dual problem, and the slack variables and dual variables after the centering correction.

[0033] S5. Solve the joint correction system to obtain the centralized correction search direction, and use the centralized correction search direction to update the initial solution to obtain the hot start initial point.

[0034] S6. Using the hot start initial point as the initial value for the original dual interior point method, solve the second-order cone programming subproblem of the current (k+1)th outer iteration to obtain the optimal trajectory of the current outer iteration.

[0035] The following is a more detailed description of the hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization, according to an embodiment of the present invention.

[0036] To illustrate the applicability and effectiveness of the symmetric neighborhood-centered online trajectory optimization hot-start method for spacecraft in online trajectory planning, the atmospheric reentry trajectory optimization problem of a common aero-vehicle (CAV) is used as an example. This invention is not limited to this specific application scenario; it is also applicable to trajectory optimization problems with similar dynamics and constraints, such as reusable launch vehicle first-stage recovery, reentry landing guidance, and spacecraft terminal guidance.

[0037] The CAV reentry process is characterized by strong nonlinearity, strong coupling, and multiple constraints. When an aircraft enters the atmosphere at hypersonic speeds, its motion must simultaneously meet dynamic constraints, aerodynamic and thermal protection constraints, and terminal accuracy requirements. To achieve a safe, economical, and controllable reentry process, trajectory optimization methods are typically used to generate optimal control commands while satisfying these constraints.

[0038] 1. Dimensionless dynamic model of CAV reentry: The motion state of the aircraft is described using a geocentric coordinate system, where the aircraft's position, velocity, and attitude-related variables are defined as follows: Position state variables: This is the distance from the spacecraft's center of mass to the Earth's center; Longitude of the Earth's core; It is the geocentric latitude; Velocity state variables: The magnitude of the speed; The track angle; For heading angle; Attitude state variables: For the angle of attack; It is the tilt angle; Therefore, the system state vector is defined as: To ensure the smoothness of the trajectory and the feasibility of the project, the rate of change of angle of attack and the rate of change of tilt angle are selected as control inputs: use =6378km as the distance to the Earth's center Dimensionless factor, , =9.81 As speed Dimensionless factor, As the Earth's rotational angular velocity Dimensionless factor, As a dimensionless factor of time, As lift and resistance The dimensionless factor allows us to establish the following dimensionless kinetic equation: Its form is as follows: Dimensionless lift and resistance The form is: in, It is a dimensional value representing the mass of an aircraft. For the dimensionless reference area of ​​the aircraft, This refers to the dimensional density of the atmosphere. and The drag coefficient can be expressed as the angle of attack. And a function of other flight parameters.

[0039] 2. Optimal control problem for CAV reentry trajectory optimization: Add the following process and state constraints: Add flight time constraints Adding process heat flow Dynamic pressure and overload constraint Add control constraints to limit the angle of attack, roll angle, and their rate of change. Add no-fly zone constraints Add initial state and terminal constraints The minimum control efficiency (i.e., the minimum rate of change of actual angle of attack and sideslip angle) is used as the performance metric. Combining the dynamic model, process constraints, initial and terminal constraints, and performance indicators, this problem is essentially an optimal control problem.

[0040] 3. CAV reentry free-time convexity approximation: In a typical hypersonic reentry vehicle (CAV) trajectory optimization problem, the termination time... Often, the free time is a free variable, meaning the task does not predefine a specific reentry time, but rather requires achieving optimal performance metrics (such as heat flux, overload, fuel consumption, or time) while satisfying path constraints and terminal conditions. However, the existence of a free termination time leads to non-convexity and scale uncertainty in the optimization problem, severely affecting the numerical stability of online solutions. Therefore, it is necessary to process and approximate the free time to transform the problem into a form suitable for sequential convexity and primal-dual interior-point methods.

[0041] Considering the free-time problem, system dynamics from the time domain Transform into , that is to say Then we have: A sequential approximation method is used, where the values ​​of each point on the reference trajectory are directly taken during the iteration process. The above equation is then applied with respect to the reference trajectory. Performing a first-order Taylor expansion, we get: in: Linearizing the heat flux density, dynamic pressure, and overload constraints, we have: Linearizing the no-fly zone constraint results in: The convexity transformation process is implemented using a first-order Taylor expansion. Therefore, to ensure the accuracy of the approximation and the convergence of the algorithm, it is necessary to introduce trust region constraints for the relevant variables. Considering the free-time problem and the fact that the system dynamics are highly sensitive to time, a trust region constraint for the terminal time is also required: 4. First-order preservation (FOH) discretization method for the CAV reentrant convex subproblem: In order to solve numerically, the infinite-dimensional continuous trajectory optimization problem needs to be transformed into a finite-dimensional discrete numerical optimization problem. The most important point is to discretize the system dynamics differential equation constraints into corresponding algebraic constraints between the state variables and control variables at each discrete point.

[0042] This invention employs a first-order hold (FOH) discretization method, which segments the control variables into a linear parameter. By defining the control variables at discrete points, the control signal at any time can be given by the control variables at two adjacent discrete points. Furthermore, through the relevant properties of the state transition matrix in linear system theory, the constraints of the linearized dynamic differential equation can be transformed into the corresponding algebraic constraint form.

[0043] Select discrete points ,in That is, a uniform discrete grid is selected, and the above constraints are applied to each discrete point.

[0044] For system dynamics, this invention employs a high-precision discretization method based on an improved first-order preservation approach, which parameterizes the control variables as piecewise linear functions: in, , .

[0045] Based on the knowledge of linear systems, if it is known (express The value of the offset between the state variable at time and the state of the reference trajectory can be used to determine the situation. The value of is: in, Let be the state transition matrix, satisfying the following properties: in addition: Pick ,Right now At the left-hand limit moment, we can obtain: Considering continuity, we have: in, It is the transmission trajectory.

[0046] The integral objective function Transform into an accumulation form and then into a cone constraint: By applying linearized process constraints to discrete points and transforming the trust region into second-order cone constraints, we obtain the discrete convex subproblem, which is then input into a second-order cone programming problem. in, To optimize the deviation between the variable and the reference trajectory, i.e. , , ; For performance metrics; Rate of change of track angle Characteristic terms, ; For state-trust-region slack variables, subscript Represents discrete points, The optimized offset is obtained by solving the problem. This allows us to obtain the optimization quantity. , , , .

[0047] The following section explains the specific details of the original dual interior point method for second-order cone programming based on the self-dual embedding model.

[0048] Within the Sequence Convex Algorithm (SCP) framework, the online trajectory optimization problem for aircraft is typically transformed into a series of convex optimization subproblems in the form of Second-Order Cone Programming (SOCP).

[0049] 1. Standard second-order cone programming primitive-dual model: The standard SOCP primitive problem addressed by this invention is expressed as: in, As decision variables, As slack variables, and For the constraint matrix, and It is a straight vector. Represents the Cartesian product of multiple second-order cones or linear cones: The corresponding dual problem is expressed as: in, and These are the dual variables corresponding to the equality constraint and the cone constraint, respectively.

[0050] 2. Self-Dual Embedding: To standardize the determination of feasibility, optimality, and infeasibility of a problem, additional variables are introduced. and variables Construct a self-dual secondary linear system: The model is through Automatic status identification feasibility Then the optimal solutions to the primal and dual problems can be obtained. ;like Then it can be determined that either the original problem or the dual problem is infeasible.

[0051] 3. Central path: Given any initial point that satisfies the following conditions: initial residual for: Initial Dual Gap for: Represents the number of second-order cones and linear cones. Its central path is: in: It represents a contemporary Euclidean number. The Nesterov-Todd (NT) scaling matrix is ​​used to scale the current point in each iteration. A unique scaling matrix can be calculated. Make the mapped variables satisfy .

[0052] all The resulting solution set defines the central path of the HSD problem (embedding the primal and dual problems into the HSD problem to allow for the detection of infeasibility). The solution corresponding to this time is the optimal solution to the original problem.

[0053] 4. Calculate the search direction and search step size: The search direction is solved using Newton's iteration, and its linearized solution system is as follows: Within the framework of the original-dual interior-point method, the Mehrotra predictor-corrector algorithm is currently recognized as the core of an efficient solution. The Mehrotra predictor-corrector algorithm is an efficient interior-point solution strategy that consists of two phases in each iteration: (1) First, a prediction step is performed to detect the current constraint curvature and estimate the potential dual gap reduction by solving the affine reduction direction; (2) Then, a correction step is performed, which uses the information obtained from the prediction step to adaptively adjust the centering parameter and introduces a second-order term to compensate for the linearization error. This significantly increases the search step size while keeping the iteration point in the neighborhood of the center path, thus significantly improving the convergence speed.

[0054] Both the prediction step and the correction step are linear systems that solve the above equation; the only difference lies in all the right-hand side terms. There are some differences.

[0055] After obtaining the final search direction The search step size needs to be calculated. Its core objective is to find a step size that is as large as possible, while strictly ensuring that the updated variable always lies within a cone. The search step size is closely related to whether the current position is close to the center path. Starting positions close to the center path often obtain larger step sizes, while initial points on the feasible set boundary cannot obtain larger search step sizes due to the interior point condition, resulting in slow convergence.

[0056] After obtaining the step size Then, the state update can be performed: The following section explains the specifics of the "hot start" strategy for the second-order cone programming problem.

[0057] 1. Similarities between the SCP algorithm and the convex subproblem: In the primal-dual interior-point method, the algorithm considers itself convergent and terminates iteration when the residuals of the primal and dual problems, as well as the duality gap, both decrease to below a preset threshold. Therefore, a "good" initial point should simultaneously possess the following characteristics: basically feasible, close to optimal, and with good centrality. When prior information about the solution is lacking, traditional interior-point solvers typically employ a cold-start strategy, generating the initial point themselves without relying on user-provided initialization information. Although this method is simple to implement and highly applicable, the resulting initial point often fails to meet the requirements of a good initial point.

[0058] In the Sequence Convexification (SCP) iteration process, the data for each current SOCP subproblem is constructed from the solution of the corresponding subproblem at the previous time step. Therefore, although the KKT system matrices of two adjacent subproblems may not be completely identical, they generally exhibit significant similarities in structure and parameters. Based on this characteristic, using the solution of the previous subproblem to construct the initial iteration point of the current subproblem can effectively improve the convergence efficiency of the current primal-dual interior-point method. The fundamental reason is that a hot-start initial point is significantly superior to a cold-start point in terms of feasibility, centrality, and near-optimality, thereby reducing the number of iterations and improving numerical stability.

[0059] 2. Definition of Hot-Start Symmetric Neighborhood: The central path of the primal-dual problem in a general sense is: in, It is a feasible solution to both the primal and dual problems. The average complementary gap.

[0060] As mentioned earlier, It represents a combination of multiple second-order cones or linear cones. The Cartesian product formed, in an ideal case, represents all cones. Corresponding slack variables and dual variables Satisfying the complementary relaxation condition A perfect center point means that the product of all complementary conditions equals ,Right now However, such conditions are difficult to satisfy, therefore constructing a symmetric neighborhood will... Limited to Within a certain range. This symmetric neighborhood is defined as follows: in, parameter The value of determines the size of the central neighborhood. The smaller the value, the larger its neighborhood.

[0061] This definition requires the complementary product of each component. It should be neither too small nor too large, that is: Therefore, the symmetric neighborhood guarantees the following properties: (1) Slack variables Appropriately away from the cone constraint boundary: through avoid Proximity to the constraint boundary leads to degradation of the active set; (2) Avoid Away from the central path: via prevent Deviating from the central path; (3) Maintain original-dual symmetry: preserve and The balance relationship; (4) Improve KKT condition number: improve the numerical stability and acceptable step size in the Newton direction.

[0062] 3. Hot-start approximate solution selection strategy: Sequence Convexification Algorithm (SCP) Part 1 During the next iteration, the internal iteration of its primary and dual interior point method will produce a series of intermediate results, such as... , The first step of the sequence convexification algorithm The number of intermediate results within the internal iterations of the original dual interior point method in the nth iteration. The Sequence Convexification Algorithm (SCP) in its nth iteration. In the second iteration, a simple strategy for selecting the hot-start approximate solution is to directly select the first one. However, these results often exhibit poor centrality, leading to low hot-start efficiency. Therefore, a new parameter is introduced. Choose its first The result of this step is used as an approximate solution.

[0063] 4. Symmetric neighborhood centralization correction system: In solving the SOCP subproblems based on SCP, a hot-start strategy is adopted, selecting an initial solution from the previous iteration. However, this initial solution often has various problems. The residuals between the primal and dual problems are not strictly zero; the dual gap may be large; slack variables and dual variables may be located near the feasible region boundary, resulting in the primal-dual interior point method having poor step size and search direction quality in the early stages of iteration, and even numerical instability.

[0064] To overcome the above problems, this invention proposes a symmetric neighborhood centering correction system, which systematically corrects the initial point during the hot start phase, so that its overall coordination tends to satisfy: the original problem is approximately feasible; the dual problem is approximately feasible; the current dual gap is maintained; and the complementary product falls into a predefined symmetric neighborhood.

[0065] (1) Symmetric neighborhood centering correction of the complementary product of slack variables and dual variables: For each complementary product If it happens to be in the symmetric neighborhood If no correction is made, the result remains unchanged. If complementary products If it falls to the left of the symmetric neighborhood, then it is corrected to... If complementary products If it falls to the right of the symmetric neighborhood, then it is corrected to... .

[0066] If the Nesterov-Todd (NT) Scaling method is applied during the correction process, then: We can obtain: in, .

[0067] That is: (2) Residual correction of the primal and dual problems The initial solution generally does not satisfy the primal and dual problems, so it is necessary to correct the residuals of the primal and dual problems.

[0068] The residual of the original problem is: The residual of the dual problem is: Introducing additional parameters Control the degree of correction applied to the residuals of the original problem and the dual problem, so that: Combination Then you can understand the complete system: By solving the above equation, the symmetric neighborhood centering search direction can be obtained: Inside the cone defined by slack variables and dual variables ( Perform a line search to obtain the maximum corrected search step size. The corrected result is: For self-dual embedding models, set The initial value is: In summary, this invention uses a symmetric neighborhood centering correction method to correct the result of the previous iteration before sequence convexization to a hot-start initial solution.

[0069] The following are the numerical analysis results for the example problem: To verify the effectiveness and engineering applicability of the symmetric neighborhood-centered online trajectory optimization hot-start method for aircraft in the embodiment of the present invention, the present invention selects a typical hypersonic reentry vehicle (CAV, Common Aero Vehicle) trajectory optimization problem as an example case, and performs numerical simulation analysis on the proposed method within the Sequence Convexity (SCP) framework.

[0070] All numerical calculations were performed on a desktop computer platform with the following configuration: Intel Core i7-11700KF processor (3.6GHz), 16GB of memory, and a 64-bit Windows 10 operating system. The program was implemented in C / C++, and the SOCP subproblems in the outer iteration of SCP were solved using the primal-dual interior-point method. To fairly compare algorithm performance, comparative experiments were conducted using a cold-start approach and a hot-start approach based on symmetric neighborhood centering correction proposed in this invention. Hot-start was enabled starting from the second SCP iteration, and its main parameter settings are shown in Table 1.

[0071] Table 1 Hot Start Parameter Selection (1) Comparative analysis of the number of iterations within SOCP Table 2 presents a comparison of the number of internal iterations of the SOCP subproblem under cold start and hot start conditions in each outer layer iteration of SCP. It can be seen that the number of iterations for the two methods is basically the same in the first two iterations; from the third iteration onwards, as the quality of the initial point for hot start gradually improves, the hot start method shows a significantly fewer number of internal point iterations in subsequent iterations.

[0072] Statistical results show that during the 2nd to 13th SCP iterations, the cold-start method SOCP required a total of 299 internal iterations, while the proposed warm-start method reduced the total number of internal iterations to 193, a reduction of 106 iterations, or approximately 35.5%. This result demonstrates that the proposed warm-start strategy can significantly reduce the number of repeated iterations in the interior-point method and effectively improve the overall solution efficiency within the sequential convexity framework.

[0073] Table 2. Number of internal iterations of the SCP algorithm SOCP (2) Analysis of overall computation time and speedup ratio Further statistical analysis of the total computation time of the SOCP subproblems for both methods during the entire SCP process is shown in Table 3. The total computation time for the cold start method is 580ms, while the total computation time is reduced to 395ms using the hot start method proposed in this invention, corresponding to a speedup of approximately: This results in a computational speedup of approximately 1.47 times and an overall computation time reduction of approximately 31.9%. These results demonstrate that the method of this invention can significantly shorten the SOCP solution time without altering the core structure of the original-dual interior point method, providing strong support for the real-time implementation of online trajectory optimization.

[0074] Table 3 Speedup of SOCP Calculation Time (3) Comparative analysis of initial mass at cold and hot start-up points To further analyze the underlying mechanism of the hot start acceleration effect, such as... Figure 2 As shown, a comparison is presented of the duality gap, primal problem residual, and dual problem residual at the initial moment of each SCP iteration between cold start and hot start. It can be observed that after multiple iterations of the sequential convexity algorithm, the initial duality gap of the initial solution given by hot start is relatively larger than that of cold start. The order of magnitude will drop rapidly to even The order of magnitude, the residuals of the primal problem and the dual problem will rapidly change from... The order of magnitude dropped rapidly to Order of magnitude. As the sequence convexity algorithm progresses, the similarity between convex subproblems increases, resulting in a more significant speedup from warm starts.

[0075] This indicates that by introducing a symmetric neighborhood centering correction strategy, the method of this invention can effectively correct the initial point to a high-quality region that is feasible for both the original and dual problems and close to the central path. This significantly improves the centrality and feasibility of the initial point, provides better starting conditions for subsequent interior point method iterations, and avoids early step size limitations and numerical ill-conditioned problems.

[0076] (4) Comprehensive analysis The results, considering the number of iterations, computation time, and initial point quality, demonstrate that the hot-start method based on symmetric neighborhood centering correction proposed in this invention significantly improves the efficiency of the original-dual interior point method in solving SOCP subproblems within the SCP framework. It not only significantly reduces the number of internal iterations but also effectively shortens the overall computation time, achieving an overall acceleration of over 30% while maintaining numerical stability. This method is particularly suitable for high real-time online trajectory optimization problems for aircraft and has significant engineering application value in reusable launch vehicle reentry guidance, autonomous recovery, and embedded real-time planning scenarios.

[0077] To achieve efficient hot start, this invention constructs a relaxation variable. With dual variables Symmetric neighborhood of complementary product Through parameters and The size of the symmetric neighborhood is controlled, and this serves as the basis for the centering correction.

[0078] For slack variables With dual variables For complementary products, this invention employs a piecewise correction strategy: when the complementary product is within a symmetric neighborhood, it remains unchanged; when the complementary product exceeds the symmetric neighborhood, it is corrected to the neighborhood boundary. This strategy effectively avoids numerical instability or deviation from the central path caused by over-correction.

[0079] This invention introduces adjustment parameters This invention coordinates and controls the correction strength of the residuals of the original problem and the dual problem, significantly reducing the level of the corrected residuals while avoiding excessive centralization of the initial point due to over-correction. Over-correction would drastically reduce the allowable step size, thus limiting the efficiency of subsequent interior-point method iterations. Through this parameterized adjustment mechanism, this invention achieves an effective balance between feasibility recovery and step size preservation, preventing excessively small step sizes due to complete residual repair while ensuring larger search step sizes and faster convergence speeds.

[0080] This invention incorporates three types of errors—the residuals of the primal problem, the residuals of the dual problem, and the deviation of the central path—into a unified correction framework. A joint correction equation system is constructed, and adjustment parameters are introduced to coordinately control the weights and correction magnitudes of each error term, thereby achieving a systematic centralization correction of the initial point. This joint correction mechanism, while ensuring the feasibility of the primal and dual problems, effectively suppresses deviations from the central path, ensuring that the corrected initial point remains consistent in terms of feasibility, duality, and centrality, providing a higher-quality starting state for subsequent interior-point method iterations.

[0081] Compared with the cold start or simple hot start methods used in traditional online trajectory optimization of aircraft, the hot start method for online trajectory optimization of aircraft based on symmetric neighborhood centering in this invention has the following significant advantages in terms of algorithm performance and engineering applicability: 1. This invention significantly improves the centrality and feasibility of the initial hot-start point, making it easier to obtain a larger search step size and faster convergence speed. Based on traditional hot-start methods, this invention introduces the concept of a symmetric neighborhood and constructs a centralized correction system by systematically correcting the complementary product of slack variables and dual variables. This ensures that the resulting initial hot-start point simultaneously possesses small residuals in the original problem, dual problem, and dual gap, and lies within the symmetric neighborhood of the central path. Compared to conventional hot-start methods that directly use the optimal solution from the previous iteration as the initial hot-start point, the initial hot-start point generated by this invention does not approach the feasible region boundary and does not lead to ill-conditioned KKT systems, thus significantly improving the centrality and feasibility of the initial hot-start point. Traditional hot-start methods often suffer from limited early search step sizes due to the initial solution being located at the constraint boundary and the complementary variables approaching zero, resulting in difficulty in achieving effective progress in the early stages of the algorithm. This invention effectively avoids the limitation of search step size by forcibly centralizing within a symmetric neighborhood and constraining only complementary products beyond the neighborhood boundary, while introducing a residual adjustment parameter to constrain the correction magnitude. This allows the algorithm to adopt a larger search step size in the initial stage. Because the hot-start initial point generated by this invention is significantly improved in terms of centrality and feasibility, and early iterations are no longer limited by step size, the algorithm can approach the central path and converge to the optimal solution more quickly. Compared with traditional hot-start methods, this invention can reduce the condition number of the KKT matrix and improve the quality of the Newton direction, thereby reducing the overall number of iterations, increasing the convergence speed, and significantly improving computational efficiency. It is particularly suitable for online trajectory optimization, high real-time tasks, and embedded computing platforms with limited computing power.

[0082] 2. This invention balances primal-dual residuals and centrality, enhancing numerical stability. Existing hot-start methods typically only push the initial solution away from the constraint boundary on one side, or only correct the primal-dual residuals, neglecting the coupling relationship between the primal problem residuals, dual problem residuals, and deviations from the central path. To address this deficiency, the proposed centralization correction system considers all three types of errors during construction and uniformly adjusts the correction magnitude through parameters, ensuring that the initial hot-start point maintains synchronous coordination in terms of feasibility, duality, and centrality. Therefore, this invention not only significantly reduces the condition number of the KKT matrix, avoiding numerical divergence during iteration, but also improves the quality of the Newton search direction, thereby enhancing the overall convergence performance of the primal-dual interior-point method.

[0083] 3. The algorithm of this invention is simple and easy to use, and has significant engineering application value and prospects. First, in terms of application background, this invention fully utilizes the characteristic of highly similar structures in consecutive SOCP subproblems in the Sequence Convexification (SCP) algorithm. By using the solution results of the previous subproblem to construct the hot-start initial point of the next subproblem, indiscriminate search can be reduced, invalid iterations can be avoided, and the efficiency of sequence solution can be significantly improved. This technology is particularly suitable for real-time guidance and trajectory planning applications such as reusable launch vehicles, reentry vehicles, and variable structure vehicles. In these tasks, the algorithm needs to frequently and quickly solve structurally similar optimization problems, thus having a significant advantage. Second, in terms of application platform, this invention significantly shortens the overall computation time of trajectory optimization by effectively reducing the number of iterations of the primal-dual interior point method and improving the efficiency of each SCP outer layer iteration. As a result, large-scale trajectory optimization problems that could originally only be solved on the ground or in offline environments can be solved online and applied in real time on airborne computers or embedded platforms. This invention demonstrates significant engineering practical value and broad application prospects in scenarios such as high-frequency commercial space launches, autonomous recovery and reentry guidance, and highly dynamic and complex constrained trajectory planning. Finally, in terms of engineering implementation, the centralized hot-start method proposed in this invention does not alter the main structure of the original-dual interior-point method or the SOCP solver; it is simply embedded into the existing solver framework as a front-end initial solution correction and generation module. This method can be directly integrated with mainstream SOCP solvers, offering advantages such as low implementation cost, minimal modification to existing systems, and ease of promotion and reproduction, thus facilitating its engineering application in existing online trajectory optimization systems for spacecraft.

[0084] The above is a detailed description of the preferred embodiments of the present invention. However, the present invention is not limited to the embodiments described. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the present invention. All such equivalent modifications or substitutions are included within the scope defined by the claims of this application.

Claims

1. A hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization, characterized in that, Includes the following steps: S1. Obtain the optimization result of the sequential convexification algorithm in solving the second-order cone programming subproblem in the kth outer layer iteration, and use the optimization result as the initial solution of the second-order cone programming subproblem in the (k+1)th outer layer iteration. The initial solution includes the original variables, slack variables, and dual variables; S2. For the initial solution, a symmetric neighborhood is introduced, which is defined as a preset range of the deviation of the complementary product of the slack variable and the dual variable from the average complementary gap. S3. Determine whether the deviation of the complementary product of each pair of slack variables and dual variables in the initial solution from the average complementary gap exceeds the symmetric neighborhood. If it does, then perform a centering correction on the complementary product that deviates from the symmetric neighborhood so that it is located on the boundary of the symmetric neighborhood, and obtain the slack variables and dual variables after centering correction. S4. Construct a joint correction system, which simultaneously includes the complementary product constraints on the residuals of the original problem, the residuals of the dual problem, and the slack variables and dual variables after the centering correction. S5. Solve the joint correction system to obtain the centralized correction search direction, and use the centralized correction search direction to update the initial solution to obtain the hot start initial point; S6. Using the hot start initial point as the initial value for the original dual interior point method, solve the second-order cone programming subproblem of the current (k+1)th outer iteration to obtain the optimal trajectory of the current outer iteration.

2. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 1, characterized in that, In step S2, the symmetric neighborhood is defined as: for any pair of slack variables With dual variables Their complementary product satisfies Where μ is the average complementary gap, The parameter controls the size of the symmetric neighborhood, and 0 < <1.

3. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 1, characterized in that, In step S3, the complementary products that deviate from the symmetric neighborhood are centered and corrected. Specifically, this includes: when the complementary products... Less than When, correct it to When complementary products Greater than When, correct it to When complementary products are located in a symmetric neighborhood [ , Within [the specified range], it remains unchanged.

4. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 1, characterized in that, In step S4, the joint correction system also introduces adjustment parameters to coordinate and control the correction intensity of the residuals of the original problem and the dual problem, so as to achieve a balance between feasibility recovery and step size preservation.

5. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 1, characterized in that, In step S4, the joint correction system is constructed based on the Nesterov-Todd scaling matrix, which is used to maintain the coupling relationship and symmetry between the original variable, dual variable and relaxed variable during the correction process.

6. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 1, characterized in that, In step S5, after solving the joint correction system to obtain the centered correction search direction, a line search is performed inside the cone defined by the slack variables and dual variables to obtain the maximum correction search step size, and the initial solution is updated using this step size.

7. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 1, characterized in that, The second-order cone programming subproblem is transformed from a continuous optimal control problem using a first-order preserved discretization method, where the control variables are parameterized as piecewise linear functions.

8. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 1, characterized in that, In step S4, the joint correction system is constructed based on a self-dual embedding model, which introduces auxiliary variables to uniformly determine the feasibility, optimality, and infeasibility of the original problem and the dual problem.

9. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 8, characterized in that, In step S5, the hot start initial point satisfies the initial conditions of the self-dual embedding model, wherein the auxiliary variables in the self-dual embedding model are all set to preset initial values ​​so that the original problem residual, the dual problem residual, and the dual gap are less than a preset threshold.

10. The hot-start method for online trajectory optimization of aircraft based on symmetric neighborhood centralization according to claim 1, characterized in that, By iteratively executing steps S1 to S6, all outer iterations of the sequential convexity algorithm are completed, and the optimal trajectory of the aircraft is obtained. In the first outer iteration of the sequential convexity algorithm, a cold start method is used to generate the initial point, and the warm start method is enabled from the second outer iteration.

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