Method for determining online abort orbit of launch vehicle based on convex difference programming

Abstract

The application discloses a launch vehicle online abort orbit determination method based on convex difference programming and belongs to the technical field of aircraft trajectory optimization. The application uses convex difference decomposition to process complex orbit entry constraints, and converts an abort orbit optimization problem into a convex difference programming problem. When solving the abort orbit optimization problem, precise relaxation is used to keep the nonlinearity of the original abort orbit optimization problem, increase the feasible region of the original problem, and make the relaxed optimization problem have better convergence. By setting the orbit entry point of the failure rocket at the perigee, an elliptical abort orbit optimization problem which is easier to solve is constructed, and the efficiency of solving the elliptical abort orbit optimization problem is improved. In order to fully utilize the remaining carrying capacity of the launch vehicle, different types of abort orbits are optimized according to different thrust loss sizes. The orbit elements of the determined abort orbit are input into the GNC system of the launch vehicle to replace the nominal orbit elements, so that the autonomy and safety of the launch vehicle in the case of responding to the thrust drop failure are significantly improved.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft trajectory optimization technology, and relates to a method for optimizing the trajectory of a launch vehicle during the ascent phase, and more particularly to a method for determining the online trajectory of a launch vehicle during the ascent phase based on convexity planning. Background Technology

[0002] In recent years, human space activities have become increasingly frequent. Companies and governments around the world have begun or are planning to build satellite internet constellations. Frequent launches have created a demand for more reliable and lower-cost launch vehicles. Currently, launch failures remain relatively common worldwide. How to save astronauts and spacecraft when a launch vehicle malfunctions during flight is a problem worthy of study. Fault-tolerant guidance, navigation, and control systems are receiving increasing attention from researchers. Existing guidance laws for the exoatmospheric ascent phase of launch vehicles, such as iterative guidance and explicit guidance, are effective in delivering payloads to the nominal orbit. Because the guidance commands given by these guidance laws are continuously updated in each guidance cycle, these guidance laws have a certain degree of fault tolerance in the event of a minor thrust loss failure in the launch vehicle. However, if a more severe thrust loss failure occurs, making the nominal orbit unreachable, these guidance laws will fail, and the launch vehicle will face the risk of crashing and ultimately orbiting too low. To avoid catastrophic damage to astronauts and spacecraft from a launch vehicle crash or an orbit that is too low, the launch mission should be aborted immediately. The remaining payload capacity of the rocket can be used to send the astronauts and spacecraft into a lower-energy, safer abort orbit. Researching online abort orbit determination methods can significantly improve the autonomy and safety of launch vehicles in dealing with thrust reduction failures; therefore, researching online abort orbit determination methods has significant practical implications.

[0003] The goal of online abort trajectory determination is to autonomously calculate a suitable abort trajectory online when a launch vehicle malfunctions, allowing astronauts and spacecraft to temporarily dock there. This provides ground support personnel with sufficient time to analyze the problem and send commands—either to continue the mission or return safely to Earth. To achieve this goal, many numerical optimization-based abort trajectory determination methods have been researched and developed in recent years. However, the dynamics of the launch vehicle's ascent phase are nonlinear, and the terminal state constraints are complex. Existing methods—such as adaptive collocation methods and directly linearized sequential convex optimization algorithms—either have limited computational efficiency and cannot guarantee good convergence, or they consider only a single scenario and cannot fully utilize the launch vehicle's remaining payload capacity. Summary of the Invention

[0004] The main objective of this invention is to provide a method for determining the online abort trajectory of a launch vehicle based on convexity programming. This method enables online abort trajectory optimization of the launch vehicle under conditions considering nonlinear system dynamics and complex mission constraints, thereby achieving online abort trajectory determination based on numerical optimization. This invention can improve the autonomy and safety of launch vehicles when dealing with thrust reduction failures.

[0005] The objective of this invention is achieved through the following technical solution.

[0006] The present invention discloses a method for determining the online abort trajectory of a launch vehicle based on convexity programming. This method establishes the dynamic equations of the ascent phase after a final stage failure in a geocentric inertial coordinate system based on the magnitude of the rocket's thrust loss rate; given a minimum orbital altitude h... min Calculate the nominal perigee height h based on the given nominal orbital elements. p With maximizing orbital altitude as the optimization objective, and constrained by circular orbit insertion conditions and the dynamic equations of the ascent phase after a launch vehicle final-stage failure, a circular abort orbit optimization problem is established. This circular abort orbit optimization problem is transformed into a convexity programming problem, which is solved using a penalized concave-convex process method to obtain the orbital elements and orbital altitude h of the highest circular abort orbit within the nominal orbital plane. c The track height h of the circular termination track is obtained from the optimized solution. c With minimum orbital height h min and perigee height h p The size relationship is divided into h c >h p h min ≤h c ≤h p h c <h min Optimize different types of abort trajectories separately; in h c >h p Under the constraints of the elliptical orbit insertion conditions and the dynamic equations of the ascent phase after a failure of the launch vehicle's final stage, and with maximizing the insertion velocity as the optimization objective, an elliptical aborted orbit optimization problem is established. This elliptical aborted orbit optimization problem is transformed into a convexity programming problem, which is solved using a penalized concavity-convexity process method to obtain the orbital elements of the elliptical aborted orbit within the nominal orbital plane; in h min ≤h c ≤h p In the case of h, the highest circle of the nominal orbital plane is the final stopping orbit; c <h minIn the case of a failure, by modifying the orbital constraints of the circular abort trajectory optimization problem, a circular abort trajectory optimization problem within the rocket's flight plane at the moment of failure is constructed. This circular abort trajectory optimization problem is then transformed into a convexity programming problem, which is solved using a penalized concave-convex process method to obtain the orbital elements and orbital height h of the highest circular abort trajectory within the rocket's flight plane at the moment of failure. c1 Compare h c1 and h min The size relationship, if h c1 <h min Then there is no stopping orbit if h c1 ≥h min The system returns the orbital elements of the circular abort orbit; it determines the launch vehicle's abort orbit based on different situations, thus realizing online abort orbit determination for launch vehicles based on convexity planning; the determined orbital elements of the abort orbit are input into the launch vehicle's GNC system to replace the nominal orbital elements, and the launch vehicle after the failure flies to the abort orbit to perform the rescue mission for astronauts and spacecraft, improving the autonomy and safety of the launch vehicle in dealing with thrust reduction failures.

[0007] The present invention discloses a method for determining the online abort trajectory of a launch vehicle based on convexity planning, comprising the following steps:

[0008] Step 1: Based on the magnitude of the launch vehicle's thrust loss rate, establish the dynamic equations for the ascent phase after the launch vehicle's final stage failure in the geocentric inertial coordinate system.

[0009] The implementation method for step one is as follows:

[0010] The dynamic equations for the ascent phase of a launch vehicle in a geocentric inertial coordinate system after a thrust loss failure are expressed as follows:

[0011]

[0012] in, It is the rocket's position vector. It is the rocket velocity vector, and T is the available thrust after a rocket malfunction. b Here, is the direction of thrust, g is the gravitational acceleration vector, and m is the rocket's mass. The formulas for calculating the thrust and mass of a rocket are as follows:

[0013] T=(1-κ)T vac (2)

[0014] m(t) = m0 - Tt / (I sp g0) (3)

[0015] In the formula, κ is the thrust loss rate, which is a constant, and T vac It is the vacuum thrust of the engine, m0 is the mass of the rocket at the moment of failure (time t0), and Isp g is the engine's specific impulse, and g0 is the gravitational acceleration at the Earth's surface.

[0016] Step 2: Given the minimum track height h min Given the nominal orbital elements, the perigee height h is calculated based on the nominal orbital elements. p With maximizing orbital altitude as the optimization objective, and constrained by the circular orbit insertion conditions and the ascent dynamics equations for the launch vehicle's final stage after a failure (constructed in step one), a circular abort orbit optimization problem is established. This circular abort orbit optimization problem is then transformed into a convexity programming problem. The penalized concave-convex process method is used to solve this problem, yielding the orbital elements and orbital altitude h of the highest circular abort orbit within the nominal orbital plane. c .

[0017] The implementation method for step two is as follows:

[0018] Step 2.1: Given the minimum orbital height h min Given the nominal orbital elements, calculate the perigee height h based on the nominal orbital elements. p .

[0019] The implementation method for step 2.1 is as follows:

[0020] Given minimum orbital altitude h min To ensure the stop track is stable, its height must be greater than h. min The orbital elements of the nominal orbit are known before the launch mission. Specifically, the orbital elements that need to be specified include: the semi-major axis a of the nominal orbit. * eccentricity e * Track inclination angle i * Right ascension of ascending node Ω * and perigee argument ω * Using formula h p =a * (1-e * )-R E Calculate the perigee height h p In the formula, R E This represents the Earth's radius.

[0021] Step 2.2: Using the circular orbit insertion conditions and the ascent dynamics equations of the launch vehicle after the final stage failure constructed in Step 1 as constraints, and with maximizing the orbital height as the optimization objective, establish the circular termination orbit optimization problem within the nominal orbital plane after the failure.

[0022] The implementation method for step 2.2 is as follows:

[0023] 2.2.1 Constraints are given for the circular termination trajectory optimization problem.

[0024] To construct an optimization problem for a circular abort trajectory within the nominal orbital plane after a failure, constraints and optimization objectives need to be defined.

[0025] 1) Thrust direction constraint: The 2-norm of the thrust direction is in the interval [t0, t... f The upper part equals 1, that is

[0026]

[0027] Where t0 and t f These are the initial time and the terminal time, respectively.

[0028] 2) Initial state constraints: The position and velocity of the rocket at the moment of launch vehicle failure (time t0) are given by the GNC system.

[0029] r(t0)=r0,v(t0)=v0 (5)

[0030] Where r0 and v0 are the position and velocity vectors of the rocket at time t0, respectively.

[0031] 3) Circular orbit insertion constraint: The launch vehicle needs to transport the astronauts and spacecraft to a circular orbit. The circular orbit insertion constraint is denoted as Φ. C =0, specifically in the following form

[0032]

[0033] Where μ is the gravitational constant, 1 h It is a unit vector in the same direction as the angular momentum of the terminated orbit. Since the terminated orbit needs to be coplanar with the nominal orbit, we have 1. h =[sini * ·sinΩ * -sini * ·cosΩ * cosi * ] T .

[0034] 2.2.2 Given an optimization objective, establish a trajectory optimization problem where the circle in the nominal track plane terminates after a fault.

[0035] When optimizing a circular termination trajectory, the optimization objective is to maximize the trajectory height, which corresponds to minimizing -||r(t). f Therefore, we obtain the following optimal control problem P0 for optimizing a circular termination trajectory. C

[0036] P0 C minimum -||r(t) f )||

[0037] Subject to Eq.(1),(4)-(6)

[0038] Question P0 C The start time t0 = 0, and the end time t f Take (m0-m) dry )I sp g0 / T, m dry It is the mass of the rocket when the fuel is exhausted. At P0 C In this context, the control vector is the thrust direction 1. b The state vector is s = [r T v T ] T .

[0039] Step 2.3: By relaxing the circular trajectory optimization problem after the fault constructed in Step 2.2, a new constraint φ is introduced. r =||r(t) f )||, perform convexity decomposition on the orbital constraints, and solve the circular termination orbit optimization problem P0 C The problem is transformed into a convexity programming problem, which is then solved using a penalized concave-convexity process method. This yields the number of orbit elements and the orbit height h of the orbit ending at the highest circle within the nominal orbital plane. c .

[0040] The implementation method for step 2.3 is as follows:

[0041] 2.3.1 Relaxation processing is applied to the optimization problem of the circular stop track after the fault constructed in step 2.2.

[0042] Relax the control constraint—that is, the thrust direction constraint (4)—as follows:

[0043] ||1 b (t)||≤1 (7) Problem P0 C The problem is relaxed to the following P1 C

[0044] P1 C minimum -||r(t) f )||

[0045] Subject to Eq.(1),(5)-(7)

[0046] Proofs using optimal control theory show that in problem P1 C In the optimal solution, constraint (7) is in [t0,t f It is active almost everywhere on the [scale]. For any optimal control quantity... It is true in almost every way, problem P1C The optimal solution is also problem P0. C The optimal solution.

[0047] 2.3.2 Introducing a new constraint φ r =||r(t) f )||.

[0048] Introduce the following new equality constraints

[0049] φ r =||r(t) f (8)

[0050] Due to the introduction of the new constraint (8), problem P1 C The objective function is equivalently rewritten as -φ r Question P1 C In the orbital insertion constraint (6), ||r(t) f )||·||v(t f )|| 2 -μ=0 is equivalently transformed into φ r ·||v(t f )|| 2 -μ=0. Therefore, the orbital insertion constraint (6) is rewritten as

[0051]

[0052] Relax the newly introduced equality constraint (8) to

[0053] φ r ≤||r(t f (10)

[0054] Because a new constraint (8) is introduced and the equality constraint (8) is replaced with a relaxation constraint (10), problem P1 C The problem is relaxed to the following P2 C

[0055] P2 C minimum -φ r

[0056] subject to Eq.(1),(5),(7),(9),(10)

[0057] Theoretical analysis reveals that problem P2 C In the optimal solution, constraint (10) is always active. This step completes the solution to problem P0. C Precise relaxation. Problems after relaxation (P2) C Compared to the original problem P0 C With a larger feasible region, however P2 C The optimal solution and P0C The optimal solutions are the same.

[0058] 2.3.3 Perform convexity decomposition on non-convex orbit constraints, and decompose P2 C This is transformed into a convexity programming problem.

[0059] For question P2 C The non-convex constraints in the mid-track constraint (9) and constraint (10) are decomposed into convex differences. Convex difference decomposition is the process of decomposing the constraint function into the difference between two convex functions. The decomposed constraints have the following form.

[0060] f(x)-g(x)≤0 (11)

[0061] Here, x is the optimization variable, and f(x) and g(x) are both convex functions.

[0062] For question P2 C The inequality constraint (10) needs to be rewritten as follows:

[0063] φ r -||r(t f )||≤0 (12)

[0064] In orbital insertion constraint (9)

[0065]

[0066] It is a linear affine constraint, not a convex constraint r T (t f )v(t f ) = 0 and φ r ·||v(t f )|| 2 -μ = 0 needs to be decomposed by convexity. First, constrain the equality r. T (t f )v(t f The equation 0 is transformed into two inequality constraints r. T (t f )v(t f )≤0 and -r T (t f )v(t f If )≤0, then the two inequality constraints are rewritten as constraint (14).

[0067]

[0068] Equality constraint φ r ·||v(t f )|| 2 -μ=0 is transformed into two inequality constraints, and then the inequality constraints are decomposed to obtain (μ / φ) r (It is a convex function)

[0069]

[0070] Due to variable φ r The introduction of φ makes the constraint φ r ·||v(t f )|| 2 -μ=0 can be decomposed into the form of constraint (15).

[0071] The dynamic equation (1) is nonlinear because the gravitational acceleration g is a nonlinear function of position r. To obtain a linear dynamic equation, the gravitational acceleration calculated in the previous iteration is used to approximate the gravitational acceleration in the current iteration, i.e.

[0072]

[0073] Where, r [k] This is the position vector profile obtained from the previous iteration of the iterative algorithm. Using the approximate gravitational model of formula (16), the dynamic equation (1) is transformed into the following linear form.

[0074]

[0075] After decomposing the orbital constraint convexity and approximating the dynamics using equation (17), problem P2 C Transform into question P3 C

[0076] P3 C minimum -φ r

[0077] subject to Eq.(5),(7),(12)-(15),(17)

[0078] The time interval [t0, t] f Divide into equal parts of length h = (t) f -t0) / N is divided into N equal parts. Let r be... j =r(t) j ), v j =v(t) j ) and 1 bj =1 b (t j ) represent time nodes t j =t0+jh, (j=0,1,…,N) position, velocity and thrust direction. After discretizing the path constraint (7) and the dynamic equation (17), problem P3 C This is then transformed into a convexity programming problem.

[0079] 2.3.4 Solving the Circular Termination Orbit Optimization Problem Using the Penalized Concave-Concave Process Method (P3) C The number of orbital elements and the orbital height h of the terminated orbit are obtained. c .

[0080] The penalized concave-convex process method is an efficient and mature approach for solving convexity programming problems. This method is used to solve the circular termination trajectory optimization problem within the nominal orbital plane (P3). C Solve P3 C The number of track elements and track height h of the track ending at the highest circle within the nominal track plane are obtained. c .

[0081] Step 3: Determine the track height h of the circular termination track based on the optimized solution. c With minimum orbital height h min and perigee height h p The size relationship is divided into h c >h p h min ≤h c ≤h p or h c <h min The launch vehicle's abort trajectory is determined online, corresponding to steps four, five, or six.

[0082] The implementation method for step three is as follows:

[0083] Based on the track height h of the circular termination track c With minimum orbital height h min and perigee height h p The size relationship is divided into h c >h p h min ≤h c ≤h p or h c <h min The abort trajectories of the launch vehicles were optimized online, and the specific comparison results corresponded to the following subsequent steps:

[0084] 1) In h c >h p In this case, proceed to step four;

[0085] 2) In h min ≤h c ≤h p In this case, proceed to step five;

[0086] 3) In h c <h min In this case, proceed to step six.

[0087] Step 4: In h c >h p Under these conditions, constrained by the elliptical orbit insertion conditions and the ascent dynamics equations for the launch vehicle's final stage after a failure (constructed in step one), and with maximizing the insertion velocity as the optimization objective, an optimization problem for the elliptical aborted orbit after a failure is established. This elliptical aborted orbit optimization problem is then transformed into a convexity programming problem. The penalized concave-convex process method is used to solve this convexity programming problem to obtain the orbital elements of the elliptical aborted orbit within the nominal orbital plane.

[0088] The implementation method for step four is as follows:

[0089] In h c >h p In this scenario, astronauts and spacecraft can be transported to a relatively high circular orbit. To further reduce orbital element deviations and facilitate subsequent orbital maneuvers, an elliptical aborted orbit with a perigee altitude equal to the nominal orbit and an apogee altitude as high as possible is optimized.

[0090] Step 4.1: Using the elliptical orbit insertion conditions and the ascent dynamics equations of the launch vehicle after the final stage failure constructed in Step 1 as constraints, and with maximizing the insertion velocity as the optimization objective, establish the elliptical abort orbit optimization problem within the nominal orbital plane after the failure.

[0091] The implementation method for step 4.1 is as follows:

[0092] 4.1.1 Constraints are given for the elliptical termination trajectory optimization problem.

[0093] In addition to satisfying the ascent dynamics equation (1) constructed in step one, the rocket after the malfunction also needs to satisfy the following constraints.

[0094] 1) Thrust direction constraint: see formula (4)

[0095] 2) Initial state constraints: see formula (5)

[0096] 3) Elliptical orbit insertion constraints: The launch vehicle needs to transport astronauts and spacecraft to an elliptical orbit. Let Φ be the elliptical orbit insertion constraint. E =0, assuming the rocket enters orbit at its perigee on its abort trajectory, then Φ E The specific form of =0 is as follows:

[0097]

[0098] Among them, 1 h It is a unit vector in the same direction as the angular momentum of the stopped trajectory. It is the perigee radius of the nominal orbit.

[0099] 4.1.2 Given an optimization objective, establish an elliptical abort orbit optimization problem within the nominal orbital plane after a fault.

[0100] When optimizing an elliptical termination trajectory, the optimization objective is to maximize the injection velocity, which corresponds to minimizing -||v(t). f Therefore, we obtain the following optimal control problem P0 for optimizing an elliptical terminated trajectory. E

[0101] P0 E minimum -||v(t) f )||

[0102] subject to Eq.(1),(4),(5),(18)

[0103] Optimal control problem P0 E The initial time t0 = 0, and the terminal time t f Take (m0-m) dry )I sp g0 / T. At P0 E In this context, the control vector is the thrust direction 1. b The state vector is s = [r T v T ] T .

[0104] Step 4.2: By relaxing the elliptical aborted trajectory optimization problem constructed in Step 4.1, and then performing convexity decomposition on the orbital constraints, the elliptical aborted trajectory optimization problem P0 is solved. E The problem is transformed into a convexity programming problem. The penalized concave-convex process method is used to solve the convexity programming problem and obtain the number of orbital elements of the elliptical terminated orbit in the nominal orbital plane.

[0105] The implementation method for step 4.2 is as follows:

[0106] 4.2.1 Relaxation processing is applied to the optimization problem of the elliptical aborted trajectory after the fault constructed in step 4.1.

[0107] Relax the control constraint—that is, the thrust direction constraint (4)—as follows:

[0108] ||1 b (t)||≤1 (19) Problem P0 E The problem is relaxed to the following P1 E

[0109] P1 E minimum -||v(t) f )||

[0110] subject to Eq.(1),(5),(18),(19)

[0111] Proofs using optimal control theory show that in problem P1 E In the optimal solution, constraint (19) is in [t0,t f It is active almost everywhere on the [scale]. For any optimal control quantity... It is true in almost every way, problem P1 E The optimal solution is also problem P0. E The optimal solution.

[0112] 4.2.2 Perform convexity decomposition on non-convex orbit constraints, and decompose P1 E This is transformed into a convexity programming problem.

[0113] Question P1 E In the orbital insertion constraint (18)

[0114]

[0115] Linear affine constraints do not need to be decomposed; only non-convex constraints r need to be decomposed. T (t f )v(t f ) = 0 and Similar to step two, constrain r T (t f )v(t f The condition ) = 0 is decomposed into two inequality constraints as shown in formula (14). Equality constraints It is decomposed into the following inequality constraints

[0116]

[0117] The problem P1 is approximated using a linear dynamic model (17). E The nonlinear dynamic equation (1) is given.

[0118] After decomposing the orbital constraint convexity and approximating the dynamics using (17), problem P1 E It was transformed into problem P2 E

[0119] P2 E minimum -||v(t) f )||

[0120] subject to Eq.(5),(14),(17),(19)-(21)

[0121] The time interval [t0, t] f Divide into equal parts of length h = (t)f -t0) / N is divided into N equal parts. Let r be... j =r(t) j ), v j =v(t) j ) and 1 bj =1 b (t j ) represent time nodes t j =t0+jh, (j=0,1,…,N) position, velocity and thrust direction. After discretizing the dynamic equation (17) and path constraint (19), problem P2 E This is then transformed into a convexity programming problem.

[0122] 4.2.3 Solving the Elliptical Termination Orbit Optimization Problem P2 using the Penalized Concave-Concave Process Method E This yields the orbital elements of the terminated orbit.

[0123] The penalized concavity / convexity process method is an efficient and mature approach for solving convexity programming problems. This method is used to solve the P2 problem of optimizing elliptical termination trajectories within the nominal orbital plane. E Solve for P2 E Get in h c >h p The number of orbital elements of an elliptical terminated orbit within the nominal orbital plane.

[0124] Step 5: In h min ≤h c ≤h p In the case of [the specific condition], the circular termination trajectory obtained in step two is the final termination trajectory.

[0125] The implementation method for step five is as follows:

[0126] In h min ≤h c ≤h p In this case, the circular termination track obtained in step two has a height greater than the minimum track height but less than the perigee height of the nominal track. This circular termination track is the final termination track in this case. The number of track elements for the highest circular termination track in this case has already been determined in step two.

[0127] Step Six: In h c <h min In this case, by modifying the circular termination trajectory optimization problem P3 in step two. C 1. Orbital insertion constraint hThe value of h is used to construct an optimization problem for the circular aborted trajectory within the rocket's flight plane at the moment of failure. This circular aborted trajectory optimization problem is transformed into a convexity programming problem, which is solved using a penalized concave-convex process method. This yields the orbital elements and orbital height h of the highest circular aborted trajectory within the rocket's flight plane at the moment of failure. c1 Compare h c1 and h min The size relationship, if h c1 <h min Then there is no stopping orbit if h c1 ≥h min Then return the number of orbital elements of the circular termination orbit solved in this step.

[0128] The implementation method for step six is ​​as follows:

[0129] Step 6.1: In h c <h min In this case, by modifying the circular termination trajectory optimization problem P3 in step two. C 1. Orbital insertion constraint h The value of h is used to construct a circular abort trajectory optimization problem within the rocket's flight plane at the moment of failure. This circular abort trajectory optimization problem is transformed into a convexity programming problem, and the penalized concavity-convexity process method is used to solve it, obtaining the orbital elements and orbital height h of the highest circular abort trajectory within the rocket's flight plane at the moment of failure. c1 .

[0130] The implementation method for step 6.1 is as follows:

[0131] In h c <h min In this case, the altitude of the highest circular abort orbit within the nominal orbital plane is lower than the lowest orbital altitude, and there is no suitable abort orbit within the nominal orbital plane. Since the launch vehicle's orbital maneuvers also consume energy, therefore, in h... c <h min In this case, it is necessary to construct an optimization problem for the circular abort trajectory within the rocket's flight plane at the moment of failure. This optimization problem for the circular abort trajectory within the rocket's flight plane at the moment of failure is similar to the optimization problem in step two (P3). C The forms are the same. Because question P3 C The purpose of constraint (13) is to constrain the plane where the stopping track is located, thus addressing problem P3. C In constraint (13), 1 h The value is changed to r0×v0 / ||r0×v0||, which yields the circular abort trajectory optimization problem within the rocket's flight plane at the moment of failure. The penalized concave-convex process method is used to solve the circular abort trajectory optimization problem described in this step, obtaining the orbital elements and orbital height h of the highest circular abort trajectory within the rocket's flight plane at the moment of failure. c1 .

[0132] Step 6.2: Compare h c1 and h min The size relationship, if h c1 <h min Then there is no stopping orbit if h c1 ≥h min Then return the number of orbital elements of the circular termination orbit solved in this step.

[0133] The implementation method for step 6.2 is as follows:

[0134] Compare h c1 and h min The size relationship, if h c1 <h min At the moment of failure, the altitude of the highest circular abort trajectory within the rocket's flight plane is lower than the minimum trajectory altitude, therefore no abort trajectory exists; if h c1 ≥h min Then return the number of orbital elements of the highest circle of the rocket's flight plane at the moment of failure, which was optimized in step 6.1.

[0135] Step 7: Based on different thrust loss scenarios, fully utilize the remaining payload capacity of the launch vehicle. Steps 4, 5, or 6 respectively determine the launch vehicle's abort trajectory, thus realizing the online abort trajectory determination of the launch vehicle based on convexity planning.

[0136] The implementation method for step seven is as follows:

[0137] Steps four, five, or six determine the abort trajectory based on the different thrust loss coefficients κ of the launch vehicle, and the abort trajectory determined in the final step is the final abort trajectory. Because the optimization problem involved in the preceding steps is solved efficiently, online abort trajectory determination for the launch vehicle based on convexity programming is achieved.

[0138] Step 8: Input the orbital elements of the aborted orbit determined in Step 7 into the launch vehicle's GNC system to replace the nominal orbital elements. After the failure, the launch vehicle will fly to the aborted orbit to carry out the rescue mission for astronauts and spacecraft, thereby improving the launch vehicle's autonomy and safety in dealing with thrust reduction failures.

[0139] The implementation method for step eight is as follows:

[0140] The orbital elements of the abort trajectory determined in step seven are input into the launch vehicle's GNC system to replace the nominal orbital elements. The rocket experiencing thrust loss begins to fly towards a lower-energy abort trajectory, temporarily suspending the spacecraft and astronauts. This provides ground support personnel with sufficient time to analyze the problem and send instructions—either to continue the mission or return safely to Earth—enabling rescue of astronauts or spacecraft on the launch vehicle experiencing thrust loss and improving the launch vehicle's autonomy and safety in responding to thrust loss failures.

[0141] Beneficial effects:

[0142] 1. This invention discloses a method for determining the online abort trajectory of a launch vehicle based on convexity programming. It uses convexity decomposition to handle complex orbital constraints, transforming the abort trajectory optimization problem into a convexity programming problem. Compared to traditional nonlinear optimization methods, this invention offers higher reliability and computational efficiency, enabling real-time computation on an onboard computer.

[0143] 2. The method for determining the online abort trajectory of a launch vehicle based on convexity programming disclosed in this invention uses precise relaxation to maintain the nonlinearity of the original abort trajectory optimization problem when solving the abort trajectory optimization problem, thereby increasing the feasible region of the original problem. Compared with directly solving the original problem, the optimization problem after relaxation has better convergence.

[0144] 3. The method for determining the online abort trajectory of a launch vehicle based on convexity programming disclosed in this invention constructs an elliptical abort trajectory optimization problem that is easier to solve by setting the orbital entry point of the faulty launch vehicle at the perigee, thereby improving the computational efficiency of solving the elliptical abort trajectory optimization problem.

[0145] 4. The present invention discloses an online method for determining the abort trajectory of a launch vehicle based on convexity programming. This method determines the trajectory height h of the circular abort trajectory obtained through optimization based on the specific thrust loss rate of the faulty rocket. c With minimum orbital height h min and perigee height h p The size relationship is divided into h c >h p h min ≤h c ≤h p h c <h min Different types of abort trajectories are optimized separately; by solving optimization problems, the circular or elliptical abort trajectories with the maximum energy are obtained, making full use of the remaining payload capacity of the launch vehicle and significantly improving the autonomy and safety of the launch vehicle in dealing with thrust reduction failures. Attached Figure Description

[0146] Figure 1This is the ascending trajectory obtained by solving the circular termination trajectory optimization problem in the embodiments of the present invention;

[0147] Figure 2 This is a graph showing the changes in height and velocity obtained by solving the circular termination trajectory optimization problem in an embodiment of the present invention.

[0148] Figure 3 This is a thrust variation diagram obtained by solving the circular termination trajectory optimization problem in an embodiment of the present invention;

[0149] Figure 4 The method for determining the terminated trajectory disclosed in this invention provides a trajectory relationship diagram for elliptical terminated trajectories;

[0150] Figure 5 The method for determining the terminated track disclosed in this invention provides a track relationship diagram for the termination of the track within the nominal track plane;

[0151] Figure 6 The method for determining the aborted trajectory disclosed in this invention provides a trajectory relationship diagram of the aborted trajectory of the launch vehicle in the in-plane circle at the moment of failure.

[0152] Figure 7 This is the ascending trajectory obtained by solving the elliptical termination trajectory optimization problem in the embodiments of the present invention;

[0153] Figure 8 This is a graph showing the changes in height and velocity obtained by solving the elliptical terminated trajectory optimization problem in an embodiment of the present invention.

[0154] Figure 9 This is a thrust variation diagram obtained by solving the elliptical termination trajectory optimization problem in an embodiment of the present invention. Detailed Implementation

[0155] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0156] The transfer orbit launch mission to the Chinese space station was selected as the simulation background. A point on the nominal ascent trajectory was taken as the state of the rocket at the moment of thrust descent failure. Then, the abort trajectory was determined according to the given thrust loss rate, thus verifying the effectiveness of the online abort trajectory determination method for launch vehicles based on convexity planning disclosed in this invention.

[0157] The method for determining the online abort trajectory of a launch vehicle based on convexity planning disclosed in this embodiment has the following specific implementation steps:

[0158] Step 1: Based on the magnitude of the rocket thrust loss rate, establish the dynamic equations for the ascent phase after the failure of the final stage of the launch vehicle in the geocentric inertial coordinate system.

[0159] The implementation method for step one is as follows:

[0160] The dynamic equations for the ascent phase of a launch vehicle in a geocentric inertial coordinate system after a thrust loss failure are expressed as follows:

[0161]

[0162] in, It is the rocket's position vector. It is the rocket velocity vector, and T is the available thrust after a rocket malfunction. b Here, is the direction of thrust, g is the gravitational acceleration vector, and m is the rocket's mass. The formulas for calculating the thrust and mass of a rocket are as follows:

[0163] T=(1-κ)T vac (twenty three)

[0164] m(t) = m0 - Tt / (I sp g0) (24)

[0165] In the formula, κ is the thrust loss rate, which is a constant, and T vac It is the vacuum thrust of the engine, m0 is the mass of the rocket at the moment of failure (time t0), and I sp κ is the engine's specific impulse, and g0 is the gravitational acceleration at the Earth's surface. In this embodiment, κ = 0.15, T vac =720 kN, m0 = 99180 kg, I sp =341.4 seconds, g0 = 9.81 meters per second squared.

[0166] Step 2: Given the minimum track height h min Given the nominal orbital elements, the perigee height h is calculated based on the nominal orbital elements. p With maximizing orbital altitude as the optimization objective, and constrained by the circular orbit insertion conditions and the ascent dynamics equations for the launch vehicle's final stage after a failure (constructed in step one), a circular abort orbit optimization problem is established. This circular abort orbit optimization problem is then transformed into a convexity programming problem. The penalized concave-convex process method is used to solve this problem, yielding the orbital elements and orbital altitude h of the highest circular abort orbit within the nominal orbital plane. c .

[0167] The implementation method for step two is as follows:

[0168] Step 2.1: Given the minimum orbital height h min Given the nominal orbital elements, calculate the perigee height h based on the nominal orbital elements. p .

[0169] The implementation method for step 2.1 is as follows:

[0170] Given minimum orbital altitude h min =160 km. To ensure the terminated orbit is stable, its altitude must be greater than h. min The orbital elements of the nominal orbit are known before the launch mission. The nominal orbital data used in this embodiment are shown in Table 1 below. Calculate the perigee altitude h. p =a * (1-e * )-R E = 220 kilometers.

[0171] Table 1 Nominal number of orbital elements

[0172]

[0173] Step 2.2: Using the circular orbit insertion conditions and the ascent dynamics equations of the launch vehicle after the final stage failure constructed in Step 1 as constraints, and with maximizing the orbital height as the optimization objective, establish the circular termination orbit optimization problem within the nominal orbital plane after the failure.

[0174] The implementation method for step 2.2 is as follows:

[0175] 2.2.1 Constraints are given for the circular termination trajectory optimization problem.

[0176] To construct an optimization problem for a circular abort trajectory within the nominal orbital plane after a failure, constraints and optimization objectives need to be defined.

[0177] 1) Thrust direction constraint: The 2-norm of the thrust direction is in [t0, t... j The upper part equals 1, that is

[0178]

[0179] Where t0 and t f These are the initial time and the terminal time, respectively.

[0180] 2) Initial state constraints: The position and velocity of the launch vehicle at the moment of failure are given by the GNC system.

[0181]

[0182] 3) Circular orbit insertion constraint: The launch vehicle needs to transport the astronauts and spacecraft to a circular orbit. The circular orbit insertion constraint is denoted as Φ. C =0, specifically in the following form

[0183]

[0184] Where, μ = 3.9860044 × 10 14cubic meters per second squared, 1 h It is a unit vector in the same direction as the angular momentum of the terminated orbit. Since the terminated orbit and the nominal orbit are coplanar, we have 1. h =[sini * ·sinΩ * -sini * ·cosΩ * cosi * ] T .

[0185] 2.2.2 Given an optimization objective, establish a trajectory optimization problem where the circle in the nominal track plane terminates after a fault.

[0186] When optimizing a circular termination trajectory, the optimization objective is to maximize the trajectory height, which corresponds to minimizing -||r(t). f Therefore, we obtain the following optimal control problem P0 for optimizing a circular termination trajectory. C

[0187] P0 C minimum -||r(t) f )||

[0188] Subject to Eq.(22),(25)-(27)

[0189] Question P0 C The start time t0 = 0, and the end time t f Take (m0-m) dry )I sp g0 / T, the mass of the rocket when the fuel is exhausted (m) dry = 25000 kg. At P0 C In this context, the control vector is the thrust direction 1. b The state vector is s = [r T v T ] T .

[0190] Step 2.3: By relaxing the circular trajectory optimization problem after the fault constructed in Step 2.2, a new constraint φ is introduced. r =||r(t) f )||, perform convexity decomposition on the orbital constraints, and solve the circular termination orbit optimization problem P0 C The problem is transformed into a convexity programming problem, which is then solved using a penalized concave-convexity process method. This yields the number of orbit elements and the orbit height h of the orbit ending at the highest circle within the nominal orbital plane. c .

[0191] The implementation method for step 2.3 is as follows:

[0192] 2.3.1 Relaxation processing is applied to the optimization problem of the circular stop track after the fault constructed in step 2.2.

[0193] Relax the control constraint—namely, the thrust direction constraint (25)—as follows:

[0194] ||1 b (t)||≤1 (28) Problem P0 C The problem is relaxed to the following P1 C

[0195] P1 C minimum -||r(t) f )||

[0196] Subject to Eq.(22),(26)-(28)

[0197] Proofs using optimal control theory show that in problem P1 C In the optimal solution, constraint (28) is in [t0,t f It is active almost everywhere on the [scale]. For any optimal control quantity... It is true in almost every way, problem P1 C The optimal solution is also problem P0. C The optimal solution.

[0198] 2.3.2 Introducing a new constraint φ r =||r(t) f )||.

[0199] Introduce the following new equality constraints

[0200] φ r =||r(t) f (29)

[0201] Due to the introduction of new constraints (29), problem P1 C The objective function is equivalently rewritten as -φ r Question P1 C In the orbital constraint (27), ||r(t) f )||·||v(t f )|| 2 -μ=0 is equivalently transformed into φ r ·||v(t f )|| 2 -μ=0. Therefore, the orbital insertion constraint (27) is rewritten as

[0202]

[0203] Relax the newly introduced equality constraint (29) to

[0204] φ r ≤||r(t f (31)

[0205] Because a new constraint (29) is introduced and the equality constraint (29) is replaced with a relaxation constraint (31), problem P1 C The problem is relaxed to the following P2 C

[0206] P2 C minimum -φ r

[0207] subject to Eq.(22),(26),(28),(30),(31)

[0208] Theoretical analysis reveals that problem P2 C In the optimal solution, constraint (31) is always active. This step completes the solution to problem P0. C Precise relaxation. Problems after relaxation (P2) C Compared to the original problem P0 C With a larger feasible region, however P2 C The optimal solution and P0 C The optimal solutions are the same.

[0209] 2.3.3 Perform convexity decomposition on non-convex orbit constraints, and decompose P2 C This is transformed into a convexity programming problem.

[0210] For question P2 C The non-convex constraints and constraints (31) in the mid-track constraint (30) are decomposed into convex differences. Convex difference decomposition is the process of decomposing the constraint function into the difference between two convex functions. The decomposed constraints have the following form.

[0211] f(x)-g(x)≤0 (32)

[0212] Here, x is the optimization variable, and f(x) and g(x) are both convex functions.

[0213] For question P2 C The inequality constraint (31) needs to be rewritten as follows:

[0214] φ r -||r(t f )||≤0 (33)

[0215] In orbital insertion constraint (30)

[0216]

[0217] It is a linear affine constraint, not a convex constraint r T (t f )v(t f ) = 0 and φ r ·||v(t f )|| 2 -μ = 0 needs to be decomposed by convexity. First, constrain the equality r. T (t f )v(t f The equation 0 is transformed into two inequality constraints r. T (t f )v(t f )≤0 and -r T (t f )v(t f If )≤0, then the two inequality constraints are rewritten as constraint (35).

[0218]

[0219] Equality constraint φ r ·||v(t f )|| 2 -μ=0 is transformed into two inequality constraints, and then the inequality constraints are decomposed to obtain (μ / φ) r (It is a convex function)

[0220]

[0221] Due to variable φ r The introduction of φ makes the constraint φ r ·||v(t f )|| 2 -μ=0 can be decomposed into the form of constraint (36).

[0222] The dynamic equation (22) is nonlinear because the gravitational acceleration g is a nonlinear function of position r. To obtain a linear dynamic equation, the gravitational acceleration calculated in the previous iteration is used to approximate the gravitational acceleration in the current iteration, i.e.

[0223]

[0224] Where, r [k] This is the position vector profile obtained from the previous iteration of the iterative algorithm. Using the approximate gravitational model of formula (37), the dynamic equation (22) is transformed into the following linear form.

[0225]

[0226] After decomposing the orbital constraint convexity and approximating the dynamics using equation (38), problem P2 C Transform into question P3C

[0227] P3 C minimum -φ r

[0228] subject to Eq.(26),(28),(33)-(36),(38)

[0229] The time interval [t0, t] f Divide into equal parts of length h = (t) f Divide (-t0) / 100 into 100 equal parts. Let r be... j =r(t) j ), v j =v(t) j ) and 1 bj =1 b (t j ) represent time nodes t j =t0+jh, (j=0,1,…,100) is the position, velocity, and thrust direction. After discretizing the path constraint (28) and the dynamic equation (38), problem P3 C This is then transformed into a convexity programming problem.

[0230] 2.3.4 Solving the Circular Termination Orbit Optimization Problem Using the Penalized Concave-Concave Process Method (P3) C The number of orbital elements and the orbital height h of the terminated orbit are obtained. c .

[0231] The penalized concave-convex process method is an efficient and mature approach for solving convexity programming problems. This method is used to solve the circular termination trajectory optimization problem within the nominal orbital plane (P3). C Solve P3 C The time taken was 38.9 milliseconds. The number of track elements for the highest circular termination track within the nominal track plane is shown in Table 2. The height of the circular termination track is h. c = 230.50 km (greater than h) p ). Figure 1 , Figure 2 and Figure 3 The ascent trajectory, altitude-velocity profile, and thrust component profile in the geocentric inertial coordinate system obtained by optimizing the circular termination trajectory in this step are given respectively.

[0232] Table 2. Number of orbital elements in a circular path.

[0233]

[0234] Step 3: Determine the track height h of the circular termination track based on the optimized solution. c With minimum orbital height h min and perigee height hp The size relationship is divided into h c >h p h min ≤h c ≤h p or h c <h min The launch vehicle's abort trajectory is determined online, corresponding to steps four, five, or six.

[0235] The implementation method for step three is as follows:

[0236] Based on the track height h of the circular termination track c With minimum orbital height h min and perigee height h p The size relationship is divided into h c >h p h min ≤h c ≤h p or h c <h min The abort trajectories of the launch vehicles were optimized online, and the specific comparison results corresponded to the following subsequent steps:

[0237] 1) In h c >h p In such cases, Figure 4 As shown, continue with step four;

[0238] 2) In h min ≤h c ≤h p In such cases, Figure 5 As shown, continue with step five;

[0239] 3) In h c <h min In such cases, Figure 6 As shown, continue with step six.

[0240] In this embodiment, h c = 230.50 km, h p = 220 kilometers, satisfying h c >h p Therefore, in this embodiment, step four is executed, and steps five and six are not executed.

[0241] Step 4: In h c >h pUnder these conditions, constrained by the elliptical orbit insertion conditions and the ascent dynamics equations for the launch vehicle's final stage after a failure (constructed in step one), and with maximizing the insertion velocity as the optimization objective, an optimization problem for the elliptical aborted orbit after a failure is established. This elliptical aborted orbit optimization problem is then transformed into a convexity programming problem. The penalized concave-convex process method is used to solve this convexity programming problem to obtain the orbital elements of the elliptical aborted orbit within the nominal orbital plane.

[0242] The implementation method for step four is as follows:

[0243] In h c >h p In this scenario, astronauts and spacecraft can be transported to a relatively high circular orbit. To further reduce orbital element deviations and facilitate subsequent orbital maneuvers, an elliptical aborted orbit with a perigee altitude equal to the nominal orbit and an apogee altitude as high as possible is optimized.

[0244] Step 4.1: Using the elliptical orbit insertion conditions and the ascent dynamics equations of the launch vehicle after the final stage failure constructed in Step 1 as constraints, and with maximizing the insertion velocity as the optimization objective, establish the elliptical abort orbit optimization problem within the nominal orbital plane after the failure.

[0245] The implementation method for step 4.1 is as follows:

[0246] 4.1.1 Constraints are given for the elliptical termination trajectory optimization problem.

[0247] In addition to satisfying the ascent dynamics equations (22) constructed in step one, the rocket after the malfunction also needs to satisfy the following constraints.

[0248] 1) Thrust direction constraint: see formula (25)

[0249] 2) Initial state constraints: see formula (26)

[0250] 3) Elliptical orbit insertion constraints: The launch vehicle needs to transport astronauts and spacecraft to an elliptical orbit. Let Φ be the elliptical orbit insertion constraint. E =0, assuming the rocket enters orbit at its perigee on its abort trajectory, then Φ E The specific form of =0 is as follows:

[0251]

[0252] Among them, 1 h It is a unit vector in the same direction as the angular momentum of the stopped trajectory. It is the perigee radius of the nominal orbit.

[0253] 4.1.2 Given an optimization objective, establish an elliptical abort orbit optimization problem within the nominal orbital plane after a fault.

[0254] When optimizing an elliptical termination trajectory, the optimization objective is to maximize the injection velocity, which corresponds to minimizing -||v(t). f Therefore, we obtain the following optimal control problem P0 for optimizing an elliptical terminated trajectory. E

[0255] P0 E minimum -||v(t) f )||

[0256] subject to Eq.(22),(25),(26),(39)

[0257] Optimal control problem P0 E The initial time t0 = 0, and the terminal time t f Take (m0-m) dry )I sp g0 / T. At P0 E In this context, the control vector is the thrust direction 1. b The state vector is s = [r T v T ] T .

[0258] Step 4.2: By relaxing the elliptical aborted trajectory optimization problem constructed in Step 4.1, and then performing convexity decomposition on the orbital constraints, the elliptical aborted trajectory optimization problem P0 is solved. E The problem is transformed into a convexity programming problem. The penalized concave-convex process method is used to solve the convexity programming problem and obtain the number of orbital elements of the elliptical terminated orbit in the nominal orbital plane.

[0259] The implementation method for step 4.2 is as follows:

[0260] 4.2.1 Relaxation processing is applied to the optimization problem of the elliptical aborted trajectory after the fault constructed in step 4.1.

[0261] Relax the control constraint—namely, the thrust direction constraint (25)—as follows:

[0262] ||1 b (t)||≤1 (40)

[0263] Question P0 E The problem is relaxed to the following P1 E

[0264] P1 E minimum -||v(t) f )||

[0265] subject to Eq.(22),(26),(39),(40)

[0266] Proofs using optimal control theory show that in problem P1 E In the optimal solution, constraint (40) is in [t0,t f It is active almost everywhere on the [scale]. For any optimal control quantity... It is true in almost every way, problem P1 E The optimal solution is also problem P0. E The optimal solution.

[0267] 4.2.2 Perform convexity decomposition on non-convex orbit constraints, and decompose P1 E This is transformed into a convexity programming problem.

[0268] Question P1 E In the orbital insertion constraint (39)

[0269]

[0270] Linear affine constraints do not need to be decomposed; only non-convex constraints r need to be decomposed. T (t f )v(t f ) = 0 and Similar to step two, constrain r T (t f )v(t f The expression ) = 0 is decomposed into two inequality constraints as shown in formula (35). Equality constraints It is decomposed into the following inequality constraints

[0271]

[0272] The problem P1 is approximated using a linear dynamic model (38). E The nonlinear dynamic equation (22) is given.

[0273] After decomposing the orbital constraint convexity and approximating the dynamics using (38), problem P1 E It was transformed into problem P2 E

[0274] P2 E minimum-||v(t) f )||

[0275] subject to Eq.(26),(35),(38),(40)-(42)

[0276] The time interval [t0, t] f Divide into equal parts of length h = (t) f Divide (-t0) / 100 into 100 equal parts. Let r be...j =r(t) j ), v j =v(t) j ) and 1 bj =1 b (t j ) represent time nodes t j =t0+jh, (j=0,1,…,100) position, velocity and thrust direction. After discretizing the dynamic equation (38) and path constraint (40), problem P2 E This is then transformed into a convexity programming problem.

[0277] 4.2.3 Solving the Elliptical Termination Orbit Optimization Problem P2 using the Penalized Concave-Concave Process Method E This yields the orbital elements of the terminated orbit.

[0278] The penalized concavity / convexity process method is an efficient and mature approach for solving convexity programming problems. This method is used to solve the P2 problem of optimizing elliptical termination trajectories within the nominal orbital plane. E Solve for P2 E It took 25.7 milliseconds to obtain the result at h. c >h p The number of orbital elements for elliptical terminated orbits within the nominal orbital plane is shown in Table 3. Figure 7 , Figure 8 and Figure 9 The ascent trajectory, altitude-velocity profile, and thrust component profile in the geocentric inertial coordinate system obtained by optimizing the elliptical termination trajectory in this step are presented respectively.

[0279] Table 3. Number of orbital elements of an elliptical terminated orbit.

[0280]

[0281] Step 5: In h min ≤h c ≤h p In this case, the circular termination trajectory obtained in step two is the final termination trajectory (step five is not performed in this embodiment).

[0282] The implementation method for step five is as follows:

[0283] In h min ≤h c ≤h p In this case, the circular termination track obtained in step two has a height greater than the minimum track height but less than the perigee height of the nominal track. This circular termination track is the final termination track in this case. The number of track elements for the highest circular termination track in this case has already been determined in step two.

[0284] Step Six: In h c <hmin In this case, by modifying the circular termination trajectory optimization problem P3 in step two. C 1. Orbital insertion constraint h The value of h is used to construct an optimization problem for the circular aborted trajectory within the rocket's flight plane at the moment of failure. This circular aborted trajectory optimization problem is transformed into a convexity programming problem, which is solved using a penalized concave-convex process method. This yields the orbital elements and orbital height h of the highest circular aborted trajectory within the rocket's flight plane at the moment of failure. c1 Compare h c1 and h min The size relationship, if h c1 <h min Then there is no stopping orbit if h c1 ≥h min Then return the number of orbital elements of the circular termination orbit obtained in this step (step six is ​​not executed in this embodiment).

[0285] The implementation method for step six is ​​as follows:

[0286] Step 6.1: In h c <h min In this case, by modifying the circular termination trajectory optimization problem P3 in step two. C 1. Orbital insertion constraint h The value of h is used to construct a circular abort trajectory optimization problem within the rocket's flight plane at the moment of failure. This circular abort trajectory optimization problem is transformed into a convexity programming problem, and the penalized concavity-convexity process method is used to solve it, obtaining the orbital elements and orbital height h of the highest circular abort trajectory within the rocket's flight plane at the moment of failure. c1 .

[0287] The implementation method for step 6.1 is as follows:

[0288] In h c <h min In this case, the altitude of the highest circular abort orbit within the nominal orbital plane is lower than the lowest orbital altitude, and there is no suitable abort orbit within the nominal orbital plane. Since the launch vehicle's orbital maneuvers also consume energy, therefore, in h... c <h min In this case, it is necessary to construct an optimization problem for the circular abort trajectory within the rocket's flight plane at the moment of failure. This optimization problem for the circular abort trajectory within the rocket's flight plane at the moment of failure is similar to the optimization problem in step two (P3). C The forms are the same. Because question P3 C The purpose of constraint (34) is to constrain the plane where the stopping track is located, thus addressing problem P3. C In constraint (13), 1 hThe value is changed to r0×v0 / ||r0×v0||, which yields the circular abort trajectory optimization problem within the rocket's flight plane at the moment of failure. The penalized concave-convex process method is used to solve the circular abort trajectory optimization problem described in this step, obtaining the orbital elements and orbital height h of the highest circular abort trajectory within the rocket's flight plane at the moment of failure. c1 .

[0289] Step 6.2: Compare h c1 and h min The size relationship, if h c1 <h min Then there is no stopping orbit if h c1 ≥h min Then return the number of orbital elements of the circular termination orbit solved in this step.

[0290] The implementation method for step 6.2 is as follows:

[0291] Compare h c1 and h min The size relationship, if h c1 <h min At the moment of failure, the altitude of the highest circular abort trajectory within the rocket's flight plane is lower than the minimum trajectory altitude, therefore no abort trajectory exists; if h c1 ≥h min Then return the number of orbital elements of the highest circle of the rocket's flight plane at the moment of failure, which was optimized in step 6.1.

[0292] Step 7: Based on the thrust loss, make full use of the remaining payload capacity of the launch vehicle. Step 4 determined the launch vehicle's abort trajectory, that is, the online abort trajectory determination of the launch vehicle based on convexity planning was realized.

[0293] The implementation method for step seven is as follows:

[0294] Step four determined the abort trajectory based on the launch vehicle's thrust loss coefficient κ = 0.15. This is because the optimization problem P3 involved in the preceding steps... C and P2 E The solution was efficient, taking a total of 64.6 milliseconds to solve the two optimization problems, thus realizing the online abort trajectory determination of the launch vehicle based on convexity programming.

[0295] Step 8: Input the orbital elements of the aborted orbit determined in Step 7 into the launch vehicle's GNC system to replace the nominal orbital elements. After the failure, the launch vehicle will fly to the aborted orbit to carry out the rescue mission for astronauts and spacecraft, thereby improving the launch vehicle's autonomy and safety in dealing with thrust reduction failures.

[0296] The implementation method for step eight is as follows:

[0297] The orbital elements of the abort trajectory determined in step seven are input into the launch vehicle's GNC system to replace the nominal orbital elements. The rocket experiencing thrust loss begins to fly towards a lower-energy elliptical abort trajectory. The spacecraft and astronauts are temporarily suspended in the abort trajectory, providing ground support personnel with sufficient time to analyze the problem and send instructions—either to continue the mission or return safely to Earth. This allows for the rescue of astronauts or spacecraft on the launch vehicle experiencing thrust loss, improving the launch vehicle's autonomy and safety in dealing with thrust loss failures.

[0298] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for determining the online abort trajectory of a launch vehicle based on convexity programming, characterized in that: Includes the following steps, Step 1: Based on the magnitude of the rocket thrust loss rate, establish the dynamic equations for the ascent phase of the launch vehicle after a final stage failure in the geocentric inertial coordinate system; The implementation method for step one is as follows: The dynamic equations for the ascent phase of a launch vehicle in a geocentric inertial coordinate system after a thrust loss failure are expressed as follows: in, It is the rocket's position vector. It is the rocket's velocity vector. It refers to the amount of usable thrust after a rocket malfunctions. It is the direction of thrust. It is the gravitational acceleration vector. This refers to the rocket's mass; the formulas for calculating the rocket's usable thrust and mass are as follows: In the formula, It is the thrust loss rate, which is a constant. It is the vacuum thrust of the engine. It was during the rocket malfunction. quality It is the engine's specific impulse. It is the gravitational acceleration at the Earth's surface; Step 2: Specify the minimum track height Given the nominal orbital elements, the perigee altitude is calculated based on the nominal orbital elements. With maximizing orbital altitude as the optimization objective, and constrained by the circular orbit insertion conditions and the ascent dynamics equations for the launch vehicle's final stage after a failure (constructed in step one), a circular abort orbit optimization problem is established. This problem is then transformed into a convexity programming problem, which is solved using a penalized concave-convex process method to obtain the orbital elements and orbital altitude of the highest circular abort orbit within the nominal orbital plane. ; Step 3: Determine the track height of the circular termination track based on the optimized solution. With minimum orbital height and perigee altitude The size relationship is divided into , or Determine the launch vehicle's abort trajectory online, corresponding to steps four, five, or six. Step Four: In In the case of elliptical orbit insertion conditions and the ascent dynamics equations of the launch vehicle after the final stage failure constructed in step one, with the goal of maximizing the insertion velocity, an elliptical aborted orbit optimization problem is established; the elliptical aborted orbit optimization problem is transformed into a convexity programming problem, and the convexity programming problem is solved using the penalized concave-convex process method to obtain the orbital elements of the elliptical aborted orbit in the nominal orbital plane. Step 5: In In this case, the circular termination trajectory obtained in step two is the final termination trajectory; Step Six: In In this case, by modifying the orbital insertion constraints of the circular abort trajectory optimization problem in step two, a circular abort trajectory optimization problem in the rocket's flight plane at the moment of failure is constructed. This circular abort trajectory optimization problem is then transformed into a convexity programming problem, which is solved using a penalized concave-convex process method. This yields the orbital elements and orbital height of the highest circular abort trajectory in the rocket's flight plane at the moment of failure. ;Compare and The size relationship, if There is no aborted track if Then return the number of orbital elements of the circle's final orbit obtained in this step; Step 7: Based on different thrust loss scenarios, make full use of the remaining payload capacity of the launch vehicle. Steps 4, 5, or 6 respectively determine the launch vehicle's abort trajectory, thus realizing the online abort trajectory determination of the launch vehicle based on convexity planning. Step 8: Input the orbital elements of the aborted orbit determined in Step 7 into the launch vehicle's GNC system to replace the nominal orbital elements. After the failure, the launch vehicle flies to the aborted orbit to carry out the rescue mission for astronauts and spacecraft, improving the launch vehicle's autonomy and safety in dealing with thrust reduction failures.

2. The method for determining the online abort trajectory of a launch vehicle based on convexity planning as described in claim 1, characterized in that: the implementation method of step two is as follows: Step 2.1: Given the minimum orbital altitude Given the nominal orbital elements, calculate the perigee altitude based on the nominal orbital elements. ; The implementation method for step 2.1 is as follows: Given minimum orbital altitude To ensure the stop track is stable, its height must be greater than [missing information]. The orbital elements of the nominal orbit are known before the launch mission. Specifically, the orbital elements that need to be specified include: the semi-major axis of the nominal orbit. eccentricity Track inclination Right ascension of ascending node and perigee argument Use formula Calculate the altitude of perigee , in the formula Indicates the Earth's radius; Step 2.2: Using the circular orbit insertion conditions and the ascent dynamics equations of the launch vehicle after the final stage failure constructed in Step 1 as constraints, and with maximizing the orbital height as the optimization objective, establish the circular termination orbit optimization problem within the nominal orbital plane after the failure. The implementation method for step 2.2 is as follows: 2.2.1 Constraints are given for the circular termination trajectory optimization problem; To construct the circular abort trajectory optimization problem within the nominal orbital plane after a failure, constraints and optimization objectives need to be given. In addition to satisfying the ascent phase dynamics equations constructed in step one, the rocket after the failure also needs to satisfy the following constraints. 1) Thrust direction constraint: The 2-norm of the thrust direction is within the interval The upper equals 1, that is in, and These are the initial time and the terminal time, respectively. 2) Initial state constraints: the moment of launch vehicle failure The rocket's position and velocity are given by the GNC system. in, and They are The rocket's position and velocity vector at any given moment; 3) Circular orbit insertion constraints: The launch vehicle needs to transport astronauts and spacecraft to a circular orbit. The circular orbit insertion constraints are denoted as... The specific form is as follows: in, It is the gravitational constant. It is a unit vector in the same direction as the angular momentum of the terminated orbit. Since the terminated orbit needs to be coplanar with the nominal orbit, therefore... ; 2.2.2 Given an optimization objective, establish a trajectory optimization problem where the circle within the nominal track plane terminates after a fault; When optimizing a circular termination trajectory, the optimization objective is to maximize the trajectory height, which corresponds to minimizing... Therefore, the optimal control problem P0 for optimizing a circular termination trajectory is obtained. C P0 C minimum Subject to Eq. , - Question P0 C The start time Terminal time Pick , It is the mass of the rocket when the fuel is exhausted; at P0 C In this context, the control vector is the thrust direction. The state vector is ; Step 2.3: By relaxing the circular abort trajectory optimization problem constructed in Step 2.2 after the fault, new constraints are introduced. The convexity decomposition of the orbital constraints is applied to solve the circular termination orbit optimization problem P0. C The problem is transformed into a convexity programming problem, which is then solved using a penalized concave-convexity process method. This yields the number of orbital elements and the orbital height at the highest circle within the nominal orbital plane. ; The implementation method for step 2.3 is as follows: 2.3.1 Relaxation processing is applied to the circular cessation trajectory optimization problem constructed in step 2.2 after the fault; Control constraints—namely, thrust direction constraints —Relaxation is defined by the following constraints Question P0 C The problem is relaxed to the following P1 C P1 C minimum Subject to Eq. , - Proofs using optimal control theory show that in problem P1 C In the optimal solution, constraints exist It is active almost everywhere; for any optimal control quantity , It is true in almost every way, problem P1 C The optimal solution is also problem P0. C The optimal solution; 2.3.2 Introducing New Constraints ; Introduce the following new equality constraints Due to the introduction of new constraints Question P1 C The objective function is equivalently rewritten as Question P1 C orbital constraints middle Equivalently transformed Therefore, orbital insertion constraints will be implemented. Rewritten as The newly introduced equality constraints relaxation Due to the introduction of new constraints And use relaxed constraints Replaced equality constraints Question P1 C The problem is relaxed to the following P2 C P2 C minimum Subject to Eq. , , , , Theoretical analysis reveals that problem P2 C In the optimal solution, constraints Always active; this step completes the work on problem P0. C Precise relaxation; problems after relaxation P2 C Compared to the original problem P0 C With a larger feasible region, however P2 C The optimal solution and P0 C The optimal solutions are the same; 2.3.3 Perform convexity decomposition on non-convex orbit constraints, and decompose P2 C This is transformed into a convexity programming problem. For question P2 C Mid-orbit constraint Non-convex constraints and constraints in Perform convexity-difference decomposition; convexity-difference decomposition is the process of decomposing a constraint function into the difference of two convex functions; the decomposed constraint has the following form. in, It is an optimization variable. and They are all convex functions; For question P2 C Inequality constraints need to be applied. Rewritten as Orbital insertion constraints middle It is a linear affine constraint, not a convex constraint. and It needs to be decomposed by convexity; first, the equality constraints are... Transform into two inequality constraints and Then the two inequality constraints are rewritten as constraints form Equality constraints Transform it into two inequality constraints, and then decompose the inequality constraints to obtain... in and It is a convex function; due to the variable The introduction of constraints Can be decomposed into constraints The form; Dynamic equations It is nonlinear because of gravitational acceleration. It's about location. The nonlinear function; in order to obtain the linear dynamic equation, the gravitational acceleration calculated in the previous iteration is used to approximate the gravitational acceleration in this iteration, that is... in, It is the position vector profile obtained from the previous iteration of the iterative algorithm; using the formula The approximate gravitational model will use the dynamic equations Transformed into the following linear form Decompose the orbital constraint convexity and apply the dynamics to the equations. After approximation, problem P2 C Transform into question P3 C P3 C minimum Subject to Eq. , , - , Time interval Divide into equal parts of length 1 of Equal parts; record , and Each is a time node The position, velocity, and thrust direction at the point, among which ; Path constraints and dynamic equations After discretization, problem P3 C This is then transformed into a convexity programming problem; 2.3.4 Solving the Circular Termination Orbit Optimization Problem Using the Penalized Concave-Concave Process Method (P3) C The number of orbital elements and the orbital height of the terminated orbit are obtained. ; The penalized concave-convex process method is an efficient and mature approach for solving convexity programming problems. This method is used to solve the circular termination trajectory optimization problem within the nominal orbital plane (P3). C Solve P3 C The number of track elements and track height are obtained for the highest circle of the nominal track plane. .

3. The method for determining the online abort trajectory of a launch vehicle based on convexity planning as described in claim 2, characterized in that: The implementation method for step three is as follows: Based on the track height of the circular termination track With minimum orbital height and perigee altitude The size relationship is divided into , or The abort trajectories of the launch vehicles were optimized online, and the specific comparison results corresponded to the following subsequent steps: 1) In In this case, proceed to step four; 2) In In this case, proceed to step five; 3) In In this case, proceed to step six.

4. The method for determining the online abort trajectory of a launch vehicle based on convexity planning as described in claim 3, characterized in that: the implementation method of step four is as follows: exist In this case, astronauts and spacecraft can be transported to a high-altitude circular orbit; in order to further reduce the deviation of orbital elements and facilitate subsequent orbital maneuvers, an elliptical aborted orbit with a perigee altitude equal to the nominal orbit and an apogee altitude as high as possible is optimized. Step 4.1: Using the elliptical orbit insertion conditions and the ascent dynamics equations of the launch vehicle after the final stage failure constructed in Step 1 as constraints, and with maximizing the insertion velocity as the optimization objective, establish the elliptical abort orbit optimization problem within the nominal orbital plane after the failure. The implementation method for step 4.1 is as follows: 4.1.1 Constraints are given for the elliptical termination trajectory optimization problem; In addition to satisfying the ascent dynamics equations constructed in step one, the rocket after the malfunction also needs to meet the following requirements. It also needs to satisfy the following constraints 1) Thrust direction constraint: see formula 2) Initial state constraints: see formula 3) Elliptical orbit insertion constraints: The launch vehicle needs to transport astronauts and spacecraft into an elliptical orbit. Let the elliptical orbit insertion constraints be denoted as... If the rocket is set to enter orbit at its perigee on its abort trajectory, then... The specific form is as follows: in, It is a unit vector in the same direction as the angular momentum of the stopped trajectory. It is the perigee radius of the nominal orbit; 4.1.2 Given an optimization objective, establish an elliptical trajectory optimization problem within the nominal orbital plane after a fault; When optimizing an elliptical termination trajectory, the optimization objective is to maximize the injection velocity, which corresponds to minimizing... Therefore, the optimal control problem P0 for optimizing an elliptical terminated orbit is obtained. E P0 E : minimum Subject to Eq. , , , Optimal control problem P0 E initial moment Terminal time Pick ; at P0 E In this context, the control vector is the thrust direction. The state vector is ; Step 4.2: By relaxing the elliptical aborted trajectory optimization problem constructed in Step 4.1, and then performing convexity decomposition on the orbital constraints, the elliptical aborted trajectory optimization problem P0 is solved. E The problem is transformed into a convexity programming problem. The penalized concave-convex process method is used to solve the convexity programming problem and obtain the number of orbital elements of the elliptical terminated orbit in the nominal orbital plane. The implementation method for step 4.2 is as follows: 4.2.1 Relaxation processing is applied to the optimization problem of the elliptical aborted trajectory after the fault constructed in step 4.1; Control constraints—namely, thrust direction constraints —Relaxation is defined by the following constraints Question P0 E The problem is relaxed to the following P1 E P1 E minimum Subject to Eq. , , , Proofs using optimal control theory show that in problem P1 E In the optimal solution, constraints exist It is active almost everywhere; for any optimal control quantity , It is true in almost every way, problem P1 E The optimal solution is also problem P0. E The optimal solution; 4.2.2 Perform convexity decomposition on non-convex orbit constraints, and decompose P1 E This is transformed into a convexity programming problem. Question P1 E orbital constraints middle Linear affine constraints do not need to be decomposed; only non-convex constraints need to be decomposed. and Similar to step two, constraints are applied. Decomposed into formulas The two inequality constraints shown; equality constraints It is decomposed into the following inequality constraints Using linear dynamics models To approximate problem P1 E Nonlinear dynamic equations ; Decompose the orbital constraint convexity and apply the dynamics. After approximation, problem P1 E It was transformed into problem P2 E P2 E minimum Subject to Eq. , , , - Time interval Divide into equal parts of length 1 of Equal parts; record , and Each is a time node The position, velocity, and thrust direction at the point, among which ; the dynamic equation and path constraints After discretization, problem P2 E This is then transformed into a convexity programming problem; 4.2.3 Solving the Elliptical Termination Orbit Optimization Problem P2 using the Penalized Concave-Concave Process Method E , thus obtaining the orbital elements of the terminated orbit; The penalized concavity / convexity process method is an efficient and mature approach for solving convexity programming problems. This method is used to solve the P2 problem of optimizing elliptical termination trajectories within the nominal orbital plane. E Solve for P2 E Get In The number of orbital elements in the nominal orbital plane where the ellipse terminates.

5. The method for determining the online abort trajectory of a launch vehicle based on convexity planning as described in claim 4, characterized in that: The implementation method for step five is as follows: exist In the case where the circular termination track obtained in step two has a height greater than the minimum track height but less than the perigee height of the nominal track, this circular termination track is the final termination track in this case; the number of track elements of the highest circular termination track in this case has already been obtained in step two.

6. The method for determining the online abort trajectory of a launch vehicle based on convexity planning as described in claim 5, characterized in that: The implementation method for step six is ​​as follows: Step 6.1: In In this case, by modifying the circular termination trajectory optimization problem P3 in step two. C Orbital insertion constraints The value of the circular aborted trajectory is used to construct an optimization problem for the circular aborted trajectory in the rocket's flight plane at the moment of failure. This circular aborted trajectory optimization problem is then transformed into a convexity programming problem, and the penalized concave-convexity process method is used to solve it. This yields the orbital elements and orbital height of the highest circular aborted trajectory in the rocket's flight plane at the moment of failure. ; The implementation method for step 6.1 is as follows: exist In the case where the altitude of the highest circular abort orbit within the nominal orbital plane is lower than the lowest orbital altitude, there is no suitable abort orbit within the nominal orbital plane; since the launch vehicle's maneuvering to change orbital planes also consumes energy, therefore... In the case of a failure, it is necessary to construct an optimization problem for the circular abort trajectory within the rocket's flight plane at the moment of failure. The optimization problem of the circular trajectory termination in the rocket's flight plane at the moment of failure and the optimization problem in step two (P3) C The forms are the same; because question P3 C constraint Its function is to constrain the plane where the stopping track is located, thus addressing problem P3. C constraints middle The value was changed to This yields the optimization problem of the circular aborted trajectory within the rocket's flight plane at the moment of failure. The penalized concave-convex process method is then used to solve the circular aborted trajectory optimization problem described in this step, obtaining the orbital elements and altitude of the highest circular aborted trajectory within the rocket's flight plane at the moment of failure. ; Step 6.2: Comparison and The size relationship, if There is no aborted track if Then return the number of orbital elements of the circle's final orbit obtained in this step; The implementation method for step 6.2 is as follows: Compare and The size relationship, if At the moment of the malfunction, the altitude of the highest circular abort trajectory within the rocket's flight plane is lower than the minimum trajectory altitude, therefore no abort trajectory exists; if Then return the number of orbital elements of the highest circle of the rocket's flight plane at the moment of failure, which was optimized in step 6.

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7. The method for determining the online abort trajectory of a launch vehicle based on convexity planning as described in claim 6, characterized in that: The implementation method for step seven is as follows: Steps four, five, or six are based on the different thrust loss coefficients of the launch vehicle. The abort trajectory was determined, and the determined abort trajectory is the final abort trajectory; because the optimization problem involved was solved efficiently, the online abort trajectory determination of the launch vehicle based on convexity programming was realized.

8. The method for determining the online abort trajectory of a launch vehicle based on convexity planning as described in claim 7, characterized in that: The implementation method for step eight is as follows: The orbital elements of the abort trajectory determined in step seven are input into the launch vehicle's GNC system to replace the nominal orbital elements. The rocket experiencing thrust loss begins to fly towards the lower-energy abort trajectory, and the spacecraft and astronauts are temporarily parked in the abort trajectory, providing sufficient time for ground support personnel to analyze the problem and send instructions—to continue the mission or return safely to Earth—to enable the rescue of astronauts or spacecraft on the launch vehicle with thrust reduction failure, and to improve the autonomy and safety of the launch vehicle in dealing with thrust reduction failure.

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