A rocket trajectory planning method based on the modified Newton method-interior point method

Abstract

This invention provides a rocket trajectory planning method based on the modified Newton's method-interior point method, with the following steps: 1. Obtaining initial conjectures for stage time points; 2. Constructing inner-level subproblems using stage time points; 3. Discretizing the inner-level subproblems using the trapezoidal rule; 4. Solving the discrete inner-level subproblems using the interior point method; 5. Calculating gradient information using the finite difference method; 6. Calculating new stage time points using the modified Newton's method; 7. Outputting the rocket trajectory planning results. Through these steps, a rocket trajectory planning method based on the modified Newton's method-interior point method is obtained. This method has both high reliability and stability, solving the problem of how to plan rocket trajectories in situations with extremely high reliability requirements. The method is scientific, has good manufacturability, and has broad application value.

Description

Technical Field

[0001] This invention relates to a rocket trajectory planning method based on the modified Newton method-interior point method, which can be applied to the rocket guidance system and belongs to the field of guidance, navigation and control technology. Background Technology

[0002] The rocket's guidance system is a crucial component of the flight control system: it is responsible for adjusting the rocket's attitude (including pitch and heading) in space, thereby controlling its position and velocity. Considering the limited and precious nature of rocket fuel, the guidance system often employs a rocket trajectory planning method to pre-plan the rocket's future position and velocity, thus conserving fuel. Given the extremely high reliability requirements of rockets, the guidance system needs a reliable and stable rocket trajectory planning method.

[0003] Rocket trajectory planning methods need to solve the following problems: determining the optimal rocket thrust direction and stage timing to complete position and velocity control tasks; this is a classic optimal control problem, in which rocket thrust direction and stage timing are two key optimization variables; rocket thrust direction refers to the direction of the thrust vector provided by the rocket engine in space, which depends on the rocket's attitude in space; stage timing refers to the ignition, extinguishing, and separation time of each stage of the rocket; (the concept of stage is introduced because rockets generally have multiple stages, so the rocket trajectory generally needs to be divided into multiple stages) In order to save fuel, rocket trajectory planning methods need to optimize the rocket thrust direction and stage timing.

[0004] Existing rocket trajectory planning methods simultaneously plan the rocket thrust direction and stage timings, i.e., simultaneously determine two key optimization variables. These methods can be categorized into three types: indirect target method, nonlinear programming method, and interior point method. The first type, the indirect target method, offers high planning accuracy but is highly dependent on initial assumptions, requiring users to provide accurate initial guesses. The second type, nonlinear programming, is currently the primary offline trajectory planning method, widely used in rocket offline trajectory planning software. However, nonlinear programming cannot guarantee reliable convergence, requiring designers to empirically adjust the method parameters based on the planning results. The third type, the interior point method, is currently the primary online trajectory planning method. The interior point method boasts extremely strong convergence and reliability and does not require users to provide suitable initial guesses, but it is difficult to determine the optimal stage timings. Summary of the Invention

[0005] (I) Purpose of the Invention

[0006] The purpose of this invention is to propose a rocket trajectory planning method based on the modified Newton's method and the interior point method, which can be applied to rocket guidance systems. The proposed rocket trajectory planning method nests the modified Newton's method and the interior point method, having an inner and outer layer structure: the outer layer uses the modified Newton's method to determine the stage timings; the inner layer uses the interior point method to determine the rocket thrust direction; that is, the optimal rocket thrust direction and stage timings are determined respectively (in sequence). The proposed rocket trajectory planning method has both high reliability and stability, and is suitable for rocket guidance systems with extremely high reliability and stability requirements.

[0007] This invention is applicable to solving a class of rocket trajectory planning problems with specific forms, such as rocket ascent trajectory planning and rocket vertical landing trajectory planning; expressed mathematically, this invention is applicable to solving rocket trajectory planning problems of the following form:

[0008]

[0009] Where t is time, r is rocket position, V is rocket velocity, m is rocket mass, t0 is initial time, r0 is initial position, V0 is initial velocity, m0 is initial mass, and F... k R is the thrust of the rocket in stage k. k It is the mass flow rate of the rocket in stage k; furthermore, f and g are arbitrary linear functions, N is the total number of stages, u is the direction of rocket thrust, and T = (t1,...,t...). N The stage time point is t1; the rocket trajectory optimization method needs to solve the above problem to determine the optimal rocket thrust direction u and the stage time point T = (t1,...,t2). N );

[0010] (II) Technical Solution

[0011] This invention discloses a rocket trajectory planning method based on the modified Newton method-interior point method, the specific steps of which are as follows:

[0012] Step 1: Obtain initial conjectures about the timing of each stage;

[0013] Request the initial conjecture T = (t1,...,t) from the user of the rocket trajectory planning method to obtain the stage time points. N );

[0014] Step 2: Construct inner-layer sub-problems at the application stage;

[0015] Application phase time point T = (t1,...,t) N This paper addresses and transforms the rocket trajectory planning problem, introduces an inner relaxation coefficient K, and constructs an inner sub-problem with a specific form.

[0016] Step 3: Discretize the inner sub-problems using the trapezoidal method;

[0017] By applying the trapezoidal method, the inner subproblem obtained in step two is discretized to obtain the discrete inner subproblem.

[0018] Step 4: Apply the interior point method to solve the discrete inner-layer subproblem;

[0019] By applying the interior point method, the discrete inner layer subproblem is solved to obtain the optimal rocket thrust direction u and inner layer relaxation coefficient K;

[0020] Step 5: Calculate gradient information using the finite difference method;

[0021] Repeat steps two through four N times, and use the finite difference method to calculate the inner layer relaxation coefficient K relative to the stage time point T = (t1,...,t...). N gradient information

[0022] Step 6: Apply the modified Newton method to calculate the new stage time point;

[0023] Based on the inner layer relaxation coefficient K and gradient information obtained in steps two through five Construct an outer subproblem with a specific form and apply the modified Newton method to calculate the new stage time point;

[0024] Step 7: Output the rocket trajectory planning results;

[0025] If the stage time points obtained in step six satisfy a specific convergence condition, then output the rocket thrust direction u obtained in step four and the stage time points T = (t1,...,t) obtained in step six. N Otherwise, proceed to step two to continue the calculation;

[0026] Specifically, the step of "requesting the initial conjecture of the stage time point from the user of the rocket trajectory planning method" mentioned in step one is as follows:

[0027] Users are required to provide initial conjectures for N time points, T = (t1,...,t...). N (i.e., the end time of N stages), where N is the total number of stages and T is an N-dimensional vector;

[0028] The "inner sub-problem with a specific form" mentioned in step two refers to the problem obtained after processing and transforming the rocket trajectory planning problem and introducing an inner relaxation coefficient K; the specific form of the inner sub-problem is as follows:

[0029]

[0030] Among them, the "trapezoidal method" mentioned in step three is a commonly used method for discretizing differential equations;

[0031] The "discrete inner-layer sub-problem" mentioned in step three has the following specific form:

[0032]

[0033] Among them, the "interior point method" mentioned in step four refers to the second-order cone programming problem solution method represented by the primal-dual homogeneous embedded interior point method, which can be used to solve discrete inner-layer subproblems;

[0034] Among them, the "finite difference method" mentioned in step five is a commonly used numerical difference method used to calculate the gradient information of the inner layer relaxation coefficient relative to the stage time point.

[0035] Among them, the "modified Newton method" mentioned in step six is ​​a commonly used method for solving nonlinear programming problems;

[0036] The "outer sub-problem with a specific form" mentioned in step six has the following specific form:

[0037]

[0038] Through the above steps, a rocket trajectory planning method based on the modified Newton method-interior point method is obtained. This method has both high reliability and stability, and solves the problem of how to plan rocket trajectories in situations with extremely high reliability requirements. The method is scientific, has good manufacturability, and has broad application value.

[0039] (III) Advantages and Efficacy of the Invention

[0040] (1) This invention applies the modified Newton method and the interior point method in a nested manner: the modified Newton method is used as the outer layer to determine the optimal stage timing; the interior point method is used as the inner layer to determine the optimal rocket thrust direction, thus having high convergence and reliability.

[0041] (2) This invention does not require special assumptions about the number of rocket trajectory stages or the magnitude of rocket thrust, and can be applied to various rocket trajectory planning problems;

[0042] (3) The method described in this invention is scientific, has good processability, and has broad application value. Attached Figure Description

[0043] Figure 1 This is a flowchart of the method described in this invention.

[0044] Figure 2 This is a diagram of rocket thrust direction.

[0045] The serial numbers, symbols, and codes in the diagram are explained as follows:

[0046] Figure 1In this context, N represents the total number of stages;

[0047] Figure 2 In the diagram, thrust direction-1, thrust direction-2, and thrust direction-3 represent the components (scalars) of the thrust direction (vector) on the first, second, and third coordinate axes, respectively. Detailed Implementation

[0048] The present invention will now be described in further detail.

[0049] This invention discloses a rocket trajectory planning method based on the modified Newton method-interior point method, the flowchart of which is shown below. Figure 1 As shown, it includes the following steps:

[0050] Step 1: Obtain the initial conjecture of the stage timing.

[0051] Request the initial conjectures (t1,...,t) for N stages from the user of the rocket trajectory planning method. N Combine them into an N-dimensional vector T = (t1,...,t) N );

[0052] Step 2: Constructing Inner Sub-problems at the Application Phase

[0053] First, the application phase time point T = (t1,...,t) N This generates the following function:

[0054] F(t)=F k ,if t k-1 ≤t≤t k (5)

[0055] R(t)=R k ,if t k-1 ≤t≤t k (6)

[0056]

[0057] Secondly, by introducing the inner layer relaxation coefficient K, we construct an inner layer subproblem of the following form:

[0058]

[0059] Step 3: Discretize the inner sub-problems using the trapezoidal rule.

[0060] Applying the trapezoidal method, the inner subproblem obtained in step two is discretized. Introducing n discrete points in each stage, and a total of Nn discrete points across N stages, the resulting discrete inner subproblem is:

[0061]

[0062] The iteration range of variable k is 1 to N, and the iteration range of variable i is 1 to n-1.

[0063] Step 4: Solve the discrete inner-layer subproblem using the interior-point method.

[0064] The interior-point method is applied to solve the discrete inner-layer subproblem. Since the discrete inner-layer subproblem is a second-order cone programming problem, the primordial-dual homogeneous embedded interior-point method is used to solve it, obtaining the optimal rocket thrust direction u and inner-layer relaxation coefficient K.

[0065] Step 5: Calculate gradient information using the finite difference method.

[0066] Repeat steps two through four N times to calculate gradient information. The input to steps two through four is the stage time point T, and the output is the inner layer relaxation coefficient K. Therefore, steps two through four can be abstracted as a function:

[0067] K = K(T) (10)

[0068] Define an N-dimensional unit vector: e1 = (1, 0, ..., 0), e2 = (0, 1, ..., 0), ..., e N = (0,0,...,N).

[0069] The gradient information of the inner layer relaxation coefficient K relative to the stage time point T is calculated using the finite difference method.

[0070]

[0071] Here, 'a' is the parameter that needs to be selected. Steps two through four were repeated N times during the calculation.

[0072] Step Six: Apply the modified Newton method to calculate the new stage time point.

[0073] Based on the inner layer relaxation coefficient K obtained in steps two through four, construct an outer layer subproblem with a specific form:

[0074]

[0075] Gradient information based on the inner layer relaxation coefficient K relative to the stage time point T Applying the modified Newton's method, a new variable X = (x1,...,x) is introduced. N As a new stage, the aforementioned outer sub-problem is transformed into:

[0076]

[0077] Where M is the parameter to be selected. Solving the above problem yields the new stage time point X = (x1,...,x...). N ).

[0078] Step 7: Output the rocket trajectory planning results

[0079] First, calculate whether the convergence condition is met:

[0080] ||XT||≤∈ (14)

[0081] Where ∈ is the parameter to be selected; then, let:

[0082] T = X (15)

[0083] Finally, if the convergence condition is met, the rocket thrust direction u and the stage time point T are output; otherwise, proceed to step two to continue the calculation.

[0084] Simulation Case:

[0085] In the simulation case: Assume the rocket's initial time t0 = 0 s, initial position r0 = [1.5, 0, 2] km, initial velocity V0 = [-75, 0, 100] m / s, initial mass m0 = 1905 kg, and the expression for the linear function f is:

[0086]

[0087] The expression for the linear function g is:

[0088] g(r) = [-3.7114, 0, 0] m / s 2 (17)

[0089] Suppose the rocket trajectory is divided into three stages, i.e., N=3. In the first and third stages, the rocket thrust is 13258 N and the flow rate is 6.743 kg / s. In the second stage, the rocket thrust is 4972 N and the flow rate is 2.529 kg / s. Assume the initial conjectures for the stage timings are 20 seconds, 40 seconds, and 60 seconds. The parameter n is chosen to be 30, the parameter a is chosen to be 1 second, and the parameter M is chosen to be:

[0090]

[0091] The parameter ∈ is selected as 0.001 seconds;

[0092] A rocket trajectory planning method based on the modified Newton method-interior point method is adopted, which converges after 27 iterations at step seven. In the rocket trajectory planning result, the rocket thrust direction is as follows: Figure 2 As shown, the time points for each stage are 41.517 seconds, 65.215 seconds, and 72.405 seconds.

[0093] Simulation results demonstrate that the proposed two-layer rocket trajectory planning method based on the modified Newton method-interior point method can plan rocket trajectories with multiple stages, and has high computational efficiency and convergence. It is suitable for rocket guidance systems with limited computing power and high reliability requirements.

Claims

1. A rocket trajectory planning method based on the modified Newton method-interior point method, characterized in that: The specific steps are as follows: Step 1: Obtain initial conjectures about the timing of each stage; Request initial conjectures for the phase timings from users of the rocket trajectory planning method. ; requires users to provide Initial conjectures at each stage , The total number of stages, for dimensional vector; Step 2: Construct inner-layer sub-problems at the application stage; Application phase timing The rocket trajectory planning problem is addressed and modified, and an inner relaxation coefficient is introduced. Construct inner subproblems with specific forms; Step 3: Discretize the inner sub-problems using the trapezoidal method; By applying the trapezoidal method, the inner subproblem obtained in step two is discretized to obtain the discrete inner subproblem. Step 4: Apply the interior point method to solve the discrete inner-layer subproblem; By applying the interior-point method to solve the discrete inner-layer subproblem, the optimal rocket thrust direction can be obtained. With inner layer relaxation coefficient The interior point method refers to a solution method for second-order cone programming problems, represented by the primal-dual homogeneous embedded interior point method, which is used to solve discrete inner-level subproblems. Step 5: Calculate gradient information using the finite difference method; repeat Steps two through four apply the finite difference method to calculate the inner layer relaxation coefficient. Relative stage time point gradient information ; Step 6: Apply the modified Newton method to calculate the new stage time point; Based on the inner layer relaxation coefficients obtained in steps two through five gradient information We construct an outer subproblem with a specific form and apply the modified Newton method to calculate the new stage time point; Step 7: Output the rocket trajectory planning results; If the stage time points obtained in step six satisfy a specific convergence condition, then output the rocket thrust direction obtained in step four. Compared with the stage time points obtained in step six ; Otherwise, proceed to step two to continue the calculation; The inner-layer subproblem with a specific form mentioned in step two refers to the processing and transformation of the rocket trajectory planning problem, and the introduction of inner-layer relaxation coefficients. The problem that followed; the specific inner sub-problems are: (1) The discrete inner-layer sub-problem described in step three is specifically as follows: Using the trapezoidal method, the inner subproblem obtained in step two is discretized, and a new subproblem is introduced in each stage. A discrete point; (2) Among them, variables The iteration range is 1 to ,variable The iteration range is 1 to ; The outer subproblem with a specific form described in step six is ​​as follows: (3) in, It is time. It's the rocket's position. It's rocket speed. It's the rocket's mass. It is the initial time. This is the initial position. It is the initial velocity. It is the initial mass. The rocket was in The magnitude of the thrust at each stage, The rocket was in The quality flow rate of the stage; in addition, and It is an arbitrary linear function.

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