CN114955007B - Method and device for fast planning of relative motion time optimal trajectory based on convex programming - Google Patents

Abstract

The application relates to a method, a device, computer equipment and a storage medium for fast planning of a relative motion time optimal track based on convex planning. The method comprises the following steps: establishing a spacecraft relative motion equation under a relative motion coordinate system, and establishing a time optimal track planning problem according to the spacecraft relative motion equation, boundary conditions and constraint conditions; setting a two-norm of the terminal error as an objective function of the minimum terminal error problem, and constructing the minimum terminal error problem according to the boundary condition, the constraint condition and the objective function; performing convex planning on the minimum terminal error problem to obtain a terminal error and a control sequence; constructing performance indexes by using the terminal errors and the control sequences, and converting the time optimal track planning problem into a double-layer optimization problem according to the performance indexes; and solving the double-layer optimization problem by using a hybrid optimization algorithm to obtain the shortest flight time. By adopting the method, the rapid track planning efficiency can be improved.

Description

Method and device for fast planning of relative motion time optimal trajectory based on convex programming

Technical Field

The present application relates to the field of spacecraft guidance and control, and in particular to a method, device, computer equipment and storage medium for rapid planning of a relative motion time optimal trajectory based on convex programming.

Background technique

Relative motion trajectory planning is the premise and basis for a series of space operations such as close-in observation, on-orbit refueling, and formation flying. The goal of time-optimal trajectory planning is to find the shortest flight time trajectory from the initial state to the target state, while satisfying the constraints of dynamics, control, etc. The shortest flight time is of great reference significance for space mission design. If the given flight time is less than the shortest flight time, there is no feasible flight trajectory. For the relative motion trajectory planning problem, it is usually handled based on direct or indirect methods. However, due to the complex nonlinear dynamic characteristics and various constraints of relative motion, the first-order optimality necessary conditions are often difficult to construct, which limits the further application of the indirect method. The direct method discretizes the control space and state space and transforms the optimal control problem into a nonlinear optimization problem. It mainly includes pseudo-spectral method, mixed integer linear programming and quadratic programming algorithm, but these algorithms are sensitive to initial values. In recent years, convex programming algorithms have been widely studied by scholars due to their high computational efficiency, and have been applied to orbit transfer, multi-UAV collaborative planning, atmospheric capture trajectory and spacecraft landing. The literature shows that this method has higher computational efficiency than pseudo-spectral method. However, the objective function of the time optimization problem is non-convex and difficult to solve directly using convex programming algorithms.

However, traditional nonlinear optimization algorithms have low solution efficiency and cannot meet the needs of fast trajectory planning.

Summary of the invention

Based on this, it is necessary to provide a method, device, computer equipment and storage medium for rapid planning of relative motion time optimal trajectory based on convex programming, which can improve the efficiency of rapid trajectory planning, in order to address the above technical problems.

A method for rapid planning of a relative motion time optimal trajectory based on convex programming, the method comprising:

Obtain boundary conditions and constraints in trajectory planning;

Establish the spacecraft relative motion equation in the relative motion coordinate system, and establish the time optimal trajectory planning problem based on the spacecraft relative motion equation, boundary conditions and constraints;

The second norm of the terminal error is set as the objective function of the minimum terminal error problem, and the minimum terminal error problem is constructed according to the boundary conditions, constraints and objective function;

Convex programming is performed on the minimum terminal error problem to obtain the terminal error and control sequence;

The terminal error and control sequence are used to construct performance indicators. According to the performance indicators, the time optimal trajectory planning problem is transformed into a two-layer optimization problem. The inner layer of the two-layer optimization problem is the minimum terminal error problem, and the outer layer is the root-finding problem.

The hybrid optimization algorithm is used to solve the double-layer optimization problem and obtain the shortest flight time.

In one embodiment, a time optimal trajectory planning problem is established based on the spacecraft relative motion equation, boundary conditions and constraints, including:

Discretize the spacecraft relative motion equation to obtain the discrete form relative motion equation in the relative motion coordinate system;

According to the discrete relative motion equations, boundary conditions and constraints, the time optimal trajectory planning problem is established.

In one embodiment, the boundary conditions include initial state constraints and terminal state constraints; the constraint conditions include control saturation constraints; according to the discrete relative motion equations, the boundary conditions and the constraint conditions, a time optimal trajectory planning problem is established, including:

The discrete form relative motion equation is x(i+1)=A _d x(i)+B _d T(i), where i=1,…N, N is the discrete step number, T represents the thrust of the spacecraft, A _d represents the state coefficient matrix, and B _d represents the control coefficient matrix;

The initial state constraints are x(t ₀ ) = x ₀ , m(t ₀ ) = m ₀ , where the spacecraft mass I _sp is the vacuum specific impulse of the thruster, g ₀ is the gravitational acceleration at sea level;

The terminal state constraint is x(t _f ) = x _f . The control saturation constraint is ||T|| ₂ ≤T _max , where T _max represents the maximum thrust of the spacecraft;

According to the discrete relative motion equation, boundary conditions and constraints, the time optimal trajectory planning problem is established as follows: Where _t0 represents the initial time and _tf represents the terminal time.

In one embodiment, the second norm of the terminal error is set as the objective function of the minimum terminal error problem, and the minimum terminal error problem is constructed according to the boundary conditions, the constraint conditions and the objective function, including:

The second norm of the terminal error is set as the objective function of the minimum terminal error problem. According to the boundary conditions, constraints and objective function, the minimum terminal error problem is constructed as min J ₂ =||x(t _f )-x _f || ₂ , where When , the minimum terminal error J ₂ decreases with the increase of t _f . , J ₂ is always zero, x(t _f ) represents the actual terminal state, x _f represents the expected terminal state, Indicates the shortest flight time.

In one embodiment, the performance indicator is constructed using the terminal error and the control sequence, including:

Constructing performance indicators using terminal errors and control sequences Among them When J ₃ (t _f )＞0, when When J ₃ (t _f )＝0, when When J ₃ (t _f ) < 0, T _max represents the maximum thrust of the spacecraft.

In one embodiment, a hybrid optimization algorithm is used to solve the double-layer optimization problem to obtain the shortest flight time, including:

The range of the solution space is narrowed down by using the bisection method until the absolute function values of the upper and lower boundaries are less than the preset values. The calculated results are then used as the initial values of the Newton method to solve the problem and obtain the shortest flight time.

In one embodiment, the range of the solution space is narrowed down by using the dichotomy method until the absolute function values of the upper and lower boundaries are less than the preset values, and then the calculation results are used as the initial values of the Newton method to solve the shortest flight time, including:

Use the binary search method to narrow the scope of the solution space until the absolute function value of the upper and lower boundaries is less than the preset value, and then use the calculation result as the initial value of Newton's method to solve the problem. The steps to obtain the shortest flight time are as follows:

Step S1: Let the upper and lower boundaries t _1b and t _ub of the dichotomy be equal to t _LB and t _UB respectively, solve the double-layer optimization problem with t _1b and t _ub as the flight time respectively, and obtain the indicators J ₃ (t _1b ) and J ₃ (t _ub ) according to the calculation results; t _LB and t _UB represent the upper and lower boundaries of the flight time;

Step S2: If the absolute values of J ₃ (t _1b ) and J ₃ (t _ub ) are greater than the preset reference values Execute step S4. Otherwise, execute step S5;

Step S3: Let t ₀ =(t _1b +t _ub )/2, and use t ₀ as the flight time to calculate the double-layer optimization problem and obtain the index J ₃ (t ₀ ). If J ₃ (t _1b )·J ₃ (t ₀ )<0, let t _1b =t ₀ , otherwise let t _ub =t ₀ , and repeat steps S1 to S3 until the absolute values of J ₃ (t _1b ) and J ₃ (t _ub ) are both less than the preset Let t ₀ = t _1b , t ₁ = t _ub , k = 0; k represents the number of iterations of Newton's method;

Step S4: Determine whether the absolute value of t ₀ -t ₁ is greater than or equal to ε and k<M. If so, execute step S6, otherwise execute step S7; ε represents the algorithm accuracy requirement; M represents the maximum number of iterations;

Step S5: Let t ₂ = t ₁ - _{J 3} (t ₁ )(t ₁ - _{t 0} )/(J ₃ (t ₁ ) - J ₃ (t ₀ )), take t ₂ as the flight time, solve the double-layer optimization problem, and calculate J ₃ (t ₂ );

Step S6: Let t ₀ = t ₁ , J ₃ (t ₀ ) = J ₃ (t ₁ ), t ₁ = t ₂ , J ₃ (t ₁ ) = J ₃ (t ₂ ), k = k + 1, and execute step S5;

Step S7: determine whether the absolute value of t ₀ -t ₁ is less than ε; if so, output t ₁ , otherwise the calculation fails; t ₁ represents the shortest flight time.

A device for rapidly planning a relative motion time optimal trajectory based on convex programming, the device comprising:

The time optimal trajectory planning problem construction module is used to obtain the boundary conditions and constraints in trajectory planning; the relative motion equation of the spacecraft is established in the relative motion coordinate system, and the time optimal trajectory planning problem is established based on the relative motion equation of the spacecraft, boundary conditions and constraints;

A minimum terminal error problem construction module is used to set the second norm of the terminal error as the objective function of the minimum terminal error problem, and to construct the minimum terminal error problem according to boundary conditions, constraints and objective function;

The problem conversion module is used to perform convex programming on the minimum terminal error problem to obtain the terminal error and control sequence; the terminal error and control sequence are used to construct performance indicators, and the time optimal trajectory planning problem is converted into a two-layer optimization problem according to the performance indicators; the inner layer of the two-layer optimization problem is the minimum terminal error problem, and the outer layer is the root-finding problem;

The problem solving module is used to solve the double-layer optimization problem using a hybrid optimization algorithm to obtain the shortest flight time.

A computer device comprises a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the following steps are implemented:

Obtain boundary conditions and constraints in trajectory planning;

Establish the spacecraft relative motion equation in the relative motion coordinate system, and establish the time optimal trajectory planning problem based on the spacecraft relative motion equation, boundary conditions and constraints;

The second norm of the terminal error is set as the objective function of the minimum terminal error problem, and the minimum terminal error problem is constructed according to the boundary conditions, constraints and objective function;

Convex programming is performed on the minimum terminal error problem to obtain the terminal error and control sequence;

The terminal error and control sequence are used to construct performance indicators. According to the performance indicators, the time optimal trajectory planning problem is transformed into a two-layer optimization problem. The inner layer of the two-layer optimization problem is the minimum terminal error problem, and the outer layer is the root-finding problem.

The hybrid optimization algorithm is used to solve the double-layer optimization problem and obtain the shortest flight time.

A computer-readable storage medium stores a computer program, which, when executed by a processor, implements the following steps:

Obtain boundary conditions and constraints in trajectory planning;

Establish the spacecraft relative motion equation in the relative motion coordinate system, and establish the time optimal trajectory planning problem based on the spacecraft relative motion equation, boundary conditions and constraints;

The second norm of the terminal error is set as the objective function of the minimum terminal error problem, and the minimum terminal error problem is constructed according to the boundary conditions, constraints and objective function;

Convex programming is performed on the minimum terminal error problem to obtain the terminal error and control sequence;

The terminal error and control sequence are used to construct performance indicators. According to the performance indicators, the time optimal trajectory planning problem is transformed into a two-layer optimization problem. The inner layer of the two-layer optimization problem is the minimum terminal error problem, and the outer layer is the root-finding problem.

The hybrid optimization algorithm is used to solve the double-layer optimization problem and obtain the shortest flight time.

The above-mentioned relative motion time optimal trajectory rapid planning method, device, computer equipment and storage medium based on convex programming, the present invention converts the time optimal control problem into a two-layer optimization problem, solves the minimum terminal error problem of the inner layer through a mature convex programming algorithm, thereby improving the calculation efficiency, constructs a new performance indicator with the terminal error and the control sequence, converts the shortest time search problem into an easy-to-solve root-finding problem, and then quickly solves the root-finding problem through the hybrid optimization algorithm with higher calculation efficiency and robustness proposed by the present invention to obtain the shortest flight time, thereby greatly improving the trajectory planning efficiency.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a schematic diagram of a process flow of a method for rapid planning of a relative motion time optimal trajectory based on convex programming in one embodiment;

FIG2 is a schematic diagram of a spacecraft relative coordinate system in one embodiment;

FIG3 is a schematic diagram of a law of performance indicators changing over time in one embodiment;

FIG4 is a schematic diagram of a framework of a two-layer optimization problem in another embodiment;

FIG5 is a diagram showing the effect of a simulation experiment of the present invention in one embodiment;

FIG6 is a schematic diagram of a state change curve of a satellite relative transfer trajectory in one embodiment;

FIG7 is a schematic diagram of a thrust component variation curve of a satellite relative transfer trajectory in one embodiment;

FIG8 is a structural block diagram of a device for rapidly planning a relative motion time optimal trajectory based on convex programming in one embodiment;

FIG. 9 is a diagram showing the internal structure of a computer device in one embodiment.

Detailed ways

In order to make the purpose, technical solution and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

In one embodiment, as shown in FIG2 , a method for fast planning of a relative motion time optimal trajectory based on convex programming is provided, comprising the following steps:

Step 102, obtain boundary conditions and constraints in trajectory planning; establish the spacecraft relative motion equation in the relative motion coordinate system, and establish the time optimal trajectory planning problem based on the spacecraft relative motion equation, boundary conditions and constraints.

The boundary conditions include the initial state constraints and terminal state constraints of the spacecraft; the constraint conditions include the control saturation constraint. According to the relative motion equation of the spacecraft, the boundary conditions and the constraint conditions, the time optimal trajectory planning problem can be established.

Step 104, setting the second norm of the terminal error as the objective function of the minimum terminal error problem, and constructing the minimum terminal error problem according to the boundary conditions, constraint conditions and the objective function.

Since the objective function of the time optimal control problem is not in convex form, the present invention takes the terminal error of the spacecraft as the objective function, thereby constructing a minimum error problem that can be transformed into a convex programming problem, and proves the law of change of the optimal terminal error with time and the thrust characteristics of the minimum error problem.

Step 106, convex programming is performed on the minimum terminal error problem to obtain the terminal error and the control sequence; the terminal error and the control sequence are used to construct a performance index, and the time optimal trajectory planning problem is converted into a two-layer optimization problem according to the performance index; the inner layer of the two-layer optimization problem is the minimum terminal error problem, and the outer layer is the root finding problem.

The control sequence of the minimum terminal error problem is solved by the convex programming algorithm. The actual terminal state of the spacecraft can be obtained by substituting the control sequence into the dynamic equation. The terminal error of the spacecraft is obtained by comparing the actual and expected terminal states. The terminal error and the control sequence are used to construct a new performance index, so that the value of the performance index is zero if and only if the given flight time is equal to the shortest flight time. Based on the above analysis, the time optimization problem can be transformed into a two-layer optimization problem, in which the inner layer is the minimum error problem and the outer layer is the root-finding problem.

Step 108, using a hybrid optimization algorithm to solve the double-layer optimization problem to obtain the shortest flight time.

The hybrid optimization algorithm uses the bisection method to narrow the scope of the solution space until the absolute function values of the upper and lower boundaries are less than the preset values, and then uses the calculated results as the initial values of the Newton method for solving the problem. It has the advantages of strong robustness of the bisection method and fast convergence speed of the Newton method. It can quickly solve the root-finding problem and obtain the shortest flight time, thereby improving the efficiency of trajectory planning.

In the above-mentioned rapid planning method for the optimal trajectory of relative motion time based on convex programming, the present invention transforms the time optimal control problem into a two-layer optimization problem, solves the minimum terminal error problem of the inner layer through a mature convex programming algorithm, thereby improving the computational efficiency, constructs a new performance indicator with the terminal error and the control sequence, transforms the shortest time search problem into an easy-to-solve root-finding problem, and then quickly solves the root-finding problem through the hybrid optimization algorithm with higher computational efficiency and robustness proposed by the present invention to obtain the shortest flight time, thereby greatly improving the trajectory planning efficiency.

In one embodiment, a time optimal trajectory planning problem is established based on the spacecraft relative motion equation, boundary conditions and constraints, including:

Discretize the spacecraft relative motion equation to obtain the discrete form relative motion equation in the relative motion coordinate system;

According to the discrete relative motion equations, boundary conditions and constraints, the time optimal trajectory planning problem is established.

In one embodiment, the boundary conditions include initial state constraints and terminal state constraints; the constraint conditions include control saturation constraints; according to the discrete relative motion equations, the boundary conditions and the constraint conditions, a time optimal trajectory planning problem is established, including:

The discrete form relative motion equation is x(i+1)=A _d x(i)+B _d T(i), where i=1,…N, N is the discrete step number, T represents the thrust of the spacecraft, A _d represents the state coefficient matrix, and B _d represents the control coefficient matrix;

The initial state constraints are x(t ₀ ) = x ₀ , m(t ₀ ) = m ₀ , where the spacecraft mass I _sp is the vacuum specific impulse of the thruster, g0 is the gravitational acceleration at sea level;

The terminal state constraint is x(t _f )=x _f , and the control saturation constraint is ||T|| ₂ ≤T _max .

According to the discrete relative motion equation, boundary conditions and constraints, the time optimal trajectory planning problem is established as follows: Where _t0 represents the initial time and _tf represents the terminal time.

The relative motion of spacecraft is usually described in a relative coordinate system, as shown in Figure 2. r _d and r _c represent the position vectors of the tracking spacecraft and the reference spacecraft in the geocentric inertial system. The origin of the relative coordinate system is located at the center of mass of the reference spacecraft, the x-axis is in the same direction as r _c , the z-axis is in the same direction as the orbital momentum, and the y-axis is determined by the right-hand rule.

The dynamic equation of the satellite in the relative coordinate system can be expressed as

Where r = [x, y, z] ^T and v = [v _x , _vy , v _z ] ^T represent the relative position and velocity of the tracking spacecraft, subscripts c and d represent the reference spacecraft and the tracking spacecraft, ω and ε are the instantaneous angular velocity and angular acceleration of the reference spacecraft, T = [T _x ,T _y ,T _z ] ^T and m are the thrust and mass of the spacecraft, and μ is the Earth's gravitational constant. Assuming that the target spacecraft is in a near-circular orbit and its orbital radius is much larger than the relative distance of the spacecraft, equation (1) can be simplified to

The scalar form of equation (2) is

Where n is the average orbital angular velocity of the reference spacecraft. Given a discrete time ΔT, equation (3) can be expressed in discrete form:

x(i+1)＝A _d x(i)+B _d T(i) (4)

Where x = [r; v] is the state vector, i = 1, 2, ... N is the discrete step number, and A _d and B _d can be expressed as

Spacecraft mass changes follow

Where I _sp is the vacuum specific impulse of the thruster, and g ₀ = 9.80665 m/s ² is the gravitational acceleration at sea level.

The relative motion process of the spacecraft needs to satisfy the initial state constraint in equation (8), the terminal state constraint in equation (9), and the control saturation constraint in equation (10).

x(t ₀ )=x ₀ , m(t ₀ )=m ₀ (8)

x(t _f )＝x _f (9)

||T|| ₂ ≤T _max (10) where T _max is the maximum thrust of the spacecraft, t ₀ is the initial time, which is usually set to zero. t _f is the terminal time to be determined, x ₀ , m ₀ and x _f are the given initial state, initial mass and expected terminal state of the spacecraft, respectively. Therefore, the time optimal trajectory planning problem can be summarized as

Satisfy constraints (4), (7) to (10).

When t<t _f , the optimal thrust amplitude of the time optimal trajectory planning problem is always ||T ^* || ₂ =T _max , and ||T ^* || ₂ ≤T _max can only be found when t=t _f and the terminal co-state value of the velocity λ _v (t _f )=0.

In one embodiment, the second norm of the terminal error is set as the objective function of the minimum terminal error problem, and the minimum terminal error problem is constructed according to the boundary conditions, the constraint conditions and the objective function, including:

The second norm of the terminal error is set as the objective function of the minimum terminal error problem. According to the boundary conditions, constraints and objective function, the minimum terminal error problem is constructed as min J ₂ =||x(t _f )-x _f || ₂ , where When , the minimum terminal error _J2 decreases with the increase of _tf . , J ₂ is always zero, x(t _f ) represents the actual terminal state, x _f represents the expected terminal state, Indicates the shortest flight time.

The objective function of the time optimal trajectory planning problem is non-convex and difficult to solve directly using a convex programming algorithm. The present invention takes the terminal error between the actual state and the desired state as the target, constructs the minimum terminal error problem under a given flight time _tf , and satisfies constraints (4), (7) to (10).

It is worth noting that the objective function and inequality constraints of the minimum terminal error problem are convex functions, and the equality constraints can be expressed in the form of affine functions, so they can be transformed into a convex programming problem.

At the same time, when When , the minimum terminal error _J2 decreases with the increase of _tf . When , J ₂ is always zero, so the shortest flight time is the inflection point where J ₂ = 0 holds. When , the amplitude of the optimal thrust for the minimum terminal error problem is always the maximum value.

In one embodiment, the performance indicator is constructed using the terminal error and the control sequence, including:

Constructing performance indicators using terminal errors and control sequences Among them When J ₃ (t _f )＞0, when When J ₃ (t _f )＝0, when When J ₃ (t _f ) < 0, T _max represents the maximum thrust of the spacecraft, as shown in FIG3 .

It is worth noting that J ₂ represents the optimization goal of convex programming, while J ₃ is just a performance indicator constructed based on thrust and terminal error, which is used to search for the shortest flight time. Through the above analysis, the initial time optimization problem is transformed into a two-layer optimization problem. The inner layer is a convex programming problem, that is, the minimum terminal error problem for a given flight time. The outer layer is a root-finding problem. The flight time t _f corresponding to the performance indicator J ₃ (t _f ) = 0 is the shortest flight time. As shown in Figure 4.

In one embodiment, a hybrid optimization algorithm is used to solve the double-layer optimization problem to obtain the shortest flight time, including:

The range of the solution space is narrowed down by using the bisection method until the absolute function values of the upper and lower boundaries are less than the preset values. The calculated results are then used as the initial values of the Newton method to solve the problem and obtain the shortest flight time.

In one embodiment, the range of the solution space is narrowed down by using the dichotomy method until the absolute function values of the upper and lower boundaries are less than the preset values, and then the calculation results are used as the initial values of the Newton method to solve the shortest flight time, including:

Use the binary search method to narrow the scope of the solution space until the absolute function value of the upper and lower boundaries is less than the preset value, and then use the calculation result as the initial value of Newton's method to solve the problem. The steps to obtain the shortest flight time are as follows:

Step S1: Let the upper and lower boundaries t _1b and t _ub of the dichotomy be equal to t _LB and t _UB respectively, solve the double-layer optimization problem with t _1b and t _ub as the flight time respectively, and obtain the indicators J ₃ (t _1b ) and J ₃ (t _ub ) according to the calculation results; t _LB and t _UB represent the upper and lower boundaries of the flight time;

Step S2: If the absolute values of J ₃ (t _1b ) and J ₃ (t _ub ) are greater than the preset reference values Execute step S4. Otherwise, execute step S5;

Step S3: Let t ₀ =(t _1b +t _ub )/2, and use t ₀ as the flight time to calculate the double-layer optimization problem and obtain the index J ₃ (t ₀ ). If J ₃ (t _1b )·J ₃ (t ₀ )<0, let t _1b =t ₀ , otherwise let t _ub =t ₀ , and repeat steps S1 to S3 until the absolute values of J ₃ (t _1b ) and J ₃ (t _ub ) are both less than the preset Let t ₀ = t _1b , t ₁ = t _ub , k = 0; k represents the number of iterations of Newton's method;

Step S4: Determine whether the absolute value of t ₀ -t ₁ is greater than or equal to ε and k<M. If so, execute step S6, otherwise execute step S7; ε represents the algorithm accuracy requirement; M represents the maximum number of iterations;

Step S5: Let t ₂ = t ₁ - _{J 3} (t ₁ )(t ₁ - _{t 0} )/(J ₃ (t ₁ ) - J ₃ (t ₀ )), take t ₂ as the flight time, solve the double-layer optimization problem, and calculate J ₃ (t ₂ );

Step S6: Let t ₀ = t ₁ , J ₃ (t ₀ ) = J ₃ (t ₁ ), t ₁ = t ₂ , J ₃ (t ₁ ) = J ₃ (t ₂ ), k = k + 1, and execute step S5;

Step S7: determine whether the absolute value of t ₀ -t ₁ is less than ε; if so, output t ₁ , otherwise the calculation fails; t ₁ represents the shortest flight time.

Aiming at the outer root-finding problem in the two-layer optimization framework, the present invention combines the advantages of Newton's method and bisection method to propose a hybrid optimization algorithm. This method first uses bisection method to narrow the scope of the solution space until the absolute function value of the upper and lower boundaries is less than the preset value, and then uses the calculation result as the initial value of Newton's method for solution. By providing high-quality initial values for Newton's method through bisection method, the hybrid optimization algorithm has the advantages of strong robustness of bisection method and fast convergence speed of Newton's method, and can accurately and quickly calculate the shortest flight time, thereby improving the efficiency of trajectory planning.

In one embodiment, the performance of the present invention is verified by numerical simulation.

The reference spacecraft is located in a near-circular orbit at an altitude of 500 km. The initial mass of the satellite is 1000 kg. The maximum thrust of the engine is 50 N, the maximum I _sp is 200 s, and the simulation process is discretized into 100 steps. The upper and lower boundary values of the flight time are 100 s and 3000 s, respectively. The convex optimization problem is solved based on the Convex Optimization Toolbox CVX, and SDPT3 is selected as the solver. The simulation computer is equipped with a 3.0GHz Intel Core i7 processor and 16Gb cache.

Assuming that the satellite maneuvers from an initial elliptical orbit to another elliptical orbit relative to the earth, without considering the influence of various perturbation factors, the satellite can fly around the target without consuming additional fuel.

The initial and final states of the satellite are

X ₀ =[1×10 ³ m,1×10 ⁴ m,0m,0m/s,-2.21m/s,2.21m/s] ^T ;

X _f =[866.03m,-1×10 ³ m,0m,-0.55m/s,-1.92m/s,0m/s] ^T ;

First, to verify the correctness of the theoretical derivation above, the minimum terminal error problem is solved for a given flight time. Sampling is performed every 10 seconds between the upper and lower boundaries of the time. The simulation results are shown in Figure 5, where J ₃₁ is the terminal error, J ₃₂ is used to measure the difference between the thrust curve and the maximum thrust, and J ₃ is the sum of J ₃₁ and J _32. As shown in Figure 5, when When the optimal terminal error J ₃₁ is greater than zero and decreases with time, When , the optimal terminal error J ₃₁ is equal to zero. When the thrust amplitude remains at T _max , When the thrust amplitude is less than T _max , the index J ₃₂ is therefore zero in the first part but not zero in the second part.

Then, the time optimal trajectory planning problem is solved using the double-layer optimization method proposed in the present invention, where the state and thrust change curves are shown in Figures 6 and 7. As can be seen from Figure 6, the method proposed in the present invention can successfully generate the satellite relative transfer trajectory. As shown in Figure 7, although the thrust component changes with time, the thrust magnitude remains at T _max .

In order to evaluate the performance of the proposed hybrid optimization algorithm, the traditional binary method, Newton method and sequential quadratic programming method are used to solve the time optimal problem with the same configuration. The SQP method can be implemented by directly calling the nonlinear programming function Fmincon in the Matlab optimization toolbox. It is worth noting that the first three methods are based on convex programming to optimize the control sequence, which is only used to search for the shortest flight time, while the SQP method directly optimizes the control sequence and the minimum flight time at the same time. Since the computational efficiency and convergence speed of the SQP method and the traditional Newton method are greatly affected by the initial value, both methods are executed 10 times with random initial values. Table 1 lists the performance index comparison of different methods.

Table 1 Performance comparison of different algorithms

It can be directly seen from Table 1 that the minimum flight time obtained by the above four methods is basically the same, which verifies the effectiveness of the hybrid method in this paper. For the calculation time criteria in Table 1, although it will be affected by various factors such as computer configuration and program integration, the calculation time of the first three methods is about two orders of magnitude less than that of the SQP method, proving that the convex programming method is absolutely superior to the nonlinear optimization method in terms of computational efficiency. The time spent by the binary method is about twice that of the hybrid method, while the time spent by the Newton method is slightly less than that of the hybrid method. It should be added that the traditional Newton method can only converge half of the 10 repeated calculations. Therefore, the hybrid optimization algorithm proposed in this paper is superior in both computational efficiency and robustness.

It should be understood that, although the various steps in the flowchart of FIG. 1 are shown in sequence according to the indication of the arrows, these steps are not necessarily executed in sequence according to the order indicated by the arrows. Unless there is a clear explanation in this article, the execution of these steps is not strictly limited in order, and these steps can be executed in other orders. Moreover, at least a portion of the steps in FIG. 1 may include a plurality of sub-steps or a plurality of stages, and these sub-steps or stages are not necessarily executed at the same time, but can be executed at different times, and the execution order of these sub-steps or stages is not necessarily to be carried out in sequence, but can be executed in turn or alternately with other steps or at least a portion of the sub-steps or stages of other steps.

In one embodiment, as shown in FIG8 , a device for rapid planning of a relative motion time optimal trajectory based on convex programming is provided, comprising: a time optimal trajectory planning problem construction module 802, a minimum terminal error problem construction module 804, a problem conversion module 806 and a problem solving module 808, wherein:

The time optimal trajectory planning problem construction module 802 is used to obtain boundary conditions and constraints in trajectory planning; establish the spacecraft relative motion equation in the relative motion coordinate system, and establish the time optimal trajectory planning problem according to the spacecraft relative motion equation, boundary conditions and constraints;

A minimum terminal error problem constructing module 804 is used to set the second norm of the terminal error as the objective function of the minimum terminal error problem, and construct the minimum terminal error problem according to the boundary conditions, the constraint conditions and the objective function;

The problem conversion module 806 is used to perform convex programming on the minimum terminal error problem to obtain the terminal error and the control sequence; the performance index is constructed using the terminal error and the control sequence, and the time optimal trajectory planning problem is converted into a two-layer optimization problem according to the performance index; the inner layer of the two-layer optimization problem is the minimum terminal error problem, and the outer layer is the root-finding problem;

The problem solving module 808 is used to solve the double-layer optimization problem using a hybrid optimization algorithm to obtain the shortest flight time.

In one embodiment, the time optimal trajectory planning problem building module 802 is further used to establish the time optimal trajectory planning problem according to the spacecraft relative motion equation, boundary conditions and constraints, including:

Discretize the spacecraft relative motion equation to obtain the discrete form relative motion equation in the relative motion coordinate system;

According to the discrete relative motion equations, boundary conditions and constraints, the time optimal trajectory planning problem is established.

In one embodiment, the boundary conditions include initial state constraints and terminal state constraints; the constraint conditions include control saturation constraints; the time optimal trajectory planning problem building module 802 is also used to establish the time optimal trajectory planning problem according to the discrete form relative motion equation, the boundary conditions and the constraint conditions, including:

The relative motion equation in discrete form is x(i+1)=A _d x(i)+B _d T(i), where i represents the discrete step number, T represents the thrust of the spacecraft, A _d represents the state coefficient matrix, and B _d represents the control coefficient matrix;

The initial state constraints are x(t ₀ ) = x ₀ , m(t ₀ ) = m ₀ , where the spacecraft mass I _sp is the vacuum specific impulse of the thruster, g0 is the gravitational acceleration at sea level;

The terminal state constraint is x(t _f ) = x _f . The control saturation constraint is ||T|| ₂ ≤T _max , where T _max represents the maximum thrust of the spacecraft;

According to the discrete relative motion equation, boundary conditions and constraints, the time optimal trajectory planning problem is established as follows: Where _t0 represents the initial time and _tf represents the terminal time.

In one embodiment, the minimum terminal error problem constructing module 804 is further used to set the second norm of the terminal error as the objective function of the minimum terminal error problem, and construct the minimum terminal error problem according to the boundary conditions, the constraint conditions and the objective function, including:

The second norm of the terminal error is set as the objective function of the minimum terminal error problem. According to the boundary conditions, constraints and objective function, the minimum terminal error problem is constructed as min J ₂ =||x(t _f )-x _f || ₂ , where When , the minimum terminal error _J2 decreases with the increase of _tf . , J ₂ is always zero, x(t _f ) represents the actual terminal state, and x _f represents the expected terminal state.

In one embodiment, the problem conversion module 806 is further used to construct a performance indicator using the terminal error and the control sequence, including:

Constructing performance indicators using terminal errors and control sequences Among them When J ₃ (t _f )＞0, when When J ₃ (t _f )＝0, when When J ₃ (t _f )＜0.

In one embodiment, the problem solving module 808 is further used to solve the double-layer optimization problem using a hybrid optimization algorithm to obtain the shortest flight time, including:

The range of the solution space is narrowed down by using the bisection method until the absolute function values of the upper and lower boundaries are less than the preset values. The calculated results are then used as the initial values of the Newton method to solve the problem and obtain the shortest flight time.

In one embodiment, the problem solving module 808 is further used to narrow the scope of the solution space by using the dichotomy method until the absolute function value of the upper and lower boundaries is less than a preset value, and then the calculation result is used as the initial value of the Newton method to solve and obtain the shortest flight time, including:

Use the binary search method to narrow the scope of the solution space until the absolute function value of the upper and lower boundaries is less than the preset value, and then use the calculation result as the initial value of Newton's method to solve the problem. The steps to obtain the shortest flight time are as follows:

Step S1: Let the upper and lower boundaries t _1b and t _ub of the dichotomy be equal to t _LB and t _UB respectively, solve the double-layer optimization problem with t _1b and t _ub as the flight time respectively, and obtain the indicators J ₃ (t _1b ) and J ₃ (t _ub ) according to the calculation results; t _LB and t _UB represent the upper and lower boundaries of the flight time;

Step S2: If the absolute values of J ₃ (t _1b ) and J ₃ (t _ub ) are greater than the preset Execute step S4. Otherwise, execute step S5;

Step S3: Let t ₀ =(t _1b +t _ub )/2, and use t ₀ as the flight time to calculate the double-layer optimization problem and obtain the index J ₃ (t ₀ ). If J ₃ (t _1b )·J ₃ (t ₀ )<0, let t _1b =t ₀ , otherwise let t _ub =t ₀ , and repeat steps S1 to S3 until the absolute values of J ₃ (t _1b ) and J ₃ (t _ub ) are both less than the preset Let t ₀ = t _1b , t ₁ = t _ub , k = 0; k represents the number of iterations of Newton's method;

Step S4: Determine whether the absolute value of t ₀ -t ₁ is greater than or equal to ε and k<M. If so, execute step S6, otherwise execute step S7; ε represents the algorithm accuracy requirement; M represents the maximum number of iterations;

Step S5: Let t ₂ = t ₁ - _{J 3} (t ₁ )(t ₁ - _{t 0} )/(J ₃ (t ₁ ) - J ₃ (t ₀ )), take t ₂ as the flight time, solve the double-layer optimization problem, and calculate J ₃ (t ₂ );

Step S6: Let t ₀ = t ₁ , J ₃ (t ₀ ) = J ₃ (t ₁ ), t ₁ = t ₂ , J ₃ (t ₁ ) = J ₃ (t ₂ ), k = k + 1, and execute step S5;

Step S7: determine whether the absolute value of t ₀ -t ₁ is less than ε; if so, output t ₁ , otherwise the calculation fails; t ₁ represents the shortest flight time.

For the specific definition of the rapid planning device for the optimal trajectory of relative motion time based on convex programming, please refer to the definition of the rapid planning method for the optimal trajectory of relative motion time based on convex programming above, which will not be repeated here. Each module in the above-mentioned rapid planning device for the optimal trajectory of relative motion time based on convex programming can be implemented in whole or in part by software, hardware and a combination thereof. The above-mentioned modules can be embedded in or independent of the processor in the computer device in the form of hardware, or can be stored in the memory in the computer device in the form of software, so that the processor can call and execute the operations corresponding to the above modules.

In one embodiment, a computer device is provided, which may be a terminal, and its internal structure diagram may be shown in FIG9. The computer device includes a processor, a memory, a network interface, a display screen, and an input device connected via a system bus. Among them, the processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The network interface of the computer device is used to communicate with an external terminal via a network connection. When the computer program is executed by the processor, a relative motion time optimal trajectory rapid planning method based on convex programming is implemented. The display screen of the computer device may be a liquid crystal display screen or an electronic ink display screen, and the input device of the computer device may be a touch layer covered on the display screen, or a key, trackball or touchpad provided on the computer device housing, or an external keyboard, touchpad or mouse, etc.

Those skilled in the art will understand that the structure shown in FIG. 9 is merely a block diagram of a partial structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than shown in the figure, or combine certain components, or have a different arrangement of components.

In one embodiment, a computer device is provided, including a memory and a processor, wherein the memory stores a computer program, and the processor implements the steps of the method in the above embodiment when executing the computer program.

In one embodiment, a computer storage medium is provided, on which a computer program is stored. When the computer program is executed by a processor, the steps of the method in the above embodiment are implemented.

Those skilled in the art can understand that all or part of the processes in the above-mentioned embodiment methods can be completed by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to memory, storage, database or other media used in the embodiments provided in this application can include non-volatile and/or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM) or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. As an illustration and not limitation, RAM is available in many forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link (Synchlink) DRAM (SLDRAM), memory bus (Rambus) direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM).

The technical features of the above embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

The above-mentioned embodiments only express several implementation methods of the present application, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the scope of the invention patent. It should be pointed out that, for a person of ordinary skill in the art, several variations and improvements can be made without departing from the concept of the present application, and these all belong to the protection scope of the present application. Therefore, the protection scope of the patent of the present application shall be subject to the attached claims.

Claims (6)

1. The method for rapidly planning the relative movement time optimal track based on convex planning is characterized by comprising the following steps:

acquiring boundary conditions and constraint conditions in track planning;

Establishing a spacecraft relative motion equation under a relative motion coordinate system, and establishing a time optimal track planning problem according to the spacecraft relative motion equation, the boundary condition and the constraint condition;

Setting a two-norm of a terminal error as an objective function of a minimum terminal error problem, and constructing the minimum terminal error problem according to the boundary condition, the constraint condition and the objective function;

Performing convex planning on the minimum terminal error problem to obtain a terminal error and a control sequence;

Constructing a performance index by utilizing the terminal error and the control sequence, and converting the time optimal track planning problem into a double-layer optimization problem according to the performance index; the inner layer of the double-layer optimization problem is the minimum terminal error problem, and the outer layer is the root finding problem;

solving the double-layer optimization problem by utilizing a hybrid optimization algorithm to obtain the shortest flight time;

According to the spacecraft relative motion equation, the boundary condition and the constraint condition, establishing a time optimal track planning problem, wherein the method comprises the following steps:

dispersing the spacecraft relative motion equation to obtain a discrete form relative motion equation in a relative motion coordinate system;

establishing a time optimal track planning problem according to the discrete form relative motion equation, the boundary condition and the constraint condition;

the boundary conditions comprise initial state constraints and terminal state constraints; the constraint conditions include controlling saturation constraints; according to the discrete form relative motion equation, the boundary condition and the constraint condition, establishing a time optimal track planning problem, wherein the method comprises the following steps:

The discrete form relative motion equation is x (i+1) =a _dx(i)+B_d T (i), wherein i=1, … N, N is the discrete step number, T represents the thrust of the spacecraft, a _d represents the state coefficient matrix, and B _d represents the control coefficient matrix;

The initial state constraint is x (t ₀)＝x₀,m(t₀)＝m₀, wherein spacecraft mass I _sp is vacuum specific impulse of the propeller, g ₀ is sea level gravity acceleration;

The terminal state constraint is x (T _f)＝x_f; the control saturation constraint is ||t| ₂≤T_max, wherein T _max represents the maximum thrust of the spacecraft;

According to the discrete form relative motion equation, the boundary condition and the constraint condition, establishing a time optimal track planning problem as Wherein t ₀ represents an initial time, and t _f represents a terminal time;

Setting the two norms of the terminal error as an objective function of the minimum terminal error problem, and constructing the minimum terminal error problem according to the boundary condition, the constraint condition and the objective function, wherein the method comprises the following steps:

Setting the two norms of the terminal error as an objective function of the minimum terminal error problem, and constructing the minimum terminal error problem as min J ₂＝||x(t_f)-x_f||₂ according to the boundary condition, the constraint condition and the objective function when At this point, the minimum termination error J ₂ decreases with increasing t _f, whenWhen J ₂ is always zero, x (t _f) represents the actual terminal state, x _f represents the desired terminal state,Representing the shortest time of flight;

Constructing a performance index using the terminal error and the control sequence, comprising:

Constructing performance index using the terminal error and control sequence Wherein whenJ ₃(t_f) is greater than 0 whenWhen J ₃(t_f) =0, whenTime J ₃(t_f)＜0,T_max represents the maximum thrust of the spacecraft.

2. The method of claim 1, wherein solving the bi-layer optimization problem using a hybrid optimization algorithm results in a minimum time of flight comprising:

And (3) reducing the range of the solution space by using a dichotomy until the absolute function value of the upper boundary and the lower boundary is smaller than a preset value, and solving the calculated result as an initial value of the Newton method to obtain the shortest flight time.

3. The method of claim 2, wherein the narrowing of the solution space by the dichotomy until the absolute function value of the upper and lower boundaries is smaller than a preset value, and solving the calculation result as an initial value of newton's method to obtain the shortest flight time comprises:

and (3) reducing the range of the solution space by using a dichotomy until the absolute function value of the upper boundary and the lower boundary is smaller than a preset value, and solving the calculated result as an initial value of the Newton method to obtain the shortest flight time, wherein the steps are as follows:

Step S1: making the upper and lower boundaries t _1b and t _ub of the dichotomy equal to t _LB and t _UB respectively, solving a double-layer optimization problem by taking t _1b and t _ub as flight time respectively, and obtaining indexes J ₃(t_1b) and J ₃(t_ub according to a calculation result; the t _LB and t _UB represent the upper and lower boundaries of time of flight;

Step S2: if the absolute values of J ₃(t_1b) and J ₃(t_ub) are greater than the preset reference value Executing the step S4, if not, executing the step S5;

Step S3: let t ₀＝(t_1b+t_ub)/2 and calculate double-layer optimization problem with t ₀ as flight time to obtain index J ₃(t₀), if J ₃(t_1b)·J₃(t₀) < 0, let t _1b＝t₀, otherwise let t _ub＝t₀, repeat steps S1 to S3 until absolute values of J ₃(t_1b) and J ₃(t_ub) are smaller than preset Let t ₀＝t_1b,t₁＝t_ub, k=0; k represents the iteration number of newton's method;

Step S4: judging whether the absolute value of t ₀-t₁ is larger than or equal to epsilon and k is smaller than M, if yes, executing step S6, otherwise, executing step S7; the epsilon represents the algorithm precision requirement; the M represents the maximum iteration number;

Step S5: let t₂＝t₁-J₃(t₁)(t₁-t₀)/(J₃(t₁)-J₃(t₀)), solve the double-layer optimization problem with t ₂ as the flight time, calculate J ₃(t₂);

step S6: causing t₀＝t₁,J₃(t₀)＝J₃(t₁),t₁＝t₂,J₃(t₁)＝J₃(t₂),k＝k+1, to perform step S5;

Step S7: judging whether the absolute value of t ₀-t₁ is smaller than epsilon; if yes, outputting t ₁, otherwise, failing to calculate; the t ₁ represents the shortest time of flight.

4. A convex programming-based rapid planning apparatus for a relative movement time optimal trajectory, for implementing the convex programming-based rapid planning method for a relative movement time optimal trajectory according to claim 1, characterized in that the apparatus comprises:

The time optimal track planning problem construction module is used for acquiring boundary conditions and constraint conditions in track planning; establishing a spacecraft relative motion equation under a relative motion coordinate system, and establishing a time optimal track planning problem according to the spacecraft relative motion equation, the boundary condition and the constraint condition, wherein the method comprises the following steps: dispersing the spacecraft relative motion equation to obtain a discrete form relative motion equation in a relative motion coordinate system; establishing a time optimal track planning problem according to the discrete form relative motion equation, the boundary condition and the constraint condition, wherein the boundary condition comprises an initial state constraint and a terminal state constraint; the constraint conditions include controlling saturation constraints; according to the discrete form relative motion equation, the boundary condition and the constraint condition, establishing a time optimal track planning problem, wherein the method comprises the following steps: the discrete form relative motion equation is x (i+1) =a _dx(i)+B_d T (i), wherein i=1, … N, N is the discrete step number, T represents the thrust of the spacecraft, a _d represents the state coefficient matrix, and B _d represents the control coefficient matrix;

The initial state constraint is x (t ₀)＝x₀,m(t₀)＝m₀, wherein spacecraft mass I _sp is vacuum specific impulse of the propeller, g ₀ is sea level gravity acceleration;

The terminal state constraint is x (T _f)＝x_f; the control saturation constraint is ||t| ₂≤T_max, wherein T _max represents the maximum thrust of the spacecraft;

According to the discrete form relative motion equation, the boundary condition and the constraint condition, establishing a time optimal track planning problem as Wherein t ₀ represents an initial time, and t _f represents a terminal time;

the minimum terminal error problem constructing module is configured to set a two-norm of a terminal error as an objective function of a minimum terminal error problem, and construct the minimum terminal error problem according to the boundary condition, the constraint condition and the objective function, and includes:

Setting the two norms of the terminal error as an objective function of the minimum terminal error problem, and constructing the minimum terminal error problem as min J ₂＝||x(t_f)-x_f||₂ according to the boundary condition, the constraint condition and the objective function when At this point, the minimum termination error J ₂ decreases with increasing t _f, whenWhen J ₂ is always zero, x (t _f) represents the actual terminal state, x _f represents the desired terminal state,Representing the shortest time of flight;

the problem conversion module is used for carrying out convex planning on the minimum terminal error problem to obtain a terminal error and a control sequence; constructing a performance index by utilizing the terminal error and the control sequence, and converting the time optimal track planning problem into a double-layer optimization problem according to the performance index; the inner layer of the double-layer optimization problem is the minimum terminal error problem, and the outer layer is the root finding problem; constructing a performance index using the terminal error and the control sequence, comprising:

Constructing performance index using the terminal error and control sequence Wherein whenJ ₃(t_f) is greater than 0 whenWhen J ₃(t_f) =0, whenTime J ₃(t_f)＜0,T_max represents the maximum thrust of the spacecraft;

And the problem solving module is used for solving the double-layer optimization problem by utilizing a hybrid optimization algorithm to obtain the shortest flight time.

5. A computer device comprising a memory and a processor, the memory storing a computer program, characterized in that the processor implements the steps of the method of any of claims 1 to 3 when the computer program is executed.

6. A computer readable storage medium, on which a computer program is stored, characterized in that the computer program, when being executed by a processor, implements the steps of the method of any of claims 1 to 3.