Method for realizing optimal collision avoidance trajectory of fuel of low-thrust spacecraft in earth-moon space

Abstract

The invention relates to an earth-moon space low-thrust spacecraft fuel optimal collision avoidance trajectory implementation method, which comprises the following steps of: constructing a kinetic model based on calculation parameters of a spacecraft collision avoidance problem, and establishing a fuel optimal performance index of the spacecraft collision avoidance problem through a homotopy method; the optimal control problem is converted into a two-point boundary value problem, targeting solving is carried out, and the state and co-state parameters of the optimal orbit are obtained; and according to the obtained state and co-state parameters of the optimal orbit, solving to obtain collision avoidance optimal control and collision probability of the spacecraft. According to the method, the fuel optimal control problem is solved by adopting the indirect method, the convergence difficulty of the low-thrust fuel optimal problem is reduced through the homotopy method, and when the thrust is calculated, the calculation efficiency is high, the precision is high, and the method is suitable for the fuel optimal collision avoidance scene of the low-thrust spacecraft for a single target in the earth-moon space.

Description

Technical Field

[0001] This invention relates to a technology in the field of aerospace control, specifically a method for achieving optimal fuel collision avoidance trajectory for low-thrust spacecraft in the Earth-Moon space. Background Technology

[0002] Collision avoidance trajectory design for low-thrust spacecraft is a key technology to ensure the on-orbit safety of low-thrust spacecraft. However, existing pulse collision avoidance strategies cannot be directly applied to low-thrust spacecraft. The convex optimization methods involved cannot solve the two-point boundary value problem or directly solve the restricted three-body problem, and cannot achieve effective collision avoidance. Summary of the Invention

[0003] This invention addresses the problem that existing technologies struggle to quickly solve the optimal control problem for low-thrust collision avoidance in a short time. It proposes a method for achieving fuel-optimal collision avoidance trajectories for low-thrust spacecraft in the Earth-Moon space. The method employs an indirect approach to solve the fuel-optimal control problem and reduces the convergence difficulty of the low-thrust fuel-optimal problem through homotopy. It boasts high computational efficiency and accuracy and is suitable for single-target fuel-optimal collision avoidance scenarios for low-thrust spacecraft in the Earth-Moon space.

[0004] This invention is achieved through the following technical solution:

[0005] This invention relates to a method for achieving optimal fuel collision avoidance trajectory for a low-thrust spacecraft in the Earth-Moon space. Based on the computational parameters of the spacecraft collision avoidance problem, a dynamic model is constructed. Then, the optimal fuel performance index for the spacecraft collision avoidance problem is established using the homotopy method. The optimal control problem is transformed into a two-point boundary value problem and solved by target practice to obtain the state and co-state parameters of the optimal trajectory. Finally, based on the obtained state and co-state parameters of the optimal trajectory, the optimal collision avoidance control and collision probability of the spacecraft are obtained.

[0006] The aforementioned dynamic model includes: a restricted three-body dynamic model of the collision-risk target in the Earth-Moon rendezvous coordinate system and a dynamic model of the spacecraft in the same coordinate system, specifically: Where: r = (x, y, z) are the position coordinates of the spacecraft or space debris in the fusion coordinate system, v = (v x ,v y ,v z T represents the velocity of the spacecraft or space debris in the rendezvous coordinate system. max The maximum thrust of the thruster is given by α, u is the engine operating amplitude (valued at [0,1]), α is the propulsion direction, m is the mass of the spacecraft, and c = I. sp g0, I sp Let g be the specific impulse of the propellant, and g0 be the gravitational acceleration at sea level at the equator.

[0007] In the described dynamic model, the two central celestial bodies are the Earth and the Moon. In the synoptic coordinate system, the coordinates of the Earth's center of mass are (1-μ,0,0), and the coordinates of the Moon's center of mass are (μ,0,0). r1 and r2 represent the distances from the spacecraft to the Earth's center of mass in the synoptic coordinate system, respectively. Distance from the Moon's center of mass

[0008] The optimal fuel performance index for spacecraft collision avoidance refers to: Where: P C The final collision probability is T; T is the nominal thrust, which is usually taken as the maximum thrust T under steady-state conditions. max λ0 is the normalization coefficient, which provides better initial value guessing during the target shooting solution process; ε is the homotopy parameter. As ε changes from 1 to 0 according to the homotopy step size, the performance index gradually changes from energy optimal to fuel optimal homotopy; W is the weighting parameter. The larger the value of W, the more the performance is biased towards minimizing the collision probability.

[0009] The homotopy step size is set in any of the following ways:

[0010] A) Subtract a fixed value from the homotopy parameter of the previous round, specifically: ε i =ε i-1 -Δε, where: ε i Let ε be the homotopy parameter for the i-th homotopy. i-1 Let be the homotopy parameter for the (i-1)th homotopy, and Δε be the arithmetic homotopy step size.

[0011] Preferably, if the homotopy calculation fails to converge in the last step, it indicates that the step size is too large. The homotopy step size needs to be reduced, and the process should return to the previous step to perform the homotopy calculation again, until the condition ε is met. i -ε i-1 When the value is less than or equal to the acceptable error, homotopy is considered complete.

[0012] B) Obtain a new homotopy parameter by proportionally reducing the homotopy parameter, specifically: ε i =k Δε ε i-1 , where: k Δε The step size is a geometric homotopy step size.

[0013] The target shooting solution specifically includes:

[0014] 1) Treat the unknown parameters in the boundary conditions as variables to be solved to construct an initial value problem: Since multiplying the performance index by a positive number does not change the essence of the problem, and this value can be any positive number, we can arbitrarily choose a positive value of λ0 to satisfy the subsequent normalization conditions. We normalize the 7 initial costate values ​​obtained by random number guessing, namely the position coordinate costates in 3 directions, the velocity coordinate costates in 3 directions, and the mass costate, as well as the positive value of λ0, so that these 8 initial costate values ​​satisfy the modulus of 1. At this time, the vector endpoints corresponding to these 8 initial costate values ​​are all located on a unit sphere, which greatly reduces the difficulty of initial value guessing and thus greatly improves the target shooting efficiency.

[0015] 2) Solve the initial value problem using numerical integration to obtain the final state;

[0016] 3) The residual function measures the difference between the terminal state and the target boundary conditions. Specifically, the residual function is: a(t) f )=f(t f )-f bc ≤10 -7 Among them, a(t) f f(t) represents the residual of the terminal state variable. f ) represents the terminal state variable, f bc These are the boundary conditions for the corresponding state variables.

[0017] 4) Use an iterative algorithm to adjust the unknown parameters in the boundary conditions so that the residual function approaches zero, thereby satisfying the boundary conditions. Specifically, start the target solution from ε1 and gradually converge until ε reaches 0. If the convergence condition is not met, improve the convergence rate by reducing the step size. Repeat steps 2-4 until all states and co-state parameters of the spacecraft collision avoidance process are output.

[0018] The collision avoidance optimal control and collision probability of the spacecraft are derived by the relationship between optimal control and costate variables, and the magnitude and direction of optimal control are calculated based on the terminal state. Technical effect

[0019] This invention solves the fuel-optimal collision avoidance problem for low-thrust spacecraft by combining the homotopy method with an indirect method. It addresses the convergence difficulty caused by discontinuous bang-bang control in the fuel-optimal solution and introduces collision avoidance constraints based on collision probability, thus solving the problem of difficult monitoring and avoidance of collisions in the Earth-Moon restricted three-body environment. Compared with existing technologies, this invention achieves high-precision low-thrust collision avoidance based on uncertain collision probability in the Earth-Moon environment, resulting in more accurate fuel-optimal collision avoidance trajectories for low-thrust spacecraft. Attached Figure Description

[0020] Figure 1 This is a flowchart of the present invention;

[0021] Figure 2 A schematic diagram of the coordinate system calculated at the instant of the collision;

[0022] Figure 3 To optimize the collision avoidance maneuver trajectory for fuel and deviate from the spacecraft's original trajectory;

[0023] Figure 4 A magnified view of the area near the potential collision location between the fuel-optimal collision avoidance maneuver trajectory and the spacecraft's original trajectory;

[0024] Figure 5 A curve showing the thrust magnitude for optimal fuel maneuvering;

[0025] Figure 6 Fuel consumption curve for optimal fuel efficiency. Detailed Implementation

[0026] like Figure 1 As shown in the figure, this embodiment relates to a method for achieving an optimal fuel collision avoidance trajectory for a low-thrust spacecraft in the Earth-Moon space. Using a set of simulated collision data from orbits near the Moon as an example, the method generates an optimal fuel collision avoidance trajectory for a low-thrust spacecraft in the Earth-Moon space, specifically including:

[0027] Step 1: Set the position and velocity r of the target at the closest moment of collision risk in the rendezvous coordinate system. s =[[0.9708920000,0.08693169497,-0.0141139994] T v s =[0.148652143405387,0.1500000000,-0.3100000000] T Set the position and velocity r of the spacecraft at the closest moment in the rendezvous coordinate system. p =[0.9708918431,0.08693169497,-0.0141139994] T v p =[0.1486521434,-0.2247801445,0.3899588750] T And mass m = 500 kg; input collision avoidance time t = 3 h, nominal thrust T = 50 mN of the small thrust engine and specific impulse I sp =3800s; Set the joint radius S of the spacecraft and the collision risk target. A =29.7m; Determine the covariance matrix of the spacecraft's position uncertainty. Where: ξ and ζ are the coordinates of the joint target vector (the vector pointing from the spacecraft center to the center of the collision risk target) in the coordinate system calculated at the instant of collision; σ ξ σ ζ To represent the variance of the covariance matrix C corresponding to the uncertainty, ρ ξζ The correlation coefficient.

[0028] Step 2: Sequentially establish the dynamic model of the collision risk target in the Earth-Moon rendezvous coordinate system. and the dynamic model of the spacecraft in the same coordinate system

[0029] Step 3: Based on collision probability calculation, in Figure 2 In the coordinate system shown, a composite performance index including homotopy parameters is established and substituted into the parameters in step 1) to obtain an initial collision probability of 99.9%, while a collision probability greater than one in ten thousand is considered a high collision risk event.

[0030] The aforementioned composite performance indicators Where: W=1 is a preset parameter that needs to be adjusted according to the weight;

[0031] The collision probability mentioned above is calculated using the short-term encounter collision probability, specifically as follows: Where: u is the ratio of the equivalent cross-sectional area to the area of ​​the 1-σ covariance ellipse in the plane of the instantaneous collision calculation coordinate system; v is the square of the Mahalanobis distance (or square Mahalanobis distance), which characterizes the similarity of the uncertainty distribution of the two target positions; m and k are summation variables, and the acceptable precision is m≥3.

[0032] Step 4: Determine the state, costate equation, and set the target equation: Where: ε starts from 1 for target acquisition and gradually converges until ε reaches 0. If the convergence condition is not met, the convergence rate is improved by reducing the step size. This iterative process continues until all states and co-state parameters of the spacecraft collision avoidance process are output. Where: R b R b T Calculate the rotation matrix and transpose matrix of the coordinate system at the instant of collision to rotate the nominal coordinate system; S, S T λ is a reduced-order identity matrix from three dimensions to two dimensions and its transpose; r , λ v , λ m These are the position, velocity, and mass covariates; denoted as mean squared error; subscripts 0 and f represent the initial and final times, respectively.

[0033] Step 5, as follows Figures 3-6 As shown, the optimal collision avoidance control of the spacecraft is solved based on the parameters obtained above. The magnitude of the control force is controlled according to the following formula: Where: switching function Using the algorithm and designed formulas described in this patent, an indirect solution is performed using a numerical integrator (code toolkits such as Matlab, Fortran, and C++). This ultimately yields the collision avoidance trajectories of the spacecraft and space debris, including a comparison with the original high-collision-risk event trajectories. Figure 3 and Figure 4 As shown, and as Figure 5 The collision avoidance control of the spacecraft shown and such Figure 6 The fuel consumption process shown has a collision probability of 2.85e-8 with the spacecraft at the end of the collision risk period, and a fuel consumption of 0.0028 kg.

[0034] Compared with existing technologies, this invention determines collision risk constraints through a collision probability calculation formula based on uncertainty, reducing false alarm rates and missed alarm rates, and improving collision risk avoidance efficiency; 2) it uses a co-state initial value normalization method for initial value guessing, greatly narrowing the range of random guessing and effectively improving the target-hitting rate of the indirect method; 3) it solves the problem of convergence difficulty caused by the discontinuity of bang-bang control in fuel optimization by using the homotopy method to solve the fuel optimization problem of low-thrust collision avoidance, thus improving convergence efficiency. This invention solves the problem of convergence difficulty caused by the discontinuity of bang-bang control in the fuel optimization solution by using the homotopy method and the indirect method to solve the fuel optimization problem of low-thrust collision avoidance. When performing optimization calculations, the calculation efficiency is high and the accuracy is high, thus making the designed low-thrust spacecraft fuel optimization collision avoidance trajectory more efficient and accurate.

[0035] The above-described specific implementations can be partially adjusted by those skilled in the art in different ways without departing from the principles and purpose of the present invention. The scope of protection of the present invention is defined by the claims and is not limited to the above-described specific implementations. All implementation schemes within the scope of the claims are bound by the present invention.

Claims

1. A method for implementing fuel-optimal collision avoidance trajectories for a low-thrust spacecraft in a lunar-moon space, characterized in that, After constructing the dynamic model based on the parameters of spacecraft collision avoidance problem, the fuel optimal performance index of spacecraft collision avoidance problem is established by homotopy method; the optimal control problem is converted into two-point boundary value problem and shooting solution is carried out to obtain the state and adjoint parameters of optimal orbit; then the optimal control of spacecraft collision avoidance and the collision probability are obtained according to the state and adjoint parameters of optimal orbit.

2. The method of claim 1, wherein the method is characterized by, The dynamics model comprises a restricted three-body dynamics model of a collision risk target in a lunar-terrestrial rendezvous coordinate system and a dynamics model of a spacecraft in the same coordinate system, and specifically comprises: Wherein: r=(x, y, z) is a position coordinate of the spacecraft or space debris in the rendezvous coordinate system, v=(v x ,v y ,v z ) is a velocity of the spacecraft or space debris in the rendezvous coordinate system, T max is the maximum thrust of the thruster, u is the engine working amplitude, the value is [0, 1], alpha is the propulsion direction, m is the mass of the spacecraft, c=I sp g0, I sp is the specific impulse of the propellant, and g0 is the equatorial sea level gravity acceleration. In the dynamic model, the two central celestial bodies are the Earth and the Moon, the coordinates of the Earth's center of mass in the coordinate system of the conjunction are (1-μ, 0, 0), the coordinates of the Moon's center of mass are (μ, 0, 0), r1 and r2 are respectively the distances of the spacecraft from the Earth's center of mass in the coordinate system of the conjunction the distance of the spacecraft from the Moon's center of mass in the coordinate system of the conjunction 3. The method of claim 1, wherein the method is characterized by, The fuel-optimal performance index of the spacecraft collision avoidance problem is: Wherein: P C is the final collision probability; T is the nominal thrust, usually taken as the maximum thrust T max in the steady state; λ0 is a normalization coefficient to provide a better initial guess in the shooting solution process; ε is a homotopy parameter, as ε changes from 1 to 0 according to the homotopy step, the performance index gradually changes from energy-optimal to fuel-optimal homotopy, and W is a weight parameter, the greater the value of W, the more the performance is biased towards minimizing the collision probability.

4. The method of claim 1, wherein the method is characterized by, The homotopy step is set by any one of the following ways: A) The homotopy parameter of the last round is subtracted by a fixed value, specifically: ε i = ε i-1 - Δε, where: ε i is the homotopy parameter at the ith homotopy, ε i-1 is the homotopy parameter at the i-1th homotopy, and Δε is the equal difference homotopy step. B) Scale down the homotopy parameter to obtain a new homotopy parameter, specifically: i = k Δε ε i-1 , where: k Δε is the homotopy step length.

5. The method of claim 4, wherein the method further comprises: If the last step homotopy does not converge, it means that the step size is too large, and the homotopy step size needs to be reduced, and then go back to the previous step to calculate the homotopy until the condition ε i -ε i-1 ≤error, it is considered that the homotopy is completed. Where error is the acceptable error.

6. The method of claim 1, wherein the method further comprises, The shooting solution specifically includes: 1) the unknown parameters in the boundary conditions are taken as variables to be solved, and an initial value problem is constructed: the value of positive λ0 is selected arbitrarily to satisfy the subsequent normalization condition, the 7 adjoint initial values obtained by random number guessing, i.e. 3 direction position coordinate adjoints, 3 direction velocity coordinate adjoints and 1 mass adjoint and the positive λ0 value are normalized to make the 8 adjoint initial values satisfy the modulus value of 1, at this time the vector endpoints corresponding to the eight adjoint initial values are located on a unit sphere; 2) the initial value problem is solved by numerical integration method to obtain the terminal state; 3) measure the difference between the terminal state and the target boundary condition by a residual function, which is specifically: a(t f ) = f(t f ) - f bc ≤ 10 -7 . Wherein a(t f ) is the residual of the terminal state quantity, f(t f ) is the terminal state quantity, f bc is the boundary condition of the corresponding state quantity. 4) the unknown parameters are adjusted by using iterative algorithm to make the residual function approach to zero, so as to satisfy the boundary conditions, specifically: take ε from 1 to carry out shooting solution, gradually converge until ε to 0, wherein if the convergence condition is not satisfied, the convergence rate is improved by reducing the step size, and the steps 2-4 are cycled until all the state and adjoint parameters of spacecraft collision avoidance process are output.

7. The method of claim 1, wherein the method further comprises: The optimal control of spacecraft collision avoidance and the collision probability are derived from the relationship between the optimal control and the adjoint variable, and the size and direction of the optimal control are calculated according to the terminal state, and the final collision probability is calculated.

Images