An Optimization Method for the Most Fuel-Saving Proximity Orbit of Nano-Satellites under Micro-Thrust

Abstract

The present invention provides an optimization method for minimizing the fuel consumption of a nano-satellite approaching an orbit under the action of micro-thrust. Taking the minimum fuel consumption of the nano-satellite approaching under the action of micro-thrust as the optimization goal, a two-point boundary value problem including the Hamiltonian function, the co-state equation, and the thrust control function is established. Using the position of the nano-satellite and its co-state variables, and the velocity and its co-state variables, a state equation is established. Using the solution of the two-point boundary value problem and the real-time value of the relative approaching distance of the nano-satellite, a measurement equation is established. The trajectory is divided into multiple segments, with the condition that the relative approaching distance is controlled within a certain range as the end condition. The volume Kalman filtering method is used to solve the state equation and the measurement equation, and the estimated values of the terminal position and velocity of each segment of filtering are used as the initial values of the next segment of filtering. Only the initial values of the co-state variables are updated for each segment of filtering. This method can obtain the optimal approaching trajectory and the thrust application situation.

Description

Technical Field

[0001] The invention belongs to the technical field of aerospace orbit dynamics and relates to a nanosatellite fuel-saving approach orbit optimization method under the action of micro-thrust. Background Art

[0002] In recent years, with the development of miniaturization technology and highly integrated technology, the development of nanosatellites has become very rapid. The mass of nanosatellites is generally 1 to 10 kg, and they are presented in a standardized structure of 3U, 6U, 9U or 12U. Nanosatellites have the advantages of small size, light weight, integration and high performance, and it is these advantages that make nanosatellites have broad space application value. No matter what kind of space application mission, orbital maneuvering technology is indispensable. Orbital maneuvering is the basis for satellites to carry out space control tasks such as space situational awareness, space attack confrontation and space defense. For satellites, especially nanosatellites, they carry limited fuel themselves. When performing different space missions, they must optimize the orbital maneuvering trajectory to meet the mission requirements. Therefore, under the action of small thrust and the condition of the most fuel-saving, how to optimize the approach trajectory of nanosatellites is a problem that needs to be solved. Summary of the invention

[0003] The purpose of the present invention is to design a nanosatellite fuel-saving approaching orbit optimization method under micro-thrust, construct an energy-saving two-point boundary value problem through the Pontryagin minimum principle, transform the two-point boundary value problem into a state estimation problem, and divide the optimized trajectory into multiple sections. The relative position and relative speed of the approaching satellite in each section of the trajectory are estimated by using a volumetric Kalman filter. The estimated values ​​of the terminal position and speed of each filtering section are used as the initial values ​​of the next filtering section. Each filtering section only updates the initial values ​​of the costate variables until the relative position enters the constraint distance range.

[0004] The specific technical scheme of the present invention is:

[0005] Step 1: Taking the minimum fuel consumption of nanosatellite approach under micro-thrust as the optimization goal, construct the Hamiltonian function H_fuel:

[0006]

[0007] Approaching the most fuel-saving performance index J: t 0 is the start time, t f is the end time;

[0008] Where r = [x, y, z] T and v = [v x ,v y ,v z ] Tare the position and velocity of the remote approaching satellite relative to the orbital coordinate system of the target satellite; ω is the orbital angular velocity of the target satellite; Thrust is the thrust of the microthruster, α is the direction in which the thrust of the microthruster is applied, and ||α||=1, m is the mass of the remote approaching satellite;

[0009] Step 2: Establish the co-state equation:

[0010]

[0011]

[0012] Step 3: Determine the direction and magnitude control function of the nanosatellite's micro-thrust:

[0013] Direction Angle:

[0014]

[0015] Thrust size:

[0016]

[0017] in:

[0018]

[0019] Step 4: Taking the optimal approach time of the nanosatellite under the action of micro-thrust as the optimization goal, the Pontryagin minimum principle is used to construct a two-point boundary value problem including the Hamiltonian function, the co-state equation, and the thrust control function:

[0020]

[0021] Under the condition of given initial state value, the integral state equation and co-state equation can be used to solve the position value, velocity value and co-state variable value that satisfy the terminal state value, and obtain the solution of the two-point boundary value problem; where f is the earth's gravity.

[0022] Step 5: Establish the state equation:

[0023]

[0024] Where w_fuel is Gaussian noise, which represents the system process noise;

[0025] Step 6: Use the solution of the two-point boundary value problem obtained in step 4 and the real-time value of the relative distance of the nanosatellite approaching to establish the measurement equation:

[0026] z_fuel=h_fuel(x)+v_fuel,

[0027] Here, z_fuel is defined as the terminal state of the system. However, since the terminal state has no constraints on the speed and co-state variables of the long-range approaching satellite, the measured value of the system is determined as z_fuel = [0,0,0] T ; v_fuel is the measurement noise and is Gaussian noise; for the state variable input x of the system at time k k ,h_fuel(x k ) is the solution of the two-point boundary value problem with this moment as the initial state value;

[0028] Step 7: Under the condition that the approaching relative distance is controlled within the constraint range, the state equation and measurement equation are solved using the cubature Kalman filter method.

[0029] (1) If the relative distance can converge within the constraint range, the solution is completed;

[0030] (2) If the relative distance decreases first and then increases, and does not enter the constraint range, the time when the relative distance minimum occurs is used as the basis for trajectory segmentation. The next filtering is performed from the time when the relative distance reaches the minimum value. The estimated values ​​of the terminal position and velocity of the previous filtering are used as the initial values ​​of the next filtering. Each filtering stage only updates the initial values ​​of the co-state variables until the relative distance converges to the constraint range.

[0031] Beneficial effects:

[0032] This method solves the problem of solving the optimal trajectory of nanosatellites under the condition of minimum fuel consumption under micro-thrust support. This method uses the cubature Kalman filter method to convert the two-point boundary value problem of the fuel-saving trajectory optimization problem into a state estimation problem, solving the problem that the two-point boundary value problem is sensitive to the initial value, and also overcoming the problem of large computational complexity of the direct optimization method, and can effectively and quickly optimize the trajectory. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is an optimization step for nanosatellite fuel to reach orbit with the least amount of fuel under micro-thrust.

[0034] Figure 2 This is the relationship between the relative distance of the filter terminal and the filtering time under the condition of the most fuel saving;

[0035] Figure 3 The trajectory of the approaching satellite in the relative coordinate system;

[0036] Figure 4 is the micro-thrust size;

[0037] Figure 5 Provides direction for micro-thrust. DETAILED DESCRIPTION

[0038] Now combined with the attached Figure 1Through specific examples, the method for optimizing the nanosatellite fuel-saving approach orbit under micro-thrust of the present invention is further described in detail.

[0039] See also Figure 1 , the optimization steps for the nanosatellite fuel-saving approach orbit under micro-thrust are:

[0040] Step 1: Taking the minimum fuel consumption of nanosatellite approach under micro-thrust as the optimization goal, construct the Hamiltonian function H_fuel:

[0041]

[0042] Approaching the most fuel-saving performance index J: t 0 is the start time, t f is the end time;

[0043] Where r = [x, y, z] T and v = [v x ,v y ,v z ] T are the position and velocity of the remote approaching satellite relative to the orbital coordinate system of the target satellite; ω is the orbital angular velocity of the target satellite; Thrust is the thrust of the microthruster, α is the direction in which the thrust of the microthruster is applied, and ||α||=1, m is the mass of the remote approaching satellite;

[0044] Step 2: Establish the co-state equation:

[0045]

[0046]

[0047] Step 3: Determine the direction and magnitude control function of the nanosatellite's micro-thrust:

[0048] Direction Angle:

[0049]

[0050] Thrust size:

[0051]

[0052] in:

[0053]

[0054] Step 4: Taking the optimal approach time of the nanosatellite under the action of micro-thrust as the optimization goal, the Pontryagin minimum principle is used to construct a two-point boundary value problem including the Hamiltonian function, the co-state equation, and the thrust control function:

[0055]

[0056] Under the condition of given initial state value, the integral state equation and co-state equation can be used to solve the position value, velocity value and co-state variable value that satisfy the terminal state value, and obtain the solution of the two-point boundary value problem; where f is the earth's gravity.

[0057] Step 5: Establish the state equation:

[0058]

[0059] Where w_fuel is Gaussian noise, which represents the system process noise;

[0060] Step 6: Use the solution of the two-point boundary value problem obtained in step 4 and the real-time value of the relative distance of the nanosatellite approaching to establish the measurement equation:

[0061] z_fuel=h_fuel(x)+v_fuel,

[0062] Here, z_fuel is defined as the terminal state of the system. However, since the terminal state has no constraints on the speed and co-state variables of the long-range approaching satellite, the measured value of the system is determined as z_fuel = [0,0,0] T ; v_fuel is the measurement noise and is Gaussian noise; for the state variable input x of the system at time k k ,h_fuel(x k ) is the solution of the two-point boundary value problem with this moment as the initial state value;

[0063] Step 7: Under the condition that the approaching relative distance is controlled within the constraint range, the state equation and measurement equation are solved using the cubature Kalman filter method.

[0064] (1) If the relative distance can converge within the constraint range, the solution is completed;

[0065] (2) If the relative distance decreases first and then increases, and does not enter the constraint range, the time when the relative distance minimum occurs is used as the basis for trajectory segmentation. The next filtering is performed from the time when the relative distance reaches the minimum value. The estimated values ​​of the terminal position and velocity of the previous filtering are used as the initial values ​​of the next filtering. Each filtering stage only updates the initial values ​​of the co-state variables until the relative distance converges to the constraint range.

[0066] The above method is illustrated by a case study. Assume that the target satellite is operating in a circular orbit at an altitude of 500km, and the initial distance between the long-range approach satellite (nanosatellite) and the target satellite is 1km. The nanosatellite conducts a small-scale rapid long-range approach to the target satellite to seek the optimal approach trajectory with the least fuel. Among them, for the thrust working mode of the microthruster, two thrust output modes are set: the maximum value is 300μN and the minimum value is 150μN. The terminal target requirement for the long-range approach of the nanosatellite is selected as ε=10m. When the terminal distance d≤ε between the long-range approach satellite and the target satellite, it means that the nanosatellite can meet the requirements of the long-range approach to the target satellite.

[0067] First, the relationship between the estimated terminal distance and the filtering time is found through simulation, such as Figure 2 As shown in the figure, it is found that the terminal distance d from the target satellite first decreases and then increases with the increase of filtering time. When the filtering time is 47s, its terminal relative distance d is 19.42m, reaching the minimum. Therefore, 47s is used as the dividing point of segmented filtering. The first segment of filtering is performed first, and the filtering time is 47s. At this time, the relative distance is 19.42m, which does not meet the requirements. Then, based on the first segment of the volumetric Kalman filtering algorithm, the final state variable estimate obtained by the first segment of the filtering algorithm is used as the initial value of the second segment of the volumetric Kalman filtering for filtering. It is worth noting that for the initial value of the second segment of the volumetric Kalman filtering, only the estimated values ​​of the position and velocity of the final value of the first segment of filtering are selected. The final value of the co-state variable of the first segment of filtering is not used as the initial value of the second segment of filtering, and so on, until the terminal state of the filtering meets the constraints. At 53s, d = 4.94m, which meets the terminal constraints. The entire segment of the approaching optimal trajectory is shown in the figure. Figure 3 As shown, the thrust magnitude and direction time are as follows Figure 4 and Figure 5 The arrow direction indicates the thrust direction, the long arrow indicates a micro-thrust of 300 μN, and the short arrow indicates a micro-thrust of 150 μN.

[0068] The present application is not limited to the contents defined in the specification and claims. Any modifications and changes known in the art fall within the scope of the present application. The specific embodiments of the specification are only exemplary descriptions of the present invention, not specific limitations of the present invention.

Claims

1. An optimization method for the most fuel-efficient approach orbit of a nanosatellite under micro-thrust, characterized in that: Step 1: Taking the most fuel-efficient approach of the nanosatellite under micro-thrust as the optimization objective, construct the Hamiltonian function \(H_{fuel}\): Proximity fuel-optimal performance index: t 0 is the start time, and t f is the end time; where r = [x, y, z] T and v = [v x , v y , v z T are the position and velocity of the remote approaching satellite relative to the orbital coordinate system of the target satellite, respectively; ω is the orbital angular velocity of the target satellite; Thrust is the magnitude of the micro-thruster thrust, α is the direction of the micro-thruster thrust application, and ||α|| = 1, m is the mass of the remote approaching satellite;​ Step 2, establish the co-state equation: Step 3, determine the control functions for the direction and magnitude of the micro-thrust of the nanosatellite: Direction angle: Thrust magnitude: Where: Step 4: Taking the time-optimal approach of the nanosatellite under micro-thrust as the optimization objective, use the Pontryagin minimum principle to construct a two-point boundary value problem including the Hamiltonian function, co-state equation, and thrust control function: Under the given initial state value conditions, the integral state equation and co-state equation can be used to solve for the position value, velocity value, and their co-state variable values that satisfy the terminal state value, and the solution of the two-point boundary value problem is obtained; where, \(f\) is the Earth's gravity; Step 5, establish the state equation: Where \(w_{fuel}\) is Gaussian noise, representing the system process noise; Step 6: Using the solution of the two-point boundary value problem obtained in Step 4 and the real-time value of the relative approach distance of the nanosatellite, establish the measurement equation: \(z_{fuel}=h_{fuel}(x)+v_{fuel}\), Among them, \(z_{fuel}\) is defined as the terminal state of the system. However, since there are no constraints on the velocity and co-state variables of the remotely approaching satellite in the terminal state, the measured value of the system is determined as \(z_{fuel} = [0, 0, 0]\). T ; \(v_{fuel}\) is the measurement noise and is Gaussian noise; for the state variable input \(x\) of the system at time \(k\). k , \(h_{fuel}(x\). k ) is the solution of the two-point boundary value problem with this moment as the initial state value. Step 7: Taking the approach relative distance being controlled within the constraint range as the condition, use the cubature Kalman filter method to solve the state equation and measurement equation, (1) If the relative distance can converge within the constraint range, the solution is completed; (2) If the relative distance first decreases and then increases and does not enter the constraint range, then use the moment when the minimum relative distance occurs as the basis for trajectory segmentation. Start the next segment of filtering from the moment when the relative distance reaches the minimum value. Use the estimated values of the terminal position and velocity of the previous segment of filtering as the initial values of the next segment of filtering. Only update the initial values of the co-state variables for each segment of filtering until the relative distance converges within the constraint range.

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