A fuel consumption optimization method for distributed satellite formation control

Abstract

The present invention discloses a distributed satellite formation control fuel consumption optimization method, comprising the following steps: step 1, describing the total fuel consumption optimization and fuel consumption balance problem of the distributed satellite formation under multiple constraints; step 2, converting the problem defined in step 1 into a convex programming requirement standard form, and introducing the terminal offset; step 3, using a convex programming solver to solve the formation total fuel consumption optimization problem with the terminal offset introduced and the total fuel consumption optimization problem considering fuel consumption balance. In this way, the total fuel consumption optimization problem and fuel consumption balance problem of the satellite formation under multiple constraints such as fixed orbit transfer time and collision avoidance are solved.

Description

Technical Field

[0001] The invention belongs to the field of distributed satellite configuration control and relates to a distributed satellite formation control fuel consumption optimization method. Technical Background

[0002] A satellite formation is a group of satellites distributed in a short range (tens of meters to tens of kilometers) and forming a certain geometric configuration. From a functional point of view, a satellite formation will be equivalent to a large satellite. Compared with large satellites, satellite formations have the advantages of strong reliability, low launch and maintenance costs, and wide coverage. Satellite formation flight is a teamwork of multiple satellites to complete mission objectives, including meteorological observation, high-resolution space interferometry, synthetic aperture, etc. With the development of formation autonomous flight and attitude control technology, the future will gradually tend to improve efficiency and performance while replacing a single large satellite with a group of small satellites to form a distributed structure to save costs.

[0003] The mission life of a satellite formation depends largely on the use of fuel. Since the formation is affected by various disturbance forces and needs to be controlled to maintain or reconstructed when necessary, minimizing fuel consumption is of great significance in orbit design and control. Applying the optimal control theory framework to the design of fuel-optimal trajectories for formation flight control problems has always been a focus of attention. Current theoretical method research usually has the following problems: ① It is difficult to solve and obtain analytical solutions for multi-satellite formation flight based on the Gaussian variational equation when considering collision avoidance constraints; ② There is little research on fuel balance problems under multiple constraints such as collision avoidance; ③ The traditional nonlinear optimization method has a large amount of calculation for solving formation fuel optimization under multiple constraints such as collision avoidance. Summary of the invention

[0004] The purpose of the present invention is to provide a distributed satellite formation control fuel consumption optimization method to solve the total fuel consumption optimization problem and fuel consumption balance problem of the satellite formation under multiple constraints such as fixed orbit transfer time and collision avoidance.

[0005] The technical solution adopted by the present invention is a distributed satellite formation control fuel consumption optimization method, comprising the following steps:

[0006] Step 1: Description of the total fuel consumption optimization and fuel consumption balance problem of distributed satellite formation under multiple constraints;

[0007] Step 2: Convert the problem defined in step 1 into a convex programming standard form and introduce the end offset;

[0008] Step 3: Use the convex programming solver to solve the optimal total fuel consumption problem of the formation with the introduction of terminal offset and the optimal total fuel consumption problem considering fuel balance.

[0009] Furthermore, the specific content of step one is:

[0010] S1.1 Establish a linear discrete relative motion model:

[0011] Define a primary satellite orbital coordinate system (LVLH), where x, y, and z are radial, along-track, and tangential directions, respectively. The dynamic equation of the motion of a single secondary spacecraft relative to the primary satellite in a near-circular orbit is expressed as:

[0012]

[0013] in, are the accelerations of the secondary satellite in radial, along-track and tangential directions, are the velocities of the secondary satellite in radial, along-track, and tangential directions, μ is the Earth's gravitational constant, R c is the orbital radius of the main star, is the average angular velocity of the primary star, T x , T y , T z They are the thrusts of the x, y, and z axes, including control force and perturbation force;

[0014] Equation (1) is linearized to obtain the HCW equation. The discretized HCW equation analytical solution can obtain the state transfer matrix (STM) based on the HCW equation:

[0015]

[0016] Among them, x k ,y k and z k are the positions of the secondary star relative to the primary star on the x, y, and z axes at time k, and are the corresponding relative velocities, and similarly, x k+1 ,y k+1 、z k+1 , are the three-axis relative position and velocity of the secondary satellite at time k+1, and T is the discretization step size.

[0017] S1.2 Obtain the initial and terminal states of the formation through the formation configuration design equation:

[0018] Customize the six numbers of the main star orbit α c , and customize the formation configuration parameters according to the needs, and obtain the initial and terminal states of the formation from the formation configuration design equation;

[0019] S1.3 defines the total fuel consumption optimization and fuel consumption balance problems under multiple constraints, including kinematic constraints, initial and terminal state constraints, maximum thrust constraints and anti-collision constraints:

[0020] Define xj (t) is the state of a single satellite in the LVLH coordinate system at time t, X(t) is the state of the formation at time t; define u j (t) is the instantaneous acceleration of satellite j at time t;

[0021] Consider a formation consisting of N (N ≥ 2) stars, with a fixed orbit transfer time (0, t f ) is controlled by applying fuel u j (t), j = 1, 2, ..., N, from the known initial state X 0 Transition to target state X f ; where t f is the reconstruction time, relatively, x j (0), x j (t f ) are the initial and final states of satellite j, j = 1, 2, ..., N.

[0022] One goal of formation maintenance or reconstruction is to minimize the total fuel consumption J of the formation. t , the objective can be described as the L1 norm of the control quantity:

[0023]

[0024] Another objective function J b To balance the fuel usage of each satellite in the formation, it is defined as:

[0025]

[0026] The orbit transfer process is constrained by the relative motion equation of formula (1), and the initial and final states are:

[0027] X(0) = X 0 ,X(t f )=X f (5)

[0028] Without loss of generality, we force the control input u of each satellite to be j (t) not exceeding the maximum thrust acceleration U max ,Right now:

[0029]

[0030] In order to ensure that there is no collision between satellites during the control process, the distance between any two satellites during the mandatory control process is always greater than the preset safety distance r. safe , that is, for Both:

[0031] ||C[x j (t)-x i(t)]||≥r safe (7)

[0032] Where C = [I 3×3 0 3×3 ];

[0033] Therefore, the problem of minimizing the total fuel consumption can be described as:

[0034]

[0035] The fuel balance problem is described as:

[0036]

[0037] The specific contents of the further step 2 are:

[0038] S2.1 Discretization objective function:

[0039] Consider a formation of N (N ≥ 2) satellites that completes orbit transfer within K steps, with a transfer time step of Δt; the state (position, speed) of satellite j at time k is x j (k), combined with x j (k) and control quantity u j (k) k∈{1,2,...,K-1};

[0040] Change J in S1.3 t and J b Discretize and write it in standard convex programming form:

[0041]

[0042]

[0043] Among them, y j is the column vector composed of the discretized single star state and control quantity, Y is the total number of y in the formation j The column vector, H j and Used to extract the control quantity u from Y j ;

[0044] S2.2 Discretize the single star dynamics and expand it to the dynamics of multiple star systems:

[0045] Under the discrete model, the dynamic equation of a single satellite j is defined as:

[0046] x j (k+1)=Ax j (k)+Bu j (k) (12),

[0047] Where A is the state transfer matrix in equation (2), and B is the control matrix. For satellite j, the relative motion equations at all discrete times 1, 2, ..., K-1 can be integrated into the following homogeneous form:

[0048]

[0049] Among them, A sd is the coefficient matrix of the dynamic equations of a single star under all discrete times. Equation (13) is expanded to the formation dynamic equation of N stars, and we get:

[0050]

[0051] in, is the coefficient matrix of the dynamic equations of the formation at all discrete times;

[0052] S2.3 Establish the initial and final states of multiple stars, and introduce the final offset into the final state:

[0053] Define a new terminal state It is the original terminal state X f With unknown offset [x r ,y r ,z r ,0,0,0] T The unknown offset can be obtained by solving the fuel consumption optimization problem in S3.1 or S3.2. That is, the new terminal state of satellite j is x j (K)+[x r ,y r ,z r ,0,0,0] T ; Define the matrix and They are used to extract the initial and final states of the formation from Y, respectively, and we get:

[0054]

[0055] S2.4 Discretized multi-satellite maximum thrust constraints:

[0056] In discrete time, the maximum thrust constraint in S1.3 is written as follows:

[0057]

[0058] Among them, the matrix Used to extract the control value u of satellite j at time k from the full state column vector Y j (k), vector 1 is determined by the thruster configuration. If thrusters are used on the x, y, and z axes, then 1 = [1, 1, ..., 1] T, length is 6N(K-1); if the thrust is only applied in the y and z directions, then 1=[0,1,1,...,0,1,1] T , the length is 6N(K-1);

[0059] S2.5 Convexify the multi-star collision avoidance constraints and discretize them:

[0060] The collision avoidance constraint (10) is a typical non-convex function that does not meet the requirements of convex programming and needs to be further rewritten into the following quadratic convex form:

[0061]

[0062] in, and are the estimated states of satellites i and j at time k, respectively. and It is obtained by solving the fuel optimization problem without collision avoidance constraints; in order to extract the relative positions of satellites i and j (i < j) from Y, the matrix C at time k can be rewritten as

[0063]

[0064] definition and At this time, the collision avoidance constraint (17) is rewritten as:

[0065]

[0066] Furthermore, the specific content of step three is:

[0067] S3.1 uses a convex programming solver to solve the optimal total fuel consumption problem of the formation with terminal offset introduced;

[0068] Among them, the total fuel consumption minimization problem in discrete time is defined as:

[0069]

[0070] S3.2 uses a convex programming solver to solve the optimal total fuel consumption problem of the formation with the introduction of terminal offset and consideration of fuel consumption balance;

[0071] Among them, the fuel consumption balance problem is defined as:

[0072]

[0073] The beneficial effects of the present invention are as follows: the present invention studies the definition of the fuel optimization problem in the framework of the primary satellite orbital coordinate system (LVLH), and the optimization problem under the collision avoidance constraint will be easier to solve. The present invention studies respectively define the formation total fuel optimization problem and the fuel balance problem considering the collision avoidance constraint and the maximum thrust constraint in the LVLH coordinate system and introducing the terminal state offset. The present invention studies the use of a convex programming method to solve the problem. The convex programming method requires the problem to be discretized. Since the formation dynamics equation is sparse after discretization, the convex programming method has a small amount of calculation and saves time when solving multi-constraint and multi-variable optimization problems; and since the local optimal solution of the convex programming problem must be the global optimal solution, it is very suitable for multi-constraint path planning and optimal control of multi-satellite formations. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figures 1-(a), 1-(b), and 1-(c) are the relative motion trajectories of the x, y, and z axes of each satellite over time when the following formation is optimized for total fuel consumption under collision avoidance constraints when the following formation changes from a 100m following distance to a 50m following distance;

[0075] Figure 2-(a) and Figure 2-(b) are the three-dimensional relative motion trajectories of the satellites when optimizing the total fuel consumption of the formation under the collision avoidance constraint with and without the addition of an offset when the following formation changes from a following distance of 100m to a following distance of 50m;

[0076] FIG3-(a) and FIG3-(b) are curves showing the total fuel consumption of each satellite changing with time when the following formation changes from a following distance of 100 m to a following distance of 50 m, with and without adding an offset when optimizing the total fuel consumption of the formation under collision avoidance constraints;

[0077] Figure 4 To change the following distance of the flight formation from 100m to 50m, optimize the total fuel consumption of the formation, and show the collision avoidance constraint effect;

[0078] Figure 5-(a), Figure 5-(b), and Figure 5-(c) are the relative motion trajectories of the x, y, and z axes of each satellite over time when the following formation changes from a following distance of 100m to a following distance of 50m under collision avoidance constraints and when the fuel consumption is balanced;

[0079] Figure 6-(a) and Figure 6-(b) are the three-dimensional relative motion trajectories of the satellites when the following formation changes from a following distance of 100m to a following distance of 50m, with and without the addition of an offset under collision avoidance constraints and balanced fuel consumption;

[0080] FIG7-(a) and FIG7-(b) are curve diagrams showing the total fuel consumption of each satellite changing with time when the following formation changes from a following distance of 100 m to a following distance of 50 m, with and without adding an offset, under the collision avoidance constraint and with balanced fuel consumption;

[0081] Figure 8 To change the following formation distance from 100m to 50m, and balance the fuel consumption, the collision avoidance constraint effect is demonstrated;

[0082] Figures 9-(a), 9-(b), and 9-(c) are the relative motion trajectories of the x, y, and z axes of each satellite over time when the total fuel consumption of the formation is optimized under the collision avoidance constraint when the space circle formation with a space circle radius of 50m is changed into a following formation with a following distance of 50m;

[0083] FIG10-(a) and FIG10-(b) are three-dimensional relative motion trajectory diagrams of each satellite when the total fuel consumption of the formation is optimized under the collision avoidance constraint with and without adding an offset when the space circle formation with a space circle radius of 50 m is changed into a following formation with a following distance of 50 m;

[0084] FIG11-(a) and FIG11-(b) are curve diagrams showing the total fuel consumption of each satellite changing with time when the total fuel consumption of the formation is optimized under the collision avoidance constraint with and without adding an offset when the space circle formation with a space circle radius of 50 m is changed into a following formation with a following distance of 50 m;

[0085] Fig.12 When the space circle formation with a space circle radius of 50m is changed into a following formation with a following distance of 50m, the total fuel consumption of the formation is optimized and the collision avoidance constraint effect is displayed;

[0086] Figures 13-(a), 13-(b), and 13-(c) are the relative motion trajectories of the x, y, and z axes of each satellite over time when the space circle formation with a space circle radius of 50m is changed into a following formation with a following distance of 50m under collision avoidance constraints and when the fuel consumption is balanced;

[0087] FIG14-(a) and FIG14-(b) are three-dimensional relative motion trajectories of satellites when the space circle formation with a space circle radius of 50 m is changed into a follow-up formation with a follow-up distance of 50 m, with and without the addition of an offset under the collision avoidance constraint and balanced fuel consumption;

[0088] FIG15-(a) and FIG15-(b) are curve diagrams showing the total fuel consumption of each satellite changing with time when the space circle formation with a space circle radius of 50 m is changed into a following formation with a following distance of 50 m, with and without adding an offset to the collision avoidance constraint and the balanced fuel consumption;

[0089] Fig.16 When the space circle formation with a space circle radius of 50m is changed into a following formation with a following distance of 50m, the collision avoidance constraint effect is demonstrated when the fuel consumption is balanced. Specific implementation plan

[0090] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0091] The present invention provides a distributed satellite formation control fuel consumption optimization method, including the following contents:

[0092] Step 1: Description of the total fuel consumption optimization and fuel consumption balance problem of distributed satellite formation under multiple constraints;

[0093] Step S1.1 establishes a linear discrete relative motion model;

[0094] Consider a formation of N satellites in a near-circular orbit. In order to control all satellites to save fuel, an additional virtual satellite is set as the master satellite, and its state is used as the formation reference point. The study of formation flight is usually described by the relative motion equation, usually using the master satellite orbital coordinate system (LVLH), where x, y, and z are radial, along-track, and tangential directions, respectively. In order to maintain generality, the dynamic equation of the motion of a single secondary spacecraft relative to the master satellite in a near-circular orbit is given, which can be expressed as:

[0095]

[0096] in, are the accelerations of the secondary satellite in radial, along-track and tangential directions, are the speeds of the secondary satellite in radial, along-track and tangential directions, μ = 398600.44 km 3 / s 2 is the Earth's gravitational constant, R c is the orbital radius of the main star, is the average angular velocity of the primary star, T x , T y , T z are the thrusts on the x, y, and z axes, including the control force and the perturbation force. The equation is expanded by Taylor and retains the first-order term (μ(R c +x) / r 3 ≈n 2 (R c -2x),μy / r 3 ≈-n 2 y,μz / r 3 ≈-n 2 z), we can get the HCW equation:

[0097]

[0098] By solving the problem of not applying thrust (T x =T y =T z=0) and discretize the analytical solution to obtain the state transfer matrix (STM) of the secondary star relative to the primary star based on the HCW equation over time, which is given as follows:

[0099]

[0100] Among them, x k ,y k and z k are the positions of the secondary star relative to the primary star at time k, x k ,y k and z k are the corresponding relative velocities, and similarly, x k+1 ,y k+1 、z k+1 , are the three-axis relative position and velocity of the secondary satellite at time k+1, and T is the discretization step size.

[0101] For ease of description, the subsequent fuel consumption optimization of the present invention uses the STM in formula (3) for configuration control, but the method is not limited to formula (3), and other relative motion models can also be used to replace the STM, such as YA STM, J2 close STM, etc.

[0102] Step S1.2 obtains the initial and terminal states of the formation through the formation configuration design equation;

[0103] Consider using Kepler's six classical orbit numbers to describe the absolute state of the satellite:

[0104] α j ={a,e,i,Ω,ω,f} j j∈{1,2,...,N} (4),

[0105] Where a is the semi-major axis of the orbit, e is the orbit eccentricity, i is the orbit inclination, Ω is the right ascension of the ascending node, ω is the argument of perigee, and f is the mean anomaly. Here N is the number of satellites. The difference in the number of orbital elements of the primary and secondary satellites is expressed as δα j The formation configuration design is usually derived using the following relative motion equations:

[0106]

[0107] Usually δa is chosen to be 0 so that the formation maintains bounded relative motion. u = f + ω is the average latitude argument. The five elements aδe, aδi, aδλ, θ selects a specific value to design a satellite formation. The reconstruction of the following formation and the space circle formation is considered in this study. Their five elements are: 0 0 aδλ 0 0 and Or θ+π, where R is the radius of the space circle.

[0108] Step S1.3 defines the total fuel optimization and fuel balance problems under multiple constraints, including kinematic constraints, initial and terminal state constraints, maximum thrust constraints and anti-collision constraints.

[0109] definition is the state of a single satellite in the LVLH framework at time t, X(t) = [x 1 (t),x 2 (t),...,x N (t)] is the state of the formation at time t. Definition is the instantaneous acceleration of satellite j at time t, are the instantaneous accelerations in the x, y and z directions respectively. Consider a formation consisting of N (N ≥ 2) stars, with a fixed orbit transfer time (0, t f ) is controlled by applying fuel u 1 (t),u 2 (t),...,u N (t), from the known initial state X 0 =[x 1 (0),x 2 (0),...,x N (0)] T Transition to target state X f =[x 1 (t f ),x 2 (t f ),...,x N (t f )] T , where t f is the reconstruction time, relatively, x j (0), x j (t f ) are the initial and final states of satellite j, j = 1, 2, ..., N. One goal of formation maintenance or reconstruction is to minimize the total fuel consumption J of the formation. t , the objective can be described as the L1 norm of the control quantity:

[0110]

[0111] Another objective function J b In order to balance the fuel usage of each satellite in the formation, that is, to minimize the maximum fuel consumption of each satellite, it is defined as:

[0112]

[0113] The orbit transfer process is constrained by the relative motion equation of formula (1), and the initial and final states are:

[0114] X(0) = X 0 ,X(t f )=X f (8)

[0115] Without loss of generality, we force the control input u of each satellite to be j (t) not exceeding the maximum thrust acceleration U max ,Right now:

[0116]

[0117] In order to ensure that there is no collision between satellites during the control process, the distance between any two satellites during the mandatory control process is always greater than the preset safety distance r. safe , that is, for Both:

[0118] ||C[x j (t)-x i (t)]||≥r safe (10)

[0119] Where C = [I 3×3 0 3×3 ]. Therefore, the problem of minimizing the total fuel consumption can be described as:

[0120]

[0121] The fuel consumption balance problem can be described as:

[0122]

[0123] Step 2: Convert the problem defined in step 1 into a convex programming standard form and introduce position offset;

[0124] S2.1 Discretization objective function;

[0125] Convex optimization problems (11) and (12) can be processed by specific solvers to effectively find the global optimal solution. The optimization problem needs to be transformed into a convex programming problem. The first step is to discretize the objective function and constraint function, and then modify them into the following standard form. Consider a formation consisting of N (N ≥ 2) satellites, which completes the orbit transfer within K steps, with a transfer time step of Δt. The state (position, velocity) of satellite j at time k is x j (k), combined with x j (k) and control quantity u j (k), define y j for:

[0126] y j =[xj (1),x j (2),...,x j (K),u j (1),u j (2),...u j (K-1)] T (13)

[0127] It is a column vector of length M=9K-3. is defined as the set of all satellite states and control quantities. Note that in the discrete case, k∈{1,2,...,K-1}.

[0128]

[0129] The fuel consumption of a single satellite is defined as the L1 norm of the satellite control quantity. The total fuel consumption objective function of the formation after discretization is It can be written as:

[0130]

[0131] Similarly, the discrete form of the fuel balance objective function is Written as:

[0132]

[0133] In order to and Written as a convex programming standard form, define H j and Used to extract the control quantity u from Y j :

[0134] H j =[0 3(K-1)×6K I 3(K-1) ] (17),

[0135]

[0136] Then you can get:

[0137]

[0138]

[0139] S2.2 Discretize single star dynamics and extend it to multi-star system dynamics;

[0140] Under the discrete model, the dynamic equation of a single satellite j is defined as:

[0141] x j (k+1)=Axj (k)+Bu j (k) (21),

[0142] Among them, A is the state transfer matrix and B is the control matrix. The A matrix of the attached results in the present invention uses the HCW state transfer matrix described in S1.1, and the B matrix is ​​in the following form:

[0143]

[0144] For satellite j, the relative motion equations for all discrete times 1, 2, ..., K-1 can be integrated into the following homogeneous form:

[0145]

[0146] in:

[0147]

[0148] Expanding equation (24) into the formation dynamics equation for N stars, we obtain:

[0149]

[0150] in:

[0151]

[0152] S2.3 Establish the initial and final states of multiple satellites, and introduce the final offset into the final state;

[0153] In this study, it is assumed that the master satellite is a virtual satellite and its state is used as a reference point. This is different from other existing assumptions (the master satellite is not controlled and other satellites use the master satellite state as a reference point). The assumption of a virtual master satellite allows the fuel consumption of the formation to be reduced by optimizing the terminal state of all N satellites. In this study, a new terminal state is defined It is the original terminal state X f With unknown offset [x r ,y r ,z r ,0,0,0] T The sum of , that is, the new terminal state of satellite j is x j (K)+[x r ,y r ,z r ,0,0,0] T , the unknown offset can be obtained by solving the optimization problem in S3.1 or S3.2. Define the matrix and They are used to extract the initial and final states of the formation from Y, respectively, and we get:

[0154]

[0155] in:

[0156]

[0157]

[0158] and:

[0159] A IC =[I 6×6 0 6×9(K-1) ] (30),

[0160] A FC =[0 6×6(K-1) I 6×6 0 6×3(K-1) ] (31),

[0161] S2.4 Discretized multi-satellite maximum thrust constraints;

[0162] In discrete time, the control input of each satellite cannot exceed the maximum input limit of the thruster within a time step Δt. In order to make the constraint satisfy the convex programming standard form, the maximum thrust constraint in S1.3 can be written as follows:

[0163]

[0164] Among them, the matrix Used to extract the control value u of satellite j at time k from the full state column vector Y j (k):

[0165]

[0166] and:

[0167]

[0168] In equation (32), vector 1 is determined by the thruster configuration. If thrusters are used on the x, y, and z axes, then 1 = [1, 1, ..., 1] T , length is 6N(K-1); if the thrust is only applied in the y and z directions, then 1=[0,1,1,...,0,1,1] T , with a length of 6N(K-1).

[0169] S2.5 Convexify the multi-satellite collision avoidance constraints and discretize them;

[0170] The collision avoidance constraint (10) is a typical non-convex function that does not meet the requirements of convex programming and needs to be further rewritten into the following quadratic convex form:

[0171]

[0172] in, and are the estimated states of satellites i and j at time k, and The closer to x i (k) and x j (k), the more accurate the convex programming is. and It is obtained by solving the fuel optimization problem without collision avoidance constraints. For more details, see S3.2. In order to extract the relative positions of satellites i and j (i < j) from Y, the matrix C at time k can be rewritten as

[0173]

[0174] definition and At this time, formula (35) can be rewritten as:

[0175]

[0176] In step 2, each constraint condition is convexified, and a total fuel consumption optimization and fuel consumption balance framework for satellite formation control is established, which is easy to solve under multiple constraints such as collision avoidance. A virtual master satellite is set to derive the satellite state (relative position, speed), and an adjustable position offset is introduced into the terminal state under the premise of meeting the formation configuration requirements, which will make the terminal state of the formation no longer limited to a fixed state. When solving the optimization problem, the orbit transfer path of each satellite is shortened, thereby reducing the total fuel consumption of the formation, and making the fuel consumption balance problem easier to optimize.

[0177] Step 3: Use the convex programming solver to solve the optimal total fuel consumption problem of the formation with the introduction of terminal offset and the optimal total fuel consumption problem considering fuel balance.

[0178] S3.1 uses a convex programming solver to solve the optimal total fuel consumption problem of the formation with terminal offset introduced;

[0179] In summary, the total fuel consumption minimization problem can be defined as:

[0180]

[0181] From the definition and properties of convex sets and convex functions, it is not difficult to see that the objective function is a convex function, and the solution space described by constraints (25), (27), (32), and (37) is also a convex function. Therefore, Problem 3 is a convex optimization problem.

[0182] In order to converge the optimal solution under the collision avoidance constraint, the present invention adopts an iterative process to solve the optimization problem under the collision avoidance constraint. This iterative process is necessary because It is solved without considering the collision avoidance constraint. If With x j ,j=1,2,...N are too different, and a non-optimal solution may be obtained. The specific operation process is: given an initial Solving the convex optimization problem (38) yields Next let And solve the convex optimization problem (38) again. This process continues until x j Convergence or maximum number of iterations reached.

[0183] There are different solvers for solving convex optimization problems, such as Sedumi, SDPT3, etc. This study uses the SDPT3 solver of the Matlab CVX toolbox. SDPT3 is used to solve cone programming problems where the constraint cone is a semidefinite cone, a second-order cone and / or a non-negative orthogonal product. It is quite effective when dealing with problems with dimensions of hundreds of dimensions. For optimal problems under collision avoidance constraints, SDPT3 can obtain more stable solutions than Sedumi. In the convex programming problem Problem 3, the dimension of the state variable depends on the number of satellites N and the time step K. To ensure the convergence of the optimal problem, the discretization step Δt within one orbital period usually needs to not exceed 2 / n. The algorithm for solving Problem 3 is as follows:

[0184]

[0185] S3.2 uses a convex programming solver to solve the optimal total fuel consumption problem of the formation with the introduction of terminal offset and consideration of fuel balance.

[0186] In summary, the fuel consumption balance problem can be defined as follows in discrete time:

[0187]

[0188] From the definition and properties of convex sets and convex functions, it is not difficult to see that the objective function is a convex function, and the solution space described by constraints (25), (27), (32), and (37) is also a convex function. Therefore, Problem 4 is a convex optimization problem.

[0189] Similar to S3.1, the solution to Problem 4 also adopts an iterative process. The specific operation process is as follows: Given the initial Solve the convex optimization problem (39) to obtain x j ,j=1,2,...N; Next, let And solve the convex optimization problem (39) again. This process continues until x j Convergence or reaching the maximum number of iterations. As described in S3.1, Problem 4 also uses the SDPT3 solver. In the convex programming problem Problem 4, the dimension of the state variable depends on the number of satellites N and the time step K. To ensure the convergence of the optimal problem, the discretization step Δt within an orbital period usually does not exceed 2 / n. The solution algorithm for Problem 4 is as follows:

[0190]

[0191]

[0192] Example 1

[0193] The present invention is not limited to the initial and terminal states of the formation. Here, taking the terminal state of the formation as a following formation as an example, two groups of simulation cases are given. The first group considers the formation reconstruction from a following formation with a following distance of 100m to a following formation with a following distance of 50m; the second group considers the formation reconstruction from a spatial circle formation to a following formation. Both groups of simulation cases consider total fuel consumption optimization Problem 3 and balanced fuel consumption Problem 4, and their solution results are expressed as: P3-Safety-Offset and P4-Safety-Offset respectively; in order to evaluate the performance improvement of the method of adding offset in the present invention, Algorithm 1 and Algorithm 2 are run for Problem 3 and Problem 4 respectively, but no offset is added, and the solution results are expressed as: P3-Safety and P4-Safety respectively; in order to show the influence of collision avoidance constraints on fuel consumption optimization results in the present invention, Algorithm 1 and Algorithm 2 are run for Problem 3 and Problem 4 respectively, but no collision avoidance constraints are added, and the solution results are expressed as: P3-Offset and P4-Offset respectively.

[0194] The simulation scenario is set as a formation of 6 satellites (N=6), and the relative motion model uses the discrete HCW equation defined by equation (3). The maximum acceleration U provided by the thruster on each axis is max =25e -6 m / s 2 , the number of orbital elements of the main star is Reconstruction time t f The time step length is 6000s, the discretization time step Δt is 30s, and the number of steps K is 201.

[0195] Case 1 (the following formation with a distance of 100m changes to a following formation with a distance of 50m)

[0196] Assume that the initial configuration is the follow-up configuration, and the follow-up distance is aδλ=100(j-1)m, that is, the initial state of satellite j is x j (0) = [0, 100 (j-1), 0, 0, 0, 0] T , j∈{1,...,6}; X 0 =[x 1 (0),x 2 (0),...,x 6 (0)] T The expected configuration is a follow-up configuration with a follow-up distance of aδλ=50(j-1)m, that is, when no offset is introduced, the terminal state of each star is x j (t f )=[0,50(j-1),0,0,0,0] T ; When the offset is introduced, x j (t f )=[x r , 50(j-])+y r , z r , 0, 0, 0] T , In order to show the effect of collision avoidance constraints, set the safety distance r safe It is 49m.

[0197] Table 1. Fuel consumption optimization results of the change from following flight formation to following flight formation

[0198]

[0199] The simulation results of Case 1 are shown in Figure 1-(a) to Figure 8 , specific data are shown in Table 1.

[0200] For Problem 3, the total fuel consumption optimization problem, the simulation results are shown in Figure 1-(a) to Figure 4 , see P3-Safety and P3-Safety-Offset in Table 1 for specific data. As shown in Figure 1-(a), Figure 1-(b), Figure 1-(c), Figure 2-(a) and Figure 2-(b), by comparing the relative motion trajectories of the P3-Safety and P3-Safety-Offset satellites, it can be seen that within the limited machine accuracy range, the end state of the formation has an obvious offset in the y direction (125.10m), and the offset in the x and z directions is 0. Figure 4 , the distance between P3-Offset satellites 1 and 2 at 5280s is 48.75m, and the collision cannot be avoided. However, the relative motion trajectories of P3-Safety and P3-Safety-Offset satellites are safe, that is, the distance between any two satellites is not less than the safety distance r safe=49m. As shown in Figure 3-(a) and Figure 3-(b), the fuel consumption of each satellite of P3-Safety and P3-Safety-Offset is also very different. In P3-Safety-Offset, satellites 1 and 6, 2 and 5, 3 and 4 are almost equal, and only three fuel consumption curves are shown in Figure 3-(a). As shown in Figure 3-(a), Figure 3-(b) and Table 1, compared with P3-Safety and P3-Safety-Offset, the maximum fuel consumption of a single satellite is reduced by 50% (satellite 6); the total fuel consumption of the formation is reduced by 40%, which shows the beneficial effect of using offsets and iterative processes.

[0201] For Problem 4, the balanced fuel consumption problem, the simulation results are shown in Figure 5-(a) to Figure 8 , see P4-Safety and P4-Safety-Offset in Table 1 for specific data. As shown in Figure 5-(a), Figure 5-(b), Figure 5-(c), Figure 6-(a) and Figure 6-(b), we can see again that in order to reduce fuel consumption, within the limited machine accuracy range, the end state of the formation produces an optimized offset (125.10m) in the y direction, and the offset in the x and z directions is 0. Figure 8 , the distance between P4-Offset satellites 1 and 2 at 5280s is 48.75m, and the collision cannot be avoided. However, the relative motion trajectories of P4-Safety and P4-Safety-Offset satellites are safe, that is, the distance between any two satellites is not less than the safety distance r safe =49m. As shown in Figure 7-(a), Figure 7-(b) and Table 1, compared with P4-Safety, the maximum fuel consumption of a single satellite in P4-Safety-Offset is reduced by 51.5%, while the total fuel usage of the formation is reduced by 19%, which illustrates the beneficial effect of using offsets and iterative processes.

[0202] By comparing P3-Safety-Offset and P4-Safety-Offset, it can be seen that P3-Safety-Offset not only balances the fuel usage, but also minimizes the total fuel usage of the formation, while P4-Safety-Offset only balances the fuel usage of the formation. Obviously, P3-Safety-Offset in Case 1 provides better optimization results than P4-Safety-Offset. This is because the initial and terminal states selected in Case 1 are both uniformly arranged follow-flight formations, which makes P3-Safety-Offset and P4-Safety-Offset get the same offset. In fact, P3-Safety-Offset and P4-Safety-Offset each have their own advantages and disadvantages. For a specific analysis, see Case 2.

[0203] Case 2 (a space circle formation with a radius of 50m becomes a following formation with a following distance of 50m)

[0204] Assume that the initial configuration is a space circle configuration, the radius of the space circle R = 50m, θ = (j-1)·π / 3, and the initial state of satellite j is x j (0) can be calculated by formula (5), (note that when calculating, the average latitude argument u=0), j∈{1,...,6}; X 0 =[x 1 (0),x 2 (0),...,x 6 (0)] T The expected configuration is a follow-up configuration with a follow-up distance of aδλ=50(j-1)m, that is, when no offset is introduced, the terminal state of each star is x j (t f )=[0,50(j-1),0,0,0,0] T ; When the offset is introduced, x j (t f )=[x r ,50(j-1)+y r ,z r ,0,0,0] T , In order to show the effect of collision avoidance constraints, set the safety distance r safe is 30m.

[0205] Table 2. Fuel consumption optimization results of space circle formation to follow-fly formation

[0206]

[0207] The simulation results of Case 2 are shown in Figure 9-(a) to Figure 16 , see Table 2 for specific data.

[0208] For Problem 3, the total fuel consumption optimization problem, the simulation results are shown in Figure 9-(a) to Figure 12 Specific data can be found in P3-Safety and P3-Safety-Offset in Table 2. As shown in Figures 9-(a), 9-(b), 9-(c), 10-(a) and 10-(b), within the limited machine accuracy range, the state of the end of the formation has an obvious offset in the y direction (-123.81m), and the offset in the x and z directions is 0. Fig.12 , the distance between P4-Offset satellites 4 and 6 in 2910s is 23.27m, and the collision cannot be avoided. However, the relative motion trajectories of P3-Safety and P3-Safety-Offset satellites are safe, that is, the distance between any two satellites is not less than the safety distance r safe=30m. As shown in Figure 11-(a), Figure 11-(b) and Table 2, compared with P3-Safety, the maximum fuel usage of a single satellite in P3-Safety-Offset is reduced by 15.8%, and the total fuel usage of the formation is reduced by 7.1%, which illustrates the beneficial effect of using offsets and iterative processes.

[0209] For Problem 4, the balanced fuel consumption problem, the simulation results are shown in Figure 13-(a) to Figure 16 , see P4-Safety and P4-Safety-Offset in Table 2 for specific data. As shown in Figures 13-(a), 13-(b), 13-(c), 14-(a) and 14-(b), in order to reduce fuel consumption, within the limited machine accuracy range, the end state of the formation produces an optimized offset (-143.63m) in the y direction, and the offset in the x and z directions is 0. Fig.16 , the distance between P4-Offset satellites 4 and 6 at 2910s is 22.48m, and the collision cannot be avoided. However, the relative motion trajectories of P4-Safety and P4-Safety-Offset satellites are safe, that is, the distance between any two satellites is not less than the safety distance r safe =30m. As shown in Figure 15-(a), Figure 15-(b) and Table 2, compared with P4-Safety, the maximum fuel consumption of a single satellite in P4-Safety-Offset is reduced by 18.2%, while the total fuel usage of the formation is reduced by 10.3%, which illustrates the beneficial effect of using offsets and iterative processes.

[0210] By comparing P3-Safety-Offset and P4-Safety-Offset, we can see that they each have their own advantages and disadvantages. P3-Safety-Offset minimizes the total fuel consumption of the formation, but does not balance the fuel use. P4-Safety-Offset balances the fuel use, but does not minimize the total fuel consumption of the formation.

[0211] Aiming at the fuel optimization problem of orbit transfer control of distributed satellite formations within a fixed reconstruction time, the present invention specifically defines the optimal total fuel consumption and fuel balance of the formation respectively; for the formation whose terminal state is the following flight configuration, a position offset is introduced, and the convex programming framework is used to solve the two optimization problems of the optimal total fuel consumption of the formation and the optimal total fuel consumption considering fuel balance under the consideration of maximum thrust constraints and anti-collision constraints. The simulation results verify that after introducing the offset, the fuel consumption of the orbit transfer process can be significantly reduced, and the fuel balance problem can also be solved, which verifies the effectiveness of the implementation plan.

[0212] The present invention defines the fuel optimization problem in the framework of the primary satellite orbital coordinate system (LVLH), and the optimization problem under the collision avoidance constraint will be easier to solve. The present invention defines the total fuel optimization problem and the fuel balance problem of the formation in the LVLH coordinate system, considering the collision avoidance constraint and the maximum thrust constraint, and introducing the terminal state offset. The present invention adopts the convex programming method to solve the problem. The convex programming method requires the problem to be discretized. Since the formation dynamics equation is sparse after discretization, the convex programming method has a small amount of calculation and saves time when solving multi-constraint and multi-variable optimization problems; and since the local optimal solution of the convex programming problem must be the global optimal solution, it is very suitable for multi-constraint path planning and optimal control of multi-satellite formations.

[0213] The present invention constructs the formation total fuel consumption optimization and fuel consumption balance problems considering multi-satellite formation, maximum thrust and inter-satellite anti-collision constraints, which can be used to optimize the formation total fuel consumption and balance the inter-satellite fuel consumption respectively; in formation control, the present invention introduces a position offset that can be changed with fuel consumption optimization at the formation terminal state, which reduces the formation total fuel consumption on the one hand and makes the fuel consumption balance advantage more significant on the other hand. For the formation total fuel consumption optimization problem, the introduction of the terminal state offset is equivalent to shortening the transfer path of the entire formation, thereby reducing the formation total fuel consumption; for the fuel consumption balance problem, the solved terminal state offset can minimize the maximum fuel consumption of a single satellite, thereby achieving the purpose of extending the formation life.

Claims

1. A distributed satellite formation control fuel consumption optimization method, characterized in that: The following steps are involved: Step 1: Description of the total fuel consumption optimization and fuel consumption balance problem of distributed satellite formation under multiple constraints; Step 2: Convert the problem defined in step 1 into a convex programming standard form and introduce the end offset; Step 3: Use the convex programming solver to solve the optimal total fuel consumption problem of the formation with the introduction of terminal offset and the optimal total fuel consumption problem considering fuel balance; The specific content of step one is as follows: S1.1 Establish a linear discrete relative motion model: Define a primary satellite orbit coordinate system LVLH, where x, y, and z are radial, along-track, and tangential directions, respectively; the dynamic equation of the motion of a single secondary spacecraft relative to the primary satellite in a near-circular orbit is expressed as: in, are the accelerations of the secondary satellite in radial, along-track and tangential directions, are the velocities of the secondary satellite in radial, along-track, and tangential directions, μ is the Earth's gravitational constant, R c is the orbital radius of the main star, is the average angular velocity of the primary star, T x , T y , T z They are the thrusts of the x, y, and z axes, including control force and perturbation force; Equation (1) is linearized to obtain the HCW equation. The discretized HCW equation analytical solution can obtain the state transfer matrix (STM) based on the HCW equation: Among them, x k ,y k and z k are the positions of the secondary star relative to the primary star on the x, y, and z axes at time k, and are the corresponding relative velocities respectively. Similarly, x k+1 ,y k+1 、z k+1 , is the relative position and velocity of the three axes of the secondary satellite at time k+1, and T is the discretization step size; S1.2 Obtain the initial and terminal states of the formation through the formation configuration design equation: Customize the six numbers of the main star orbit α c , and customize the formation configuration parameters according to the needs, and obtain the initial and terminal states of the formation from the formation configuration design equation; S1.3 defines the total fuel consumption optimization and fuel consumption balance problems under multiple constraints, including kinematic constraints, initial and terminal state constraints, maximum thrust constraints and anti-collision constraints: Define x j (t) is the state of a single satellite in the LVLH coordinate system at time t, X(t) is the state of the formation at time t; define u j (t) is the instantaneous acceleration of satellite j at time t; Consider a formation consisting of N (N ≥ 2) stars, with a fixed orbit transfer time (0, t f ) is controlled by applying fuel u j (t), j = 1, 2, ..., N, from the known initial state X0 to the target state X f ; where t f is the reconstruction time, relatively, x j (0), x j (t f ) are the initial and final states of satellite j, j = 1, 2, ..., N; One goal of formation maintenance or reconstruction is to minimize the total fuel consumption J of the formation. t , the objective can be described as the L1 norm of the control quantity: Another objective function J b To balance the fuel usage of each satellite in the formation, it is defined as: The orbit transfer process is constrained by the relative motion equation of formula (1), and the initial and final states are: X(0)=X0,X(t f )=X f (5), Without loss of generality, we force the control input u of each satellite to be j (t) not exceeding the maximum thrust acceleration U max ,Right now: In order to ensure that there is no collision between satellites during the control process, the distance between any two satellites during the mandatory control process is always greater than the preset safety distance r. safe , that is, for i,j∈{1,2,...,N} all have: ||C[x j (t)-x i (t)]||≥r safe (7), Where C = [I 3×3 0 3×3 ]; Therefore, the problem of minimizing the total fuel consumption can be described as: The fuel balance problem is described as: The specific content of step 2 is as follows: S2.1 Discretization objective function: Consider a formation of N (N ≥ 2) satellites that completes orbit transfer within K steps, with a transfer time step of Δt; the state (position, speed) of satellite j at time k is x j (k), combined with x j (k) and control quantity u j (k) k∈{1,2,...,K-1}; Change J in S1.3 t and J b Discretize and write it in standard convex programming form: Among them, y j is the column vector composed of the discretized single star state and control quantity, Y is the total number of y in the formation j The column vector, H j and Used to extract the control quantity u from Y j ; S2.2 Discretize the single star dynamics and expand it to the dynamics of multiple star systems: Under the discrete model, the dynamic equation of a single satellite j is defined as: x j (k+1)=Ax j (k)+Bu j (k) (12), Where A is the state transfer matrix in equation (2), and B is the control matrix. For satellite j, the relative motion equations at all discrete times 1, 2, ..., K-1 can be integrated into the following homogeneous form: Among them, A sd is the coefficient matrix of the dynamic equations of a single star under all discrete times. Equation (13) is expanded to the formation dynamic equation of N stars, and we get: in, is the coefficient matrix of the dynamic equations of the formation at all discrete times; S2.3 Establish the initial and final states of multiple stars, and introduce the final offset into the final state: Define a new terminal state It is the original terminal state X f With unknown offset [x r ,y r ,z r ,0,0,0] T The unknown offset can be obtained by solving the fuel consumption optimization problem in S3.1 or S3.2, that is, the new terminal state of satellite j is x j (K)+[x r ,y r ,z r ,0,0,0] T ; Define the matrix and They are used to extract the initial and final states of the formation from Y, respectively, and we get: S2.4 Discretized multi-satellite maximum thrust constraints: In discrete time, the maximum thrust constraint in S1.3 is written as follows: Among them, the matrix Used to extract the control value u of satellite j at time k from the full state column vector Y j (k), vector 1 is determined by the thruster configuration. If thrusters are used on all three axes, then 1 = [1,1,...,1] T , length is 6N(K-1); if the thrust is only applied in the y and z directions, then 1=[0,1,1,...,0,1,1] T , the length is 6N(K-1); S2.5 Convexify the multi-star collision avoidance constraints and discretize them: The collision avoidance constraint (10) is a typical non-convex function that does not meet the requirements of convex programming and needs to be further rewritten into the following quadratic convex form: in, and are the estimated states of satellites i and j at time k, respectively. and It is obtained by solving the fuel optimization problem without collision avoidance constraints; in order to extract the relative positions of satellites i and j (i < j) from Y, the matrix C at time k can be rewritten as definition and At this time, the collision avoidance constraint (17) is rewritten as: The specific content of step three is as follows: S3.1 uses a convex programming solver to solve the optimal total fuel consumption problem of the formation with terminal offset introduced; Among them, the total fuel consumption minimization problem in discrete time is defined as: S3.2 uses a convex programming solver to solve the optimal total fuel consumption problem of the formation with the introduction of terminal offset and consideration of fuel consumption balance; Among them, the fuel consumption balance problem is defined as:

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