CN108614420B - Satellite cluster level satellite fault-tolerant control method based on nonlinear programming - Google Patents

Abstract

The invention relates to a fault-tolerant control method for a satellite cluster level satellite, which designs a reasonable and effective fault-tolerant control strategy for a satellite cluster level satellite formation system based on a microsatellite formation technology, so that a formation configuration can be adjusted by utilizing a nonlinear dynamic programming technology under the condition that a formation satellite fails, and the task requirement is continuously completed, and the fault-tolerant control method for the satellite cluster level satellite based on the nonlinear programming is characterized in that after the satellite fails in the formation process, the optimal leaving and entering positions of a failed satellite and a backup satellite are determined by determining the initial and final positions of the failed satellite and the backup satellite, considering inter-satellite collision and time constraint and taking fuel consumption as performance indexes, a Gaussian pseudospectral method is utilized to determine the optimal leaving and entering positions of the failed satellite and the backup satellite, and respective optimal trajectory planning is obtained, the entry of the backup satellite can be completely filled up the position of the failed satellite, and a new formation is formed together with healthy satellites, the performance requirements of the formation are realized. The fault-tolerant control method is mainly applied to fault-tolerant control occasions of the satellite at the satellite cluster level.

Description

Satellite cluster level satellite fault-tolerant control method based on nonlinear programming

Technical Field

The invention relates to a fault-tolerant control method for a satellite cluster level satellite based on nonlinear programming, which is mainly applied to the process of fault occurrence and formation reconfiguration of formation satellites during in-orbit work and belongs to the field of aerospace satellite formation control. In particular to a satellite cluster-level satellite fault-tolerant control method based on nonlinear programming.

Background

In recent years, with the increasing complexity of space mission requirements, the traditional single satellite is far from meeting mission requirements. The vigorous development of the microsatellite technology enables a plurality of satellites to form a formation to cooperatively work to jointly complete a complex task target, and becomes the mainstream direction of the development of the space technology. The multi-satellite formation flying means that a plurality of satellites maintain a space configuration which changes along with time, each satellite realizes system coupling through an inter-satellite link, common navigation and integral cooperative control are carried out through a cooperative strategy, tasks are completed through mutual cooperation, and a huge 'virtual satellite' is formed on the overall performance, so that the multi-satellite formation flying system has the advantages that a single satellite is incomparable, and the functions which are difficult to realize by the traditional single satellite can be realized.

Formation flying becomes the enabling technology of many future space missions due to the characteristics of long base line, flexible configuration, low cost, short development period and rapid launching and networking. However, in the flight process of the satellite, due to the fact that satellite resources and manual intervention capacity are limited, the space monitoring environment is severe, uncertainty factors are large and the like, a satellite flywheel or a thruster and the like are prone to failure, and therefore a certain deviation is generated between the actual input torque and the control command torque, and the control performance of the system is affected. Patent CN201610026039.X researches a reconfigurable actuator configuration design method for satellite attitude control system faults to reconstruct the satellite attitude control system actuator faults and ensure that a satellite can still stably operate when the actuator fails. The patent CN201610831816.8 provides an active fault-tolerant control method based on an iterative learning observer aiming at the problems of limited input saturation and external disturbance of an actuator fault in the satellite attitude control process, and the stability of an attitude control system when the actuator fault occurs in an in-orbit working satellite is ensured. The patent CN201610080866.7 is designed to a fault-tolerant controller based on a fault characteristic model aiming at the problem of the actuator fault of a rigid-flexible liquid spacecraft. However, if a certain satellite fails during the formation operation process and the single satellite fault-tolerant control still cannot meet the task requirement of the satellite formation, in order to ensure the overall performance of the satellite formation, the satellite fault-tolerant control in the planetary cluster level is required, so that the function of the whole formation is recovered, which has very important significance for the development of aerospace and space detection technologies in China.

In order to prevent collision in the process of formation flight of the satellites and consider that the satellite resources are limited, the effective reduction of fuel consumption is very important, and the satellite trajectory planning by using an optimization means has very strong engineering significance. The existing common optimization method is mainly divided into an indirect method and a direct method, wherein the indirect method is the most main method applied to optimization in the early stage, and the core idea is to firstly deduce a first-order necessity condition of an optimal control problem based on the Ponderria minimum principle, convert the problem into a two-point edge value problem or a multi-point edge value problem and solve the problem. The conventional direct method has a certain problem because it is difficult to derive the first-order requirement of the solution. The pseudo-spectral method is a direct method which is newly appeared and widely researched by people, overcomes the defect that the direct method is difficult to deduce the first-order necessity condition of a solution, is easier to solve compared with an indirect method, has an exponential convergence speed, and is the most suitable method for solving the complex optimal control problem in the existing method.

Through the search of the prior art, a research patent of formation satellite fault-tolerant control based on Gaussian pseudo-spectral method nonlinear dynamic programming is not found, and the research in the direction is in the front-edge exploration research stage in China or abroad. The invention provides a new solution strategy for beneficially exploring the deep space exploration-oriented satellite formation fault-tolerant control and provides necessary technical reserve for the aerospace industry of China.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention aims to design a reasonable and effective satellite formation fault-tolerant control strategy for a deep space exploration system based on a microsatellite formation technology, so that formation satellites can adjust formation configuration by utilizing a nonlinear dynamic programming technology under the condition of failure, and the task requirement is continuously completed. The invention provides a novel satellite formation fault-tolerant strategy aiming at the problem of satellite faults in the deep space exploration task based on satellite formation, researches the trajectory planning of the formation reconfiguration of the fault star leaving formation and the backup star entering formation in the maneuver, realizes the reconstruction of the configuration of the whole satellite formation under the fault, has universality for the satellite formation of any configuration, and provides a novel idea for the adoption of the microsatellite formation to realize the beneficial exploration of the deep space exploration technology and the high-precision flight of the microsatellite formation under the satellite fault. According to the technical scheme, after a satellite fails in the formation process, the initial and end positions of a failed star and a backup star are determined, inter-satellite collision and time constraint are considered, fuel consumption is taken as a performance index, the optimal leaving and entering positions of the failed star and the backup star are determined by using a Gaussian pseudo-spectrum method, respective optimal trajectory plans are obtained, the situation that the backup star enters can be completely filled up, a new formation is formed together with a healthy star, and the performance requirement of the formation is met.

Specifically, considering from three stages, the control means and functions of each stage are implemented as follows:

stage one: satellite formation configuration design and fly-by-flight trajectory equation determination

(1) Defining a distributed satellite relative coordinate system

1) Center of earth inertial coordinate system

Defining a coordinate origin O as the earth center, taking an equatorial plane as a datum plane, leading an X axis to point to a spring equinox along the intersection line of the earth equatorial plane and a ecliptic plane, leading a Z axis to point to the north pole direction, and forming a right-hand coordinate system by the Y axis, the X axis and the Z axis in the equatorial plane;

2) orbital coordinate system

Defining a coordinate origin as a main satellite mass center, pointing an x-axis to a main satellite direction along the earth center, pointing a y-axis to the x-axis direction in a main satellite orbit plane and pointing to the main satellite motion direction, and keeping a z-axis to the main satellite orbit plane and obeying a right-hand rule;

(2) determining flight path equations for formation satellites

Under the condition of horizontal round formation, in a horizontal plane, the distance between the kth accompanying spacecraft and the reference spacecraft keeps a fixed distance, namely a subsatellite point round formation, and the constraint conditions are met:

y_k ²+z_k ²＝r² (1)

and the projection of the horizontal circle formation plane in the xz plane can be obtained:

z_k＝±2x_k (2)

wherein x is_k,y_k,z_kShowing the relative sitting of the kth orbiting satellitePosition coordinates in the frame. The "+" and "-" in equation (2) indicate that in each horizontal circular formation there are two planes, the projections of which in the xz plane are both straight and intersect along the y-axis, but one plane is inclined at 26.565 degrees and the other at-26.565 degrees. Further derivation of formula (2) to yield

Therefore, the initial value of the horizontal circle formation of the kth flying satellite meets the following conditions:

z_k0＝x_k0

the relative position and relative speed between the slave star and the master star under the horizontal round formation are as follows:

wherein x is_k,y_k,z_kThe relative position of the kth orbiting flying satellite and the main satellite,

the relative velocity of the kth orbiting flying satellite and the main satellite,

the initial condition of the satellite is that n is the average motion angular velocity of the reference spacecraft, and the two initial conditions are represented by the formulas (5) and (6) by the radius r of a horizontal circle and the phase angle theta of the initial position of the kth flying satellite_kRepresents:

therefore, only the radius of the track and the initial phase angle θ are determined_kObtaining the initial position of the kth satellite so as to obtain the position of the satellite in the horizontal circle formation at any time;

and a second stage: failed star leaving formation stage

In the process of leaving the fault star, the selection of the initial and final value positions directly influences the consumption of fuel and the realization of collision constraint, under the condition of collision constraint, multiple optimization results of different positions are compared based on a Gaussian pseudo-spectral method and a sequential quadratic programming algorithm, a relation curve between the phase angle of the position where the fault star leaves the formation and the fuel consumption is solved, and the fault star leaving track which saves the fuel most is determined;

and a third stage: backup star entering formation phase

Under the condition that the maneuvering initial position of the backup star is fixed, the position of the backup star entering the formation directly influences the fuel consumption and the implementation of collision constraint, under the condition of collision constraint, multiple optimization results of different final value positions are compared based on a Gaussian pseudo-spectral method and a sequence quadratic programming algorithm, a relation curve between a phase angle of the backup star entering the formation position and the fuel consumption is solved, and the most fuel-saving backup star butt-joint position and the best maneuvering starting time are determined.

The two-step refinement of the stage is as follows:

1) constraint conditions

Constraint conditions of the satellite formation configuration optimization process comprise edge value constraint, inter-satellite collision avoidance constraint, output torque constraint and configuration optimization time constraint;

and (3) edge value constraint:

initial value: simply by determining the radius of orbit and the initial phase angle theta of the kth satellite_kAnd obtaining the initial position of the satellite, so that the position of the satellite in the horizontal circular formation at any time can be obtained, and assuming that the orbit radius is r, the initial position of the fault star i is located on the horizontal circular formation with the radius r, and the initial x-direction position and speed of the fault star i are:

wherein, theta_iThe initial phase angle of the fault star i, n is the average motion angular velocity of the reference spacecraft, and r is the horizontal circular formation radius. Based on the initial conditions of (5) to (6) and the horizontal round formation equation, obtaining the arbitrary position and speed of the fault star i on the horizontal round formation:

final value: suppose the final position phase angle of the fault star i is alpha_iAt a distance 2r from the central primary star, the final position and velocity of the failed star i can be determined as:

wherein x is_if,y_if,z_ifIs the final value position of the failed star i,

the final value speed of the fault star i;

collision avoidance restraint: in the process of the satellite formation movement, a collision avoidance strategy based on a safety domain is adopted, the safety distance is set to be d in consideration of the size of a satellite platform and the size of the formation configuration, and collision avoidance of the main star and the fault star and collision avoidance constraint between the fault star and the healthy star are set to be d

Wherein x is_j,y_j,z_jFor formation of satellites other than the failed satelliteRelative distance of the remaining healthy star to the primary star.

Thrust force restraint:

because the output amplitude of the thruster is limited, the output amplitudes of 6 thrusters in three directions are required not to exceed the maximum amplitude, which is expressed as:

wherein u is_xk,u_yk,u_zkRepresents the thrust of the kth satellite in the x, y and z directions, u_maxRepresenting the maximum amplitude of the thruster;

and (3) time constraint:

and limiting the leaving optimization time of the fault star i according to specific conditions:

tf≤t_max (14)

2) performance index

The performance index is the minimum total energy consumption of the failed star i, and is as follows:

minJ＝∫(|u_xi|+|u_yi|+|u_zi|) (15)

wherein u is_xi,u_yi,u_ziThrust accelerations of the fault star i in the x direction, the y direction and the z direction are respectively;

therefore, the performance index and each constraint condition of the fuel economy need to be comprehensively considered, and the fault star i arrival phase angle is determined to be alpha based on the Gaussian pseudo-spectrum method combined with the sequence quadratic programming_iAnd an optimum off-position phase angle theta having a radius of 2r_iAnd designing an optimal track, specifically, solving an optimal control problem by adopting a Gaussian pseudo-spectrum method: discretizing the state variable and the control variable on a series of discrete Gaussian points, constructing Lagrange interpolation polynomial on approximate state variable and control variable by taking the points as nodes, transforming a differential equation of a motion trail of a relative dynamic model describing satellite motion into a set of algebraic constraints by deriving the Lagrange interpolation polynomial on the approximate state variable with respect to time, calculating an integral term existing in a desired performance index by Gaussian integration, and calculating the integral term based on the aboveAnd the constraint conditions, the performance indexes and the like are continuously and circularly optimized by using a Gaussian pseudo-spectrum method to obtain the optimal position of the fault star.

(1) Converting the time interval of the optimal control problem to the time interval applying Gauss pseudo-spectrum method through time domain conversion:

wherein t is₀Representing the initial time of trajectory optimization, t_fRepresents the termination time of the trajectory optimization, and tau represents the time interval satisfying the Gaussian pseudo-spectral method, and the arbitrary trajectory optimization time interval is converted into [ -1,1]Preparing for discretization of a Gaussian pseudo-spectrum method;

(2) the matching points are selected from LG points of K order, namely roots of Legendre polynomials of K order:

(3) selecting nodes: here, discrete K LG points and an initial time point tau are used₀As discrete nodes

Wherein g (τ) ═ 1+ τ) P_N(τ)，X(τ_i) T represents the state variable X_iA discrete point. The above equation gives a discretization method of the state variables. Performing polynomial summation on the state variable discrete points, wherein coefficients of the polynomial are Lagrange interpolation polynomials, so that the state variable is discretized;

(4) the expression of discretization of the control quantity is as follows:

in the formula, U (τ)_i) T represents the state variable U_iThe discretization method of the control quantity is consistent with that of the state variable, and the control variable is discretized by summing polynomials of the control variable discretization points;

(5) and (3) final value constraint in a discrete state:

for gaussian pseudo-spectral method, the discretization process of the state variable and the control variable does not contain a terminal point, and here, the terminal value constraint is discretized through the processing of integral, and the discretization method is as follows:

wherein w_kA weighting coefficient representing discretization of the integral object f;

(6) since the state variable derivatives exist in the satellite attitude dynamics and kinematic model constraints, the method for discretizing the state variable derivatives is given here as follows:

polynomial weighting coefficient D for discretization of state variable derivative_kiThat is by direct p-Lagrangian polynomial L_i(τ) is derived.

The three stages are detailed as follows:

(1) determining optimal entry formation locations for backup stars

1) Constraint conditions

And (3) edge value constraint: initial value: the initial position of the backup star assumes a distance of 3r from the virtual center star and a phase angle of β, and the initial position and speed of the backup star are as follows:

wherein x is_b0,y_b0,z_b0To back up the initial relative positions of the star and the primary star,

the initial relative velocity of the backup star to the primary star.

Final value: the final position of the backup star is located on a horizontal circular formation with radius r, and the final phase angle is set to be mu (0 degrees to 360 degrees), then the final position and speed of the backup star can be determined as:

wherein x is_bf,y_bf,z_bfTo back up the relative positions of the end values of the stars and the primary star,

the relative speed of the backup star and the final value of the main star is used;

collision avoidance restraint: considering the size of the satellite platform and the size of the formation configuration, the set safety distance is dm, and the collision avoidance constraints of the main star and the backup star and the healthy star are as follows:

wherein x is_b,y_b,z_bRepresenting the relative position of the backup star and the main star under the orbit coordinate system of the main star

Thrust force restraint:

because the output amplitude of the thruster is limited, the output amplitudes of 6 thrusters in three directions are required not to exceed the maximum amplitude, which is expressed as:

wherein u is_xk,u_yk,u_zkRepresents the thrust of the kth satellite in the x, y and z directions, u_maxRepresenting the maximum amplitude of the thruster;

and (3) time constraint:

because the minimum fuel is used as a performance index, in order to avoid overlarge optimization time, the backup star entering optimization time needs to be limited according to specific conditions:

tf≤t_max (27)

2) performance index

In order to reduce the fuel consumption in the optimization process of leaving the formation of the backup stars, the total energy consumption of the backup stars is the minimum performance index, which is as follows:

minJ＝∫(|u_xb|+|u_yb|+|u_zb|) (28)

wherein u is_xb,u_yb,u_zbThe thrust acceleration of the backup star in the x direction, the y direction and the z direction is respectively.

Therefore, the optimal final value position of the backup star, namely the termination phase angle β, needs to be determined based on a gaussian pseudo-spectral method in combination with a quadratic sequence planning algorithm, considering the performance index of fuel economy and each constraint condition under the condition of a fixed final value position;

(2) determining selection of time for backup stars to enter formation

Supposing that the phase of the optimal position of the backup star entering the formation is determined to be mu in the last step_bestUnder the condition of determining the position of the starting position and the position of the entering formation point of the motor, the entering time of the backup star and the motor time t thereof_jAnd the orbit period T of the central star, the phase of the virtual fault star in the process of the backup star reaching the selected position is

Time t_j(ii) a The entry time of the backup star is represented by the phase μ of the virtual position of the failed star, that is, the backup star starts to enter when the failed star runs to which phase under the condition that the failed star still exists:

according to the divided task stages, constraint conditions, performance indexes and the like of satellite trajectory optimization in each stage are calculated by combining satellite orbit information, formation configuration information and the like, so that formation satellites are better guided to achieve a set task target.

The invention has the characteristics and beneficial effects that:

the satellite fault-tolerant control technology based on the nonlinear programming comprehensively considers the problems of track planning and determination of fault satellites leaving formation and backup satellites entering formation when faults occur in the satellite formation maneuvering process, discusses the optimal position selection of the fault satellites leaving formation and the backup satellites entering formation, considers the constraint conditions such as collision and the like, determines the maneuvering starting time of the satellites, can enable the backup satellites and the healthy satellites to form an ideal formation, and can realize the most fuel saving under the condition of avoiding collision. The technology has good engineering practicability and application value, can effectively perform fault-tolerant control on satellite formation, and has important significance for improving the aerospace detection technology of China.

Social and economic benefits: the satellite formation fault-tolerant control technology provided by the invention has important reference significance for solving the problem of current micro-satellite formation flight, and can provide technical reserve for realizing formation cooperative control of on-orbit satellites. China is a few countries developing satellite formation cooperative control and application research internationally, the breakthrough of the technology can greatly enrich astronomical observation means, and the technology has important social value and military value in the fields of national economic construction and national defense safety.

Description of the drawings:

FIG. 1 is a flow chart of the steps of a method of implementing the present invention.

FIG. 2 illustrates a geocentric inertial frame and a satellite orbital frame.

FIG. 3 is a diagram of a horizontal round formation configuration.

Fig. 4 is an overall schematic diagram of fault-tolerant control of satellite formation.

FIG. 5 is a graph of energy consumption versus initial phase angle for a failed star.

FIG. 6 is a schematic diagram of a failed star departure formation.

FIG. 7 is a two-dimensional and three-dimensional trajectory graph of each star during the process of formation of a failed star departure.

FIG. 8 is a graph of the location, velocity thrust and distance from the primary star for the failed star.

FIG. 9 is a graph of phase angle versus fuel consumption for a backup star entering formation.

Fig. 10 is a schematic diagram of backup star entry formation.

FIG. 11 is a backup of two-dimensional and three-dimensional trajectory plots of stars during the formation of the star entry.

Figure 12 is a graph of the position, velocity thrust and distance from the primary star for the backup star.

Detailed Description

Aiming at the situation that in the process of completing a deep space exploration task by a satellite in a sub-satellite horizontal round formation configuration, the whole formation is difficult to meet the required task precision requirement under the condition that a certain component such as a satellite actuator, a sensor or an on-satellite radar in the satellite formation fails, the invention firstly provides a satellite formation fault-tolerant control strategy based on nonlinear programming; the method comprises the steps of determining the optimal departure formation position of a failed star by taking the overall performance of a satellite formation as an implementation target and considering the problem of fuel consumption, enabling the failed star to exit the formation, timely supplementing a new backup star to enter the formation, and solving the optimal position of the backup star entering the formation and the optimal starting time of the maneuver so as to ensure that the position of the failed star can be completely supplemented and the function of the satellite formation can be recovered.

The invention solves the problem of fault-tolerant control of satellite formation by combining a Gaussian pseudo-spectral method with a sequence quadratic programming method for the first time. The problems of collision, exceeding of a flywheel or a thruster torque constraint and the like easily occur in the formation flight process of the satellite, the satellite resources are very limited, and the problem of fuel consumption needs to be taken into consideration, so that the satellite trajectory planning by using an optimization means has strong engineering significance. The Gaussian pseudo-spectrum method is widely applied as an optimization algorithm, can overcome the defect that the traditional method is difficult to derive the first-order necessary condition of the solution, and is easier to solve. After the satellite breaks down in the formation process, the initial and end positions of the fault star and the backup star are determined, constraints such as inter-star collision and time are considered, fuel consumption is used as a performance index, the optimal leaving and entering positions of the fault star and the backup star are determined by using a Gaussian pseudo-spectral method, the optimal trajectory plans of the fault star and the backup star are obtained, the situation that the backup star enters can be completely filled up, a new formation is formed together with the healthy star, and the performance requirements of the formation are met.

In the following, three stages are considered, and the control means and functions of each stage are implemented as follows:

stage one: satellite formation configuration design and fly-by-flight trajectory equation determination

(1) Defining a distributed satellite relative coordinate system

1) Center of earth inertial coordinate system

Defining a coordinate origin O as the earth center, taking an equatorial plane as a reference plane, leading an X axis to a vernality point along an intersection line of the earth equatorial plane and a ecliptic plane, leading a Z axis to a north pole direction, and forming a right-hand coordinate system with the X axis and the Z axis in the equatorial plane by a Y axis.

2) Orbital coordinate system

And defining a coordinate origin as a main satellite mass center, pointing the x axis to the main satellite direction along the earth center, perpendicular to the x axis direction in the main satellite orbit plane and pointing to the main satellite motion direction on the y axis, and perpendicular to the main satellite orbit plane on the z axis, and obeying the right-hand rule.

(2) Determining flight path equations for formation satellites

Under the condition of horizontal circle formation, in a horizontal plane (yz plane), the distance between the kth accompanying spacecraft and a reference spacecraft is kept at a fixed distance, namely the kth accompanying spacecraft is formed into a circle with a point below a satellite, and the constraint conditions are met:

y_k ²+z_k ²＝r² (1)

and the projection of the horizontal circle formation plane in the xz plane can be obtained:

z_k＝±2x_k (2)

wherein x is_k,y_k,z_kTo representThe position coordinates of the kth orbiting satellite in the relative coordinate system. The "+" and "-" in equation (2) indicate that in each horizontal circular formation there are two planes, the projections of which in the xz plane are both straight and intersect along the y-axis, but one plane is inclined at 26.565 degrees and the other at-26.565 degrees. Further derivation of the formula (2) gives

Therefore, the initial value of the horizontal circle formation of the kth flying satellite meets the following conditions:

z_k0＝x_k0

the relative position and relative speed between the slave star and the master star under the horizontal round formation are as follows:

wherein x is_k,y_k,z_kThe relative position of the kth orbiting flying satellite and the main satellite,

the relative velocity of the kth orbiting satellite and the main satellite.

And n is the average moving angular velocity of the reference spacecraft, which is the initial condition of the satellite. From equations (5) and (6), it can be seen that the horizontal circular formation has only these two degrees of freedom. The two initial conditions can be vividly represented by the radius r of a horizontal circle and the phase angle theta of the initial position of the kth orbiting satellite_kRepresents:

therefore, only the radius of the track and the initial phase angle θ are determined_kThe initial position of the kth satellite, and thus the position of the satellite in the horizontal round formation at any time, is obtained.

And a second stage: failed star leaving formation stage

In the process of leaving the fault star, the selection of the initial and final value positions directly influences the consumption of fuel and the realization of collision constraint, under the condition of collision constraint, multiple optimization results of different positions are compared based on a Gaussian pseudo-spectrum method and a sequential quadratic programming algorithm, a relation curve between the phase angle of the position where the fault star leaves the formation and the fuel consumption is solved, and the fault star leaving track which saves the fuel most is determined.

1) Constraint conditions

The constraint conditions of the satellite formation configuration optimization process comprise edge value constraint, inter-satellite collision avoidance constraint, output torque constraint and configuration optimization time constraint.

And (3) edge value constraint:

initial value: as known from the preliminary knowledge of the horizontal round formation, the orbit radius and the initial phase angle theta of the kth satellite are determined_kThe initial position of the satellite, and thus the position of the satellite in the horizontal round formation at any time, is obtained. Assuming that the track radius is r, the initial position of the fault star i is located on the horizontal circular formation with the radius r, and the position and the speed of the available fault star i in the initial x direction are as follows:

wherein, theta_iThe initial phase angle of the fault star i, n is the average motion angular velocity of the reference spacecraft, and r is the horizontal circular formation radius. Based on the initial conditions of (5) to (6) and the horizontal round formation equation, the arbitrary position and speed of the fault star i on the horizontal round formation can be obtained:

final value: suppose the final position phase angle of the fault star i is alpha_iAt a distance 2r from the central primary star, the final position and velocity of the failed star i can be determined as:

wherein x is_if,y_if,z_ifIs the final value position of the failed star i,

is the final speed of the failed star i.

Collision avoidance restraint: and in the process of the formation movement of the satellites, a collision avoidance strategy based on a safety domain is adopted. Considering the size of the satellite platform and the size of the formation configuration, the safety distance is set to be d, and the collision avoidance of the main star and the fault star and the collision avoidance constraint between the fault star and the healthy star are

Wherein x is_j,y_j,z_jThe relative distance of the remaining healthy stars in the formation satellites, except for the failed star, from the primary star.

Thrust force restraint:

because the output amplitude of the thruster is limited, the output amplitudes of 6 thrusters in three directions are required not to exceed the maximum amplitude, which is expressed as:

wherein u is_xk,u_yk,u_zkRepresents the thrust of the kth satellite in the x, y and z directions, u_maxRepresenting the maximum amplitude of the thruster.

And (3) time constraint:

since the minimum fuel is used as a performance index, in order to avoid the optimization time from being too large, the time for the fault star i to leave the optimization time needs to be limited according to specific conditions:

tf≤t_max(14)

2) performance index

In order to reduce the fuel consumption in the optimization process of leaving the formation of the failed star i, the total energy consumption of the failed star i is taken as a performance index, and the following steps are carried out:

minJ＝∫(|u_xi|+|u_yi|+|u_zi|) (15)

wherein u is_xi,u_yi,u_ziThe thrust acceleration of the fault star i in the x direction, the y direction and the z direction is respectively.

Therefore, the performance index and each constraint condition of the fuel economy need to be comprehensively considered, and the fault star i arrival phase angle is determined to be alpha based on the Gaussian pseudo-spectrum method combined with the sequence quadratic programming_iAnd an optimum off-position phase angle theta having a radius of 2r_iAnd designing an optimal trajectory.

Solving an optimal control problem by adopting a Gaussian pseudo-spectrum method: discretizing the state variable and the control variable on a series of Gaussian points after discretization, and constructing Lagrangian interpolation polynomial by taking the points as nodes to approximate the state variable and the control variable. The differential equations describing the motion trajectory of the relative dynamical model of satellite motion can be converted into a set of algebraic constraints by deriving the derivative of the approximate state variable with respect to time for the lagrange interpolating polynomial, the integral term present in the desired performance indicator being calculated by gaussian integration. And (4) continuously and circularly optimizing by using a Gaussian pseudo-spectrum method based on the constraint conditions, the performance indexes and the like to obtain the optimal position of the fault star.

(1) Converting the time interval of the optimal control problem to the time interval applying Gauss pseudo-spectrum method through time domain conversion:

wherein t is₀Representing the initial time of trajectory optimization, t_fRepresenting the termination time of the trajectory optimization. τ represents a time interval satisfying the gaussian pseudo-spectrometry. Converting an arbitrary trajectory optimization time interval to [ -1,1 ] by the above equation]And preparing for discretization of the Gaussian pseudo-spectrum method.

(2) The matching points are selected from LG points of K order, namely roots of Legendre polynomials of K order:

(3) selecting nodes: here, discrete K LG points and an initial time point tau are used₀As discrete nodes

Wherein g (τ) ═ 1+ τ) P_N(τ)，X(τ_i) T represents the state variable X_iA discrete point. The above equation gives a discretization method of the state variables. And carrying out polynomial summation on the state variable discrete points, wherein the coefficients of the polynomial are Lagrange interpolation polynomials, so that the state variable is discretized.

(4) The expression of discretization of the control quantity is as follows:

in the formula, U (τ)_i) T represents the state variable U_iA discrete point. The discretization method of the control quantity is consistent with the state variable, and the control variable is discretized by summing polynomials of discrete points of the control variable.

(5) And (3) final value constraint in a discrete state:

for gaussian pseudo-spectral method, the discretization process of the state variable and the control variable does not contain a terminal point, and here, the terminal value constraint is discretized through the processing of integral, and the discretization method is as follows:

wherein w_kA weight coefficient representing the discretization of the integral object f.

(6) Since the state variable derivatives exist in the satellite attitude dynamics and kinematic model constraints, the method for discretizing the state variable derivatives is given here as follows:

it can be seen that the state variable derivative discretization polynomial weighting coefficient D_kiThat is by direct p-Lagrangian polynomial L_i(τ) is derived.

The specific process of the discretization of the Gaussian pseudo-spectrum method is given, the process comprises a discretization method of state quantity, control quantity, derivative of the state quantity and final value constraint, and the nonlinear programming problem after discretization is solved through a sequential quadratic programming algorithm to obtain the required optimal track.

And a third stage: backup star entering formation phase

The backup star maneuvering aims to enable the backup star to enter an original horizontal round formation within a specified time after the fault star leaves, completely fill up the virtual position of the fault star, form a corresponding formation with the remaining healthy stars, and continue to complete corresponding tasks. Therefore, under the condition that the maneuvering initial position of the backup satellite is fixed, the position of the backup satellite entering the formation directly influences the fuel consumption and the implementation of collision constraint, under the condition of collision constraint, multiple optimization results of different final value positions are compared based on a Gaussian pseudo-spectrum method and a sequence quadratic programming algorithm, a relation curve between the phase angle of the backup satellite entering the formation position and the fuel consumption is solved, and the most fuel-saving backup satellite butt-joint position and the best maneuvering starting time are determined.

(1) Determining optimal entry formation locations for backup stars

1) Constraint conditions

And (3) edge value constraint: initial value: the initial position of the backup star assumes a distance of 3r from the virtual center star and a phase angle of β, and the initial position and speed of the backup star are as follows:

wherein x is_b0,y_b0,z_b0To back up the initial relative positions of the star and the primary star,

the initial relative velocity of the backup star to the primary star.

Final value: the final position of the backup star is located on a horizontal circular formation with radius r, and the final phase angle is set to be mu (0 degrees to 360 degrees), then the final position and speed of the backup star can be determined as:

wherein x is_bf,y_bf,z_bfTo back up the relative positions of the end values of the stars and the primary star,

the relative speed of the final value of the backup star and the main star is used.

Collision avoidance restraint: considering the size of the satellite platform and the size of the formation configuration, the set safety distance is dm, and the collision avoidance constraints of the main star and the backup star and the healthy star are as follows:

wherein x is_b,y_b,z_bRepresenting the relative positions of the backup star and the primary star in the orbital coordinate system of the primary star. At this time, the failed star has left the formation and is maneuvered to a position directly in front of the central star without colliding with the remaining stars, so collision constraints with the failed star are not considered here.

Thrust force restraint:

because the output amplitude of the thruster is limited, the output amplitudes of 6 thrusters in three directions are required not to exceed the maximum amplitude, which is expressed as:

wherein u is_xk,u_yk,u_zkRepresents the thrust of the kth satellite in the x, y and z directions, u_maxRepresenting the maximum amplitude of the thruster.

And (3) time constraint:

because the minimum fuel is used as a performance index, in order to avoid overlarge optimization time, the backup star entering optimization time needs to be limited according to specific conditions:

tf≤t_max (27)

2) performance index

In order to reduce the fuel consumption in the optimization process of leaving the formation of the backup stars, the total energy consumption of the backup stars is the minimum performance index, which is as follows:

minJ＝∫(|u_xb|+|u_yb|+|u_zb|) (28)

wherein u is_xb,u_yb,u_zbThe thrust acceleration of the backup star in the x direction, the y direction and the z direction is respectively.

Therefore, it is necessary to determine the optimal final value position of the backup star, i.e., the termination phase angle β, by considering the performance index of fuel economy and each constraint condition under the condition of determining the fixed final value position based on the gaussian pseudo-spectral method in combination with the quadratic sequence planning algorithm.

(2) Determining selection of time for backup stars to enter formation

Supposing that the phase of the optimal position of the backup star entering the formation is determined to be mu in the last step_bestUnder the condition of determining the position of the starting position and the position of the entering formation point of the motor, the entering time of the backup star and the motor time t thereof_jAnd the orbital period T of the central star. The backup star arrives at the selected location process (time t)_j) The phase of the middle virtual fault star is

The entry time of the backup star is represented by the phase μ of the virtual position of the failed star, that is, the backup star starts entering when the failed star runs to which phase under the condition that the failed star still exists, so that the following results are obtained:

according to the divided task stages, constraint conditions, performance indexes and the like of satellite trajectory optimization in each stage are calculated by combining satellite orbit information, formation configuration information and the like, so that formation satellites are better guided to achieve a set task target.

The method solves the problem of fault-tolerant control of formation satellites based on a Gaussian pseudo-spectrum combined sequence quadratic programming method for the first time, determines the optimal positions of departure of fault stars and entry of backup stars by taking energy consumption as an optimization target, ensures that the backup stars can completely fill the positions of the fault stars, forms new formation with healthy stars, and realizes reconstruction of formation functions. The satellite formation fault-tolerant strategy can be suitable for any satellite formation configuration situation, has no requirement on the number of satellite groups, and has strong flexibility and universality.

The implementation steps of the non-linear programming based star-cluster-level satellite fault-tolerant control method provided by the invention are specifically described below by taking an example of a horizontal circle formation configuration design of one virtual master star and four slave stars as a sub-star point and combining with the accompanying drawings. The specific operation steps are as follows:

stage one: satellite formation configuration design and fly-by-flight trajectory equation determination

(1) Defining a distributed satellite relative coordinate system

Fig. 2 is a schematic view of the geocentric inertial coordinate system and the orbit coordinate system, which is specifically expressed as follows:

1) center of earth inertial coordinate system

The coordinate origin O is the earth center, the equatorial plane is used as a reference plane, the X axis is along the intersection line of the earth equatorial plane and the ecliptic plane and points to the spring equinox, the Z axis points to the north pole direction, and the Y axis, the X axis and the Z axis form a right-hand coordinate system in the equatorial plane.

2) Orbital coordinate system

The origin of coordinates is the center of mass of the main satellite, the x axis points to the direction of the main satellite along the center of the earth, the y axis is perpendicular to the x axis direction in the orbit plane of the main satellite and points to the direction of the movement of the main satellite, and the z axis is perpendicular to the orbit plane of the main satellite and obeys the right hand rule.

(2) Determining flight path equations for formation satellites

In the case of a horizontal round formation, in the horizontal plane (y)_zPlane), the distance between the accompanying spacecraft and the reference spacecraft is kept at a fixed distance, and the constraint condition is met:

y_N ²+z_N ²＝r²(N＝1,2,3,4) (30)

and can obtain:

z_N＝±2x_N (31)

i.e. representing the projection of the horizontal circular formation plane in the xz-plane. Wherein "+" and "-" in (31) are consistent in the subsequent processing, so the first one, i.e. the inclination of 26.565 degrees, is selected, and the other one will not be described again.

The relative positions and relative speeds between the four slave stars and the master star under the horizontal round formation are respectively as follows:

the initial conditions of four satellites can be visualized by using the radius r of the horizontal circle and the phase angle theta of the initial position of each satellite_NRepresents:

FIG. 3 is a projection of the horizontal circle formation in xy and yz planes. As can be seen from FIG. 3, under the horizontal round formation, the projections of the satellite formation on the xy plane are the major axis r and the minor axis r

The projection on the yz plane of the ellipse of (2) is a circle with radius r.

And a second stage: failed star leaving formation stage

In the invention, the fault star is supposed to move to a position 2r (r is the horizontal circle radius of the projection of the formation on the yz plane) right in front of the central main star, and the fault star and the central star form a serial formation. Under the condition that the final position of the maneuvering is fixed, the selection of the initial position directly influences the consumption of fuel and the implementation of collision constraint, under the condition of the collision constraint, multiple optimization results of different initial positions are compared based on a Gaussian pseudo-spectrum method and a sequential quadratic programming algorithm, a relation curve between a phase angle of a fault star leaving formation position and the fuel consumption is solved, and the fault star leaving position which saves the fuel most is determined.

(1) Constraint conditions

And (3) edge value constraint: initial value: without loss of generality, assuming that the satellite 1 fails, the initial x-direction position and velocity of the failed satellite 1 can be obtained as follows:

based on the initial conditions and the horizontal round formation equation of (35), the arbitrary position and speed of the fault star 1 on the horizontal round formation can be obtained:

for convenience of trajectory planning and fuel saving, the final position of the fault star 1 is assumed to be located at 2r right in front of the central star and on the serial formation formed by the central star, namely, the phase angle is alpha₁At 270 °, the final position and velocity of the fault star 1 may be determined as:

collision avoidance restraint: assuming that the safe distance between the satellite platform and the formation configuration is set to be 10m, the collision avoidance of the fault star and the main star and the collision avoidance between the sub-stars are constrained to be

Thrust force restraint:

because the output amplitude of the thruster is limited, the output amplitudes of 6 thrusters in three directions are required not to exceed the maximum amplitude, which is expressed as:

wherein u is_x1,u_y1,u_z1Representing the thrust of the fault star 1 in the x, y and z directions, u_maxRepresenting the maximum amplitude of the thruster, u in the simulation process_maxTake 0.01m/s²。

And (3) time constraint:

because the minimum fuel is used as a performance index, in order to prevent the optimization time from being too large, the fault star leaving optimization time needs to be limited according to specific conditions:

tf≤t_max (41)

the given optimization time in the simulation process is 2000s, namely the fault star 1 is required to reach the final value point within 2000s, and the time constraint can be adjusted arbitrarily according to the task requirements.

(2) Performance index

In order to reduce the fuel consumption in the optimization process of leaving formation of the failed satellite, the total energy consumption of the failed satellite is taken as a performance index, and the following steps are carried out:

minJ＝∫(|u_x1|+|u_y1|+|u_z1|) (42)

wherein u is_x1,u_y1,u_z1The thrust acceleration of the fault star 1 in the x direction, the y direction and the z direction is respectively.

Therefore, it is necessary to determine an initial position, i.e., an initial phase angle θ, at which the failed star 1 is optimal in the case of a fixed final position, in consideration of the performance index of fuel economy and each constraint condition.

The discretization process based on the gaussian pseudo-spectrum method is referred to as the second technical scheme, and is not described herein again. The invention selects an SNOPT solver under the TOMLAB company flag, and utilizes a Gaussian pseudo-spectrum method combined with sequence quadratic programming to perform continuous cycle optimization based on the constraint conditions, the performance indexes and the like, and the result is shown in FIG. 5.

FIG. 5 is a plot of the phase angle θ (0 θ ≦ 360) of the failed star from the formation position versus fuel consumption for the failed star. As can be seen from fig. 5, when the phase angle θ is 156 °, the faulty star is moved away from the fuel consumption least, and therefore the position where the faulty star leaves the formation is determined, i.e., the initial position of the faulty star trajectory optimization.

Fig. 6 is a schematic diagram of the faulty star departure formation, in which the orange dotted circles indicate the initial positions of the respective satellites and the white solid circles indicate the end positions of the respective satellites after 2000 s. As can be seen from fig. 6, when the satellite 1 reaches the optimal departure position obtained by optimization, the satellite 1 performs the trajectory maneuver, the whole time takes 2000s, that is, after 2000s, the satellite 1 reaches the position marked in fig. 6, and in the 2000s, the rest satellites naturally fly aroundOver angle of

Each arriving at the position marked by the white solid circle in the figure.

Fig. 7 is a two-dimensional and three-dimensional trajectory diagram of each satellite during the process of formation of the failed satellite, which includes the maneuver optimization curve during the process of 2000s of the failed satellite and the motion curves of the remaining three healthy satellites. In the figure, the four thick points are the initial positions of the four satellites respectively, and the lines corresponding to the colors represent the motion trajectories of the four satellites within 2000 s. As can be seen from fig. 7, when the satellite 1 fails and naturally orbits to the optimal departure phase, the failed satellite departs along the red triangular trajectory optimized in fig. 7, and the remaining healthy satellites continue to orbit along the horizontal circular formation. It can also be seen from the two-dimensional diagram in fig. 7 that the failed star will eventually follow the red triangular trajectory to the desired position, i.e. 2r directly in front of the central star. The running track diagram of each satellite in fig. 7 obtained by simulation completely coincides with the expected movement of each satellite in the running track diagram of fig. 6, and the expected result is met.

FIG. 8 is a graph of the location, velocity thrust and distance from the primary star for a failed star. As can be seen from fig. 8, the position of the fault star 1 and the distance from the main star are completely the same as those described in fig. 6 and 7, and the speed and the thrust both satisfy the respective constraints, which can satisfy the actual needs of the engineering.

And a third stage: backup star entering formation phase

The backup star maneuvering aims to enable the backup star to enter an original horizontal round formation within a specified time after the fault star leaves, completely fills the virtual position of the fault star, and forms a corresponding formation with the remaining healthy stars.

(1) Constraint conditions

And (3) edge value constraint:

initial value: the initial position of the backup star before maneuvering can be set at will according to task requirements, and here, the initial position of the backup star is assumed to be located right behind the virtual center star and on a serial formation formed by the backup star and the center star, so that the initial position and the speed of the backup star are as follows:

final value: the final position and speed of the backup star can be determined as follows, if the final position of the backup star is positioned on a horizontal circle with the radius r and the phase angle is mu (0 degrees to 360 degrees), the final position and speed of the backup star can be determined as follows:

collision avoidance restraint: considering the size of the satellite platform and the size of the formation configuration, the set safety distance is 10m, and the collision avoidance of the backup star and the main star and the collision avoidance between the sub-stars are constrained to be

Notably, the failed star has left the formation and has maneuvered to a position directly in front of the central star without colliding with the remaining stars, so collision constraints with the failed star are not considered here.

Thrust force restraint:

u in simulation_maxTake 0.01m/s²。

And (3) time constraint:

because the minimum fuel is used as a performance index, in order to avoid overlarge optimization time, the backup satellite optimization time needs to be limited according to specific conditions:

tf≤t_max (47)

without loss of generality, the optimization time given in the simulation process is 2000s, namely the backup star is required to reach the final point within 2000s, and the time constraint can be adjusted at will according to the task requirements.

(2) Performance index

In order to reduce the fuel consumption of backup stars in the formation optimization process, the total energy consumption of the backup stars is the minimum performance index, which is as follows:

minJ＝∫(|u_xb|+|u_yb|+|u_zb|) (48)

wherein u is_xb,u_yb,u_zbThe thrust acceleration of the backup star in the x direction, the y direction and the z direction is respectively.

(3) Time of day selection for entering formation

The best time to enter the formation will be explained in detail in the subsequent simulation results.

(4) Simulation results and analysis

FIG. 9 is a plot of phase angle θ (0 ≦ θ ≦ 360) of the backup star entering the formation versus fuel consumption. As can be seen from fig. 9, when the backup star moves to the position where the phase angle θ is 90 ° in the horizontal round formation, the fuel consumption is the least. Therefore, the position of the backup star entering the formation is determined, namely the optimal final position of the backup star track is determined.

Fig. 10 is a schematic diagram of backup star formation, in which an orange dotted circle represents the initial position of each satellite, and a white solid circle represents the end position 2000s after each satellite. Analysis shows that the time of the backup star to reach the optimal position of the formation is 2000s, namely the backup star needs to reach the virtual position where the fault star 1 is supposed to reach in 2000s, and the blank of the star 1 is filled. Since the orbit period is 6298s, it can be known that the phase angle of the virtual star 1 moving during 2000s of the backup star entering the formation is

Therefore, the backup star starts to maneuver when each healthy star in the figure naturally flies to the initial position marked in the figure, the virtual position of the fault star 1 is just filled after the backup star is added into the formation, the rest stars reach the positions marked by the white solid line circles in the figure, and four stars just form a horizontal circle formation.

Fig. 11 is a two-dimensional and three-dimensional trajectory diagram of each satellite during the backup star entering formation, which includes the maneuver optimization curve during the backup star 2000s and the motion curves of the remaining three healthy stars. In the figure, the four thick points are the initial positions of the four satellites respectively, and the lines corresponding to the colors represent the motion trajectories of the four satellites within 2000 s. As can be seen from fig. 11, when the virtual position of the failed star 1 is:

and when the backup star starts to maneuver along the red triangular track, the backup star can be ensured to completely fill the blank of the failed star 1. Since several stars are uniformly distributed, the phase difference between the stars is

As can be clearly seen from the two-dimensional track diagram in fig. 11, the backup star just reaches the position where the fault star 1 originally reaches according to the horizontal circle constraint, and the position of the fault star is completely filled up, so that the horizontal circle formation configuration is maintained, and the task requirement is met. The running track diagram of each satellite in fig. 11 obtained by simulation completely coincides with the expected movement of each satellite in the running track diagram of fig. 10, and the expected result is met.

FIG. 12 is a graph of the position, velocity thrust, and distance from the primary star for the backup star. As can be seen from fig. 12, the positions of the backup satellites and the distances between the backup satellites and the main satellites are completely the same as those described in fig. 10 and 11, and the speed and the thrust both satisfy respective constraints, which can satisfy the actual needs of the engineering.

Claims (3)

1. A fault-tolerant control method of a satellite cluster level based on nonlinear programming is characterized in that after a satellite breaks down in the formation process, the initial and end positions of a broken star and a backup star are determined, inter-satellite collision and time constraint are considered, fuel consumption is taken as a performance index, the optimal leaving and entering positions of the broken star and the backup star are determined by a Gaussian pseudo-spectrum method, respective optimal trajectory planning is obtained, the situation that the backup star enters to completely fill up the position of the broken star is ensured, a new formation is formed together with a healthy star, and the performance requirement of the formation is realized; specifically, considering from three stages, the control means and functions of each stage are implemented as follows:

stage one: satellite formation configuration design and fly-by-flight trajectory equation determination

(1) Defining a distributed satellite relative coordinate system

1) Center of earth inertial coordinate system

Defining a coordinate origin O as the earth center, taking an equatorial plane as a datum plane, leading an X axis to point to a spring equinox along the intersection line of the earth equatorial plane and a ecliptic plane, leading a Z axis to point to the north pole direction, and forming a right-hand coordinate system by the Y axis, the X axis and the Z axis in the equatorial plane;

2) orbital coordinate system

Defining a coordinate origin as a main satellite mass center, pointing an x-axis to a main satellite direction along the earth center, pointing a y-axis to the x-axis direction in a main satellite orbit plane and pointing to the main satellite motion direction, and keeping a z-axis to the main satellite orbit plane and obeying a right-hand rule;

(2) determining flight path equations for formation satellites

Under the condition of horizontal round formation, in a horizontal plane, the distance between the kth accompanying spacecraft and the reference spacecraft keeps a fixed distance, namely a subsatellite point round formation, and the constraint conditions are met:

y_k ²+z_k ²＝r² (1)

and the projection of the horizontal circle formation plane in the xz plane can be obtained:

z_k＝±2x_k (2)

wherein x is_k,y_k,z_kThe position coordinates of the kth orbiting flying satellite in a relative coordinate system are shown, the + and-in the formula (2) indicate that two planes exist in each horizontal circle formation, the projections of the planes in the xz plane are straight lines and intersect along the y axis, but one plane is inclined by 26.565 degrees, the other plane is inclined by-26.565 degrees, and the formula (2) is further differentiated to obtain

Therefore, the initial value of the horizontal circle formation of the kth flying satellite meets the following conditions:

z_k0＝x_k0

the relative position and relative speed between the slave star and the master star under the horizontal round formation are as follows:

wherein x is_k,y_k,z_kThe relative position of the kth orbiting flying satellite and the main satellite,

is the relative velocity, x, of the kth orbiting satellite to the main satellite_k0,

The initial condition of the satellite is that n is the average motion angular velocity of the reference spacecraft, and the two initial conditions are represented by the formulas (5) and (6) by the radius r of a horizontal circle and the phase angle theta of the initial position of the kth flying satellite_kRepresents:

therefore, only the radius of the track and the initial phase angle θ are determined_kObtaining the initial position of the kth satellite so as to obtain the position of the satellite in the horizontal circle formation at any time;

and a second stage: failed star leaving formation stage

In the process of leaving the fault star, the selection of the initial and final value positions directly influences the consumption of fuel and the realization of collision constraint, under the condition of collision constraint, multiple optimization results of different positions are compared based on a Gaussian pseudo-spectral method and a sequential quadratic programming algorithm, a relation curve between the phase angle of the position where the fault star leaves the formation and the fuel consumption is solved, and the fault star leaving track which saves the fuel most is determined;

and a third stage: backup star entering formation phase

Under the condition that the maneuvering initial position of the backup star is fixed, the position of the backup star entering the formation directly influences the fuel consumption and the implementation of collision constraint, under the condition of collision constraint, multiple optimization results of different final value positions are compared based on a Gaussian pseudo-spectral method and a sequence quadratic programming algorithm, a relation curve between a phase angle of the backup star entering the formation position and the fuel consumption is solved, and the most fuel-saving backup star butt-joint position and the best maneuvering starting time are determined.

2. The fault-tolerant control method for the satellite at the star cluster level based on the nonlinear programming as claimed in claim 1, characterized in that the two stages are refined as follows:

1) constraint conditions

Constraint conditions of the satellite formation configuration optimization process comprise edge value constraint, inter-satellite collision avoidance constraint, output torque constraint and configuration optimization time constraint;

and (3) edge value constraint:

initial value: simply by determining the radius of orbit and the initial phase angle theta of the kth satellite_kAnd obtaining the initial position of the satellite, so that the position of the satellite in the horizontal circular formation at any time can be obtained, and assuming that the orbit radius is r, the initial position of the fault star i is located on the horizontal circular formation with the radius r, and the initial x-direction position and speed of the fault star i are:

wherein, theta_iIs the initial phase angle of the failed star i, n is the average moving angular velocity of the reference spacecraft, r is the levelAnd (3) obtaining the random position and speed of the fault star i on the horizontal circular formation based on the initial conditions of (5) to (6) and the horizontal circular formation equation:

final value: suppose the final position phase angle of the fault star i is alpha_iAt a distance 2r from the central primary star, the final position and velocity of the failed star i can be determined as:

wherein x is_if,y_if,z_ifIs the final value position of the failed star i,

the final value speed of the fault star i;

collision avoidance restraint: in the process of the satellite formation movement, a collision avoidance strategy based on a safety domain is adopted, the safety distance is set to be d in consideration of the size of a satellite platform and the size of the formation configuration, and collision avoidance of the main star and the fault star and collision avoidance constraint between the fault star and the healthy star are set to be d

Wherein x is_j,y_j,z_jRelative distances between the remaining healthy stars except the fault star in the formation satellite and the main star;

thrust force restraint:

because the output amplitude of the thruster is limited, the output amplitudes of 6 thrusters in three directions are required not to exceed the maximum amplitude, which is expressed as:

wherein u is_xk,u_yk,u_zkRepresents the thrust of the kth satellite in the x, y and z directions, u_maxRepresenting the maximum amplitude of the thruster;

and (3) time constraint:

and limiting the leaving optimization time of the fault star i according to specific conditions:

t_f≤t_max (14)

2) performance index

The performance index is the minimum total energy consumption of the failed star i, and is as follows:

min J＝∫(|u_xi|+|u_yi|+|u_zi|) (15)

wherein u is_xi,u_yi,u_ziThrust accelerations of the fault star i in the x direction, the y direction and the z direction are respectively;

therefore, the performance index and each constraint condition of the fuel economy need to be comprehensively considered, and the fault star i arrival phase angle is determined to be alpha based on the Gaussian pseudo-spectrum method combined with the sequence quadratic programming_iAnd an optimum off-position phase angle theta having a radius of 2r_iAnd designing an optimal track, specifically, solving an optimal control problem by adopting a Gaussian pseudo-spectrum method: discretizing a state variable and a control variable on a series of discrete Gaussian points, constructing a Lagrange interpolation polynomial on an approximate state variable and a control variable by taking the points as nodes, deriving a derivative of the Lagrange interpolation polynomial on the approximate state variable with respect to time, converting a differential equation of a motion trajectory of a relative dynamic model describing satellite motion into a group of algebraic constraints, calculating an integral term existing in an expected performance index by Gaussian integration, and performing continuous cycle optimization by using a Gaussian pseudo-spectral method based on the constraint condition, the performance index and the like to obtain an optimal position where a fault star leaves;

(1) converting the time interval of the optimal control problem to the time interval applying Gauss pseudo-spectrum method through time domain conversion:

wherein t is₀Representing the initial time of trajectory optimization, t_fRepresents the termination time of the trajectory optimization, and tau represents the time interval satisfying the Gaussian pseudo-spectral method, and the arbitrary trajectory optimization time interval is converted into [ -1,1]Preparing for discretization of a Gaussian pseudo-spectrum method;

(2) the matching points are selected from LG points of K order, namely roots of Legendre polynomials of K order:

(3) selecting nodes: here, discrete K LG points and an initial time point tau are used₀As discrete nodes

Wherein g (τ) ═ 1+ τ) P_N(τ)，X(τ_i) T represents the state variable X_iThe state variable discretization method is given by the above formula, the state variable discretization points are subjected to polynomial summation, and coefficients of the polynomial are Lagrange interpolation polynomials, so that the state variable is discretized;

(4) the expression of discretization of the control quantity is as follows:

in the formula, U (τ)_i) T represents the controlled variable U_iThe discretization method of the control quantity is consistent with the state variable, and the control quantity is discretized by a polynomial of the discrete points of the control variableSumming to discretize the control variable;

(5) and (3) final value constraint in a discrete state:

for gaussian pseudo-spectral method, the discretization process of the state variable and the control variable does not contain a terminal point, and here, the terminal value constraint is discretized through the processing of integral, and the discretization method is as follows:

wherein w_kA weighting coefficient representing discretization of the integral object f;

(6) since the state variable derivatives exist in the satellite attitude dynamics and kinematic model constraints, the method for discretizing the state variable derivatives is given here as follows:

polynomial weighting coefficient D for discretization of state variable derivative_kiThat is by direct p-Lagrangian polynomial L_i(τ) is derived.

3. The fault-tolerant control method for the satellite at the star cluster level based on the nonlinear programming as claimed in claim 2, characterized in that the three stages are refined as follows:

(1) determining optimal entry formation locations for backup stars

1) Constraint conditions

And (3) edge value constraint: initial value: the initial position of the backup star assumes a distance of 3r from the virtual center star and a phase angle of β, and the initial position and speed of the backup star are as follows:

wherein x is_b0,y_b0,z_b0To back up the initial relative positions of the star and the primary star,

the initial relative speed of the backup star and the main star;

final value: the final position of the backup star is located on a horizontal circular formation with radius r, and the final phase angle is set to be mu (0 degrees to 360 degrees), then the final position and speed of the backup star can be determined as:

wherein x is_bf,y_bf,z_bfTo back up the relative positions of the end values of the stars and the primary star,

the relative speed of the backup star and the final value of the main star is used;

collision avoidance restraint: considering the size of the satellite platform and the size of the formation configuration, the set safety distance is d m, and the collision avoidance constraints of the main star and the backup star and the healthy star are as follows:

wherein x is_b,y_b,z_bRepresenting the relative position of the backup star and the main star under the orbit coordinate system of the main star

Thrust force restraint:

because the output amplitude of the thruster is limited, the output amplitudes of 6 thrusters in three directions are required not to exceed the maximum amplitude, which is expressed as:

wherein u is_xk,u_yk,u_zkRepresents the thrust of the kth satellite in the x, y and z directions, u_maxRepresenting the maximum amplitude of the thruster;

and (3) time constraint:

because the minimum fuel is used as a performance index, in order to avoid overlarge optimization time, the backup star entering optimization time needs to be limited according to specific conditions:

t_f≤t_max (27)

2) performance index

In order to reduce the fuel consumption in the optimization process of leaving the formation of the backup stars, the total energy consumption of the backup stars is the minimum performance index, which is as follows:

min J＝∫(|u_xb|+|u_yb|+|u_zb|) (28)

wherein u is_xb,u_yb,u_zbThrust acceleration of the backup satellite in the x direction, the y direction and the z direction is respectively;

therefore, the optimal final value position of the backup star, namely the termination phase angle β, needs to be determined based on a gaussian pseudo-spectral method in combination with a quadratic sequence planning algorithm, considering the performance index of fuel economy and each constraint condition under the condition of a fixed final value position;

(2) determining selection of time for backup stars to enter formation

Supposing that the phase of the optimal position of the backup star entering the formation is determined to be mu in the last step_bestUnder the condition of determining the position of the starting position and the position of the entering formation point of the motor, the entering time of the backup star and the motor time t thereof_jAnd the orbit period T of the central star, the phase of the virtual fault star in the process of the backup star reaching the selected position is

Time t_j(ii) a The entry time of the backup star is represented by the phase μ of the virtual position of the failed star, that is, the backup star starts to enter when the failed star runs to which phase under the condition that the failed star still exists:

according to the divided task stages, constraint conditions, performance indexes and the like of satellite trajectory optimization in each stage are calculated by combining satellite orbit information, formation configuration information and the like, so that formation satellites are better guided to achieve a set task target.

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