Lossless convexity method for gravitational wave spacecraft formation attitude alignment random optimal control

Abstract

The invention discloses a lossless convexity method for gravitational wave spacecraft formation attitude alignment random optimal control, and the method comprises the steps: setting related parameters in a task scene, including the rotational inertia of a spacecraft, the maximum thrust moment, the initial state and target state of the spacecraft, the pointing constraint, and a disturbance coefficient matrix; establishing a random spacecraft attitude dynamic model; utilizing opportunity constraints to describe a plurality of constraints with uncertainty, including moment constraints, taboo constraints and forced constraints; converting a non-convex part in the constraint condition into a linear matrix inequality (LMIs) by adopting the Schel complement theorem, and adding a virtual control variable and a penalty term to obtain a convex problem which can be directly solved by using the CVX; the invention aims to realize random optimal control of gravitational wave spacecraft formation attitude alignment through a convex optimization method.

Description

Technical Field

[0001] The present invention belongs to the field of aerospace technology, and specifically refers to a lossless convexification method for random optimal control of gravitational wave spacecraft formation attitude alignment. Background Art

[0002] The detection of gravitational waves in space requires the establishment of a communicative laser link between spacecraft. To ensure accurate laser link establishment and proper communication, high spacecraft attitude accuracy is required. Uncertainties in the space environment, such as solar pressure and noise disturbances from thrusters and other actuators, can affect spacecraft attitude accuracy. Stochastic systems incorporate random noise based on a deterministic system model and are suitable for scenarios where environmental uncertainty must be considered. In stochastic systems where states or parameters have probabilistic distributions, chance constraints are a widely used form of constraint that enhances system flexibility and is particularly useful in managing uncertainty.

[0003] Existing techniques use iterative relaxation and penalty methods based on semidefinite programming (SDP) to sequentially approximate the quadratically constrained quadratic programming (QCQP) problem, achieving good iterative feasibility and solution search directions. The second-order terms involved in the variables are then convexified using hybrid second-order cone programming (SOCP) techniques. However, this convex optimization method does not account for the environmental uncertainties and random noise perturbations that may exist during spacecraft formation attitude planning. Therefore, designing a non-destructive convexification method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment is of great significance for the detection of gravitational waves in space. Summary of the Invention

[0004] In view of the shortcomings of the above-mentioned technologies, the present invention provides a lossless convexification method for random optimal control of gravitational wave spacecraft formation attitude alignment.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] The present invention provides a lossless convexification method for random optimal control of gravitational wave spacecraft formation attitude alignment, comprising the following steps:

[0007] Step 1: Establish a quaternion-based spacecraft attitude dynamics model and set the spacecraft's mission parameters, including moment of inertia, initial quaternion and angular velocity, terminal quaternion and angular velocity, maximum control torque, and pointing constraint;

[0008] Step 2: Adding a noise term representing uncertainty to the spacecraft attitude dynamics model to establish a spacecraft random attitude dynamics model;

[0009] Step 3: Express the moment constraint and directional constraint with uncertainty through the relatively loose constraint form of chance constraint;

[0010] Step 4: Use the convex relaxation technique of Schur's complement theorem and the method of introducing dummy variables to convexify the non-convex constraints and obtain a convex problem equivalent to the original problem.

[0011] Furthermore, the step 1 specifically includes: setting the relevant parameters under the mission scenario, including the spacecraft moment of inertia J, the maximum thrust torque , initial time t0, terminal time t f , the initial state of the spacecraft x0 (including quaternion q0 and angular velocity ω0), the terminal state of the spacecraft x f (including quaternion q f and angular velocity ω f ); and parameters describing the pointing constraint, including the direction vector of the sensor in the body coordinate system , the direction vector of the bright object in the inertial coordinate system , the direction vector of the center to be observed in the body coordinate system , the corresponding sensor field of view half angle and .

[0012] Furthermore, the step 2 specifically includes establishing the spacecraft attitude dynamics equation:

[0013] (11);

[0014] in is the unit quaternion, represents the first derivative of the unit quaternion with respect to time, is the spacecraft angular velocity, is the spacecraft moment of inertia, represents the first-order derivative of the spacecraft angular velocity with respect to time, and the superscript T represents the transpose. is the spacecraft moment of inertia, is the thrust torque. Indicates the rotation angle, 、 、 Respectively represent the direction of rotation around the X, Y, and Z axes, 、 、 Represent the angular velocity components around the X, Y, and Z axes respectively, 、 、 Respectively represent the thrust torque of X, Y, and Z axes; 、 、 Respectively represent the moment of inertia of the rigid body around the X, Y, and Z axes; represents the state variable of the system, t represents time, Represent the attitude dynamics equations related to state, thrust torque and time;

[0015] The linear stochastic system is obtained by linearizing the above dynamic equations through the first-order Taylor expansion:

[0016] (12);

[0017] in is the state vector, and the coefficient matrix is ​​defined as:

[0018] (13);

[0019] Take time Divide into N equal parts:

[0020] (14);

[0021] The thrust torque is discretized using zero-order hold:

[0022] (15);

[0023] definition , the discrete linear random system is:

[0024] (16);

[0025] in, is the system status, is the system matrix, which describes the connection between the internal state variables of the system. is the input matrix, which represents the effect of the input on each state variable, is the thrust torque, is the uncertainty matrix, subscript k represents the state corresponding to time k, and subscript k+1 represents the state corresponding to time k+1. Using state feedback to control the spacecraft attitude to reach the target state, the control law is:

[0026] (17);

[0027] in, is the feedback gain matrix, is the feedforward gain matrix, Indicates the set thrust torque value.

[0028] Furthermore, step 3 specifically includes: introducing chance constraints to relax the constraints into a probabilistic form. In random systems where states or parameters have a probabilistic distribution, chance constraints are a widely used form of constraints that can enhance the flexibility of the system and are very useful in dealing with uncertainty. Thrust torque amplitude constraint:

[0029] (18);

[0030] in, represents the expected probability, Then, the probability form of the constraint is transformed into a more general constraint form:

[0031] (19);

[0032] (20);

[0033] in, is the dimension of the moment, represents the coefficient related to the expected probability and the dimension of the variable.

[0034] Correspondingly, the opportunity constraint of the pointing constraint is of the form:

[0035] (twenty one);

[0036] in, , is the mean matrix, The covariance is 0 and the mean is Gaussian distribution. Convert it to the general hard constraint form:

[0037] (twenty two);

[0038] in, is the inverse cumulative distribution function of the standard normal distribution, Represents vectorization.

[0039] Furthermore, step 4 specifically includes: using the convex relaxation technique of Schur's complement theorem and the method of introducing dummy variables to convexify the moment constraint and pointing constraint converted in step 3. A non-negative auxiliary variable is used to replace the non-convex term in the moment constraint:

[0040] (twenty three);

[0041] in, represents the mean value of the moment at time k, express Reference value of

[0042] By Schur's complement theorem, the moment constraint can be equivalent to:

[0043] (twenty four);

[0044] in, is a non-negative auxiliary variable, represents the Schur complement of a block matrix;

[0045] The torque constraint can be written in the form of a linear inequality:

[0046] (25);

[0047] in, , represents the reference moment. To obtain a convex problem, Taylor expansion is performed at the reference value of :

[0048] (26);

[0049] in, represents the relaxation term; Represents the mean value of the reference moment.

[0050] Add a penalty In the objective function.

[0051] definition , Satisfying the mean The covariance is The pointing constraint is equivalent to:

[0052] (27);

[0053] The second term of Schur's complement theorem can be transformed into a second-order cone constraint:

[0054] (28);

[0055] Add a penalty In the objective function.

[0056] The beneficial effects of the present invention compared with the prior art are as follows: the lossless convexification method for random optimal control of gravitational wave spacecraft formation attitude alignment proposed in the present invention makes up for the defect that uncertainty is not considered in the traditional attitude alignment process; through nonlinear random systems and state feedback control, the lossless convexification of random optimal control of gravitational wave spacecraft formation attitude alignment is realized; the chance constraint is more in line with the system whose state or parameters have a probability distribution form, thereby enhancing the flexibility of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 Schematic diagram of the process of the present invention. DETAILED DESCRIPTION

[0058] In order to facilitate understanding by those skilled in the art, the present invention is further described below with reference to examples and drawings. The contents mentioned in the embodiments are not intended to limit the present invention.

[0059] Reference Figure 1 As shown in the figure, a lossless convexification method for random optimal control of gravitational wave spacecraft formation attitude alignment is proposed. The steps are as follows:

[0060] 1) Set the spacecraft initial state, pointing constraints, torque constraints, and terminal state constraints.

[0061] Set the relevant parameters of the spacecraft and its mission scenario, including: spacecraft moment of inertia J, maximum thrust torque , initial time t0, terminal time t f , the initial state of the spacecraft x0 (including quaternion q0 and angular velocity ω0), the target state of the spacecraft x f (including quaternion q f and angular velocity ω f ), and parameters describing the pointing constraint, including the direction vector of the sensor in the body coordinate system , the direction vector of the bright object in the inertial coordinate system , the direction vector of the center to be observed in the body coordinate system , the corresponding sensor field of view half angle and .

[0062] 2) Establish a random attitude dynamics model of the spacecraft.

[0063] A quaternion-based spacecraft attitude dynamics equation is established, and random terms are added to obtain the corresponding random spacecraft attitude dynamics equation. The random system is discretized, and the torque is controlled according to the zero-order hold law. State feedback is used to control the spacecraft attitude to reach the target state.

[0064] 3) Introduce chance constraints to reformulate torque constraints and directional constraints.

[0065] To handle the uncertainty in the system, hard constraints are relaxed into probabilistic forms and chance constraints are converted into equivalent general constraints that are tractable.

[0066] 4) Convexify the non-convex terms in the original problem.

[0067] By using Schurbe's theorem and the method of introducing dummy variables, the non-convex terms of the constraints in step 3 are convexified to obtain a convex problem equivalent to the original problem.

[0068] The following is an example of the spacecraft attitude alignment process in the LISA mission:

[0069] Step 1. First, in this problem, assume that both the initial and target states of the spacecraft are known. The spacecraft state can be measured by satellite sensors, so this assumption is reasonable. Assume that the spacecraft is equipped with a field-emission electric thruster (FEEP) that can provide a maximum thrust torque of 7 mN∙m. FEEP has been used in the LISA Pathfinder mission, so the maximum torque assumption is reasonable. The spacecraft initial quaternion q0 = [0.73224, 0.07897, 0.42632, 0.52521], the spacecraft initial angular velocity ω0 = [0.08606, 0.10219, 0.06650] rad / s, and the target quaternion q f =[0.74458,-0.068725,-0.428083,-0.507564], spacecraft target angular velocity ω f =[0,0,0] rad / s, spacecraft moment of inertia J=diag(551.25,450.83,450.83) kg∙m 2 , initial time , terminal time , time step The parameters of the pointing constraint are set as follows: the direction vector f1 of taboo cone 1 = [0.5000, -0.8660, 0], the direction vector f2 of taboo cone 2 = [0.2432, 0.9077, -0.3420], the direction vector f3 of taboo cone 3 = [0.4924, 0.0868, -0.8660], the field of view half angle is 30 deg, the direction vector m of the mandatory cone is [-0.7660, 0.6428, 0], the field of view half angle is 60 deg, and the expected probability is 0.95.

[0070] Step 2: Establish the random attitude dynamics equation based on quaternion:

[0071] (29);

[0072] in is the state variable, is the thrust torque control quantity,

[0073] (30);

[0074] (31);

[0075] The specific form of the coefficient matrix is:

[0076] (32);

[0077] definition , the discrete linear random system is:

[0078] (33);

[0079] in, is the system status, is the system matrix, which describes the connection between the internal state variables of the system. is the input matrix, which represents the effect of the input on each state variable, is the thrust torque, is the uncertainty matrix, subscript k represents the state corresponding to time k, and subscript k+1 represents the state corresponding to time k+1. Use state feedback to control the spacecraft attitude to reach the target state. Use state feedback to control the spacecraft attitude to reach the target state, and the control law is:

[0080] (34);

[0081] in, is the feedback gain matrix, is the feedforward gain matrix, Indicates the set thrust torque value.

[0082] Step 3: Introduce opportunity constraints. The moment constraints and general directional constraints are:

[0083] (35);

[0084] (36);

[0085] in, represents the expected probability, Indicates the maximum thrust torque, , is the mean matrix, The covariance is 0 and the mean is Gaussian distribution. Convert it to the general hard constraint form:

[0086] (37);

[0087] (38);

[0088] in, is the dimension of the moment, represents the coefficient related to the expected probability and the dimension of the variable.

[0089] (39);

[0090] in, is the inverse cumulative distribution function of the standard normal distribution, Represents vectorization.

[0091] Step 4: Convexify the constraints. Convexify the directional constraints and torque constraints converted into general hard constraints in step 3. Write the torque constraints in the form of linear inequalities:

[0092] (40);

[0093] in, represents the reference torque, represents the mean value of the moment at time k, express To obtain a convex problem, Taylor expansion is performed at the reference value of :

[0094] (41);

[0095] in, is a non-negative auxiliary variable, , represents the relaxation term, represents the mean value of the reference moment;

[0096] definition , Satisfying the mean The covariance is The pointing constraint is equivalent to:

[0097] (42);

[0098] The second term can be transformed into a second-order cone constraint (SOC) through Schur's complement theorem:

[0099] (43);

[0100] Add a penalty and In the objective function.

[0101] As can be seen from the above examples, the present invention establishes a discrete random attitude dynamics model of a spacecraft; utilizes state feedback to control the uncertainty in the initial state; relaxes the constraints into a probabilistic form through chance constraints; and adopts Schur's complement theorem and the introduction of dummy variables to convexify the non-convex terms in the constraints to obtain a convex problem that can be directly solved.

[0102] The present invention has many specific application paths. The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements can be made without departing from the principles of the present invention. These improvements should also be considered as the scope of protection of the present invention.

Claims

1. A lossless convexification method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment, characterized by: include: Step 1: Establish a quaternion-based spacecraft attitude dynamics model and set the spacecraft's mission parameters, including moment of inertia, initial quaternion and angular velocity, terminal quaternion and angular velocity, maximum control torque, and pointing constraint; Step 2: Adding a noise term representing uncertainty to the spacecraft attitude dynamics model to establish a spacecraft random attitude dynamics model; Step 3: Express the moment constraint and directional constraint with uncertainty through chance constraint; Step 4: Use the convex relaxation technique of Schur's complement theorem and the method of introducing dummy variables to convexify the non-convex constraints and obtain a convex problem equivalent to the original problem.

2. The lossless convexification method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1 is characterized in that: In step 1, the quaternion-based attitude dynamics equation is established as follows: (1)； in, is the unit quaternion, the superscript T represents the transpose, represents the first derivative of the unit quaternion with respect to time, is the spacecraft angular velocity, is the spacecraft moment of inertia, represents the first-order derivative of the spacecraft angular velocity with respect to time, is the thrust torque; ； in, Indicates the rotation angle, 、 、 Respectively represent the direction of rotation around the X, Y, and Z axes, 、 、 Represent the angular velocity components around the X, Y, and Z axes respectively, 、 、 Respectively represent the thrust torque of X, Y, and Z axes; Define the state vector , the attitude dynamics equation is expressed as: ； in, 、 、 Respectively represent the moment of inertia of the rigid body around the X, Y, and Z axes, represents the state variable of the system, t represents time, Represent the attitude dynamics equations related to state, thrust torque and time; The initial and terminal state constraints of the spacecraft are in the form of: ； in, They represent the initial time and terminal time respectively; Pointing constraints include taboo constraints and mandatory constraints, which are expressed as follows: ； in, represents the direction vector of the sensor in the body coordinate system, represents the direction vector of the bright celestial body in the inertial coordinate system, Represents the direction vector of the center to be observed in the body coordinate system, and denote the corresponding sensor field of view half angles respectively; reformulated in a more general form: ； in, 、 They correspond to real symmetric matrices in quadratic forms respectively; ； in, represents the direction vector of the tabu constraint, represents the direction vector of the sensor, represents the direction vector of the enforced constraint, Represents the three-dimensional identity matrix.

3. The lossless convexification method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1 is characterized in that: In step 2, a random attitude dynamics model of the spacecraft is established, including: a multi-dimensional Brownian motion for describing uncertainty and the initial state that satisfies the Gaussian distribution , including the mean and covariance .

4. The lossless convexification method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1, characterized in that: In step 2, based on the nonlinear random system , after first-order Taylor expansion and discretization, we can get the discrete-time linear random system: ； in, ； in, is the system matrix, which describes the connection between the internal state variables of the system. is the input matrix, which represents the effect of the input on each state variable, represents the effect of random disturbance on the state variables, represents random perturbations, represents the thrust torque, Indicates from arrive The state transfer matrix, subscript k represents the state corresponding to time k, and subscript k+1 represents the state corresponding to time k+1.

5. The lossless convexification method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1, characterized in that: The chance constraint in step 3 relaxes the constraint into a probabilistic form, allowing a small probability of violating the constraint; the probability form of the moment constraint and the pointing constraint is: ； Among them, P represents the probability of an event occurring, represents the maximum value of thrust torque, and p is the expected probability.

6. The lossless convexification method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1, characterized in that: The chance constraint satisfying the Gaussian distribution in step 4 is equivalent to a more conservative constraint form; the Schur complement theorem is used to transform the nonlinear constraints in the optimization problem into linear matrix inequalities, and a penalty term is added to the objective function to ensure convergence; a dummy variable is introduced to replace the square root term, and the amplitude constraint of the control torque is expanded at the reference torque value. A penalty term is also added to the objective function to ensure that all constraints are met.

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