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

Abstract

This invention discloses a lossless convexification method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment, comprising: setting relevant parameters under the mission scenario, including the spacecraft's moment of inertia, maximum thrust torque, initial and target states, pointing constraints, and perturbation coefficient matrix; establishing a stochastic spacecraft attitude dynamics model; describing multiple constraints with uncertainties using chance constraints, including torque constraints, taboo constraints, and mandatory constraints; using the Schur complement theorem to transform the non-convex parts of the constraints into linear matrix inequalities (LMIs), and adding virtual control variables and penalty terms to obtain a convex problem that can be directly solved using CVX; this invention aims to achieve stochastic optimal control of gravitational wave spacecraft formation attitude alignment through convex optimization methods.

Description

Technical Field

[0001] This invention belongs to the field of aerospace technology, specifically referring to a lossless convexity method for stochastic optimal control of the attitude alignment of gravitational wave spacecraft formations. Background Technology

[0002] During space-based gravitational wave detection, establishing communicable laser links between spacecraft is required. To ensure accurate establishment and normal communication, high attitude accuracy is crucial for the spacecraft. Uncertainties in the space environment, such as solar radiation pressure and noise disturbances generated by thrusters and other actuators, can all affect the attitude accuracy of the spacecraft. Stochastic systems incorporate random noise into a deterministic system model, making them suitable for scenarios where environmental uncertainties need to 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 very useful in handling uncertainties.

[0003] Existing techniques employ an iterative relaxation and penalty method based on semidefinite programming (SDP) to successively approximate a quadratic constrained quadratic programming (QCQP) problem, thus obtaining good iterative feasibility and solution search direction. Subsequently, a hybrid second-order cone programming (SOCP) technique is used to convexize the second-order terms involved in the variables. However, this convex optimization method does not consider the environmental uncertainties and random noise disturbances that may exist during spacecraft formation attitude planning. Therefore, designing a lossless convexization method for stochastic optimal control of spacecraft formation attitude alignment for gravitational wave detection is of great significance. Summary of the Invention

[0004] To address the shortcomings of the aforementioned technologies, this invention provides a lossless convexity method for stochastic optimal control of the attitude alignment of gravitational wave spacecraft formations.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] The present invention provides a lossless convexity transformation method for stochastic optimal control of attitude alignment in gravitational wave spacecraft formations, comprising the following steps:

[0007] Step 1: Establish a spacecraft attitude dynamics model based on quaternions 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 constraints.

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

[0009] Step 3: Express the moment constraints and orientation constraints containing uncertainty through the relatively loose constraint form of chance constraints;

[0010] Step 4: Using the convex relaxation technique of Schuler's complement theorem and the method of introducing dummy variables, the non-convex constraint terms are made convex to obtain the equivalent convex problem of the original problem.

[0011] Furthermore, step 1 specifically includes: setting relevant parameters for the mission scenario, including the spacecraft's moment of inertia J and maximum thrust torque. Initial time t0, terminal time t f The initial state of the spacecraft x0 (including quaternion q0 and angular velocity ω0), and the final state of the spacecraft x f (Including quaternion q) f and angular velocity ω f ); and parameters describing the pointing constraints, including the sensor's orientation vector in the body coordinate system. The direction vector of a bright celestial body in an inertial coordinate system The direction vector of the center of observation in the body coordinate system The corresponding sensor field of view half angle and .

[0012] Further, step 2 specifically includes establishing the spacecraft attitude dynamics equations:

[0013] (11);

[0014] in For unit quaternions, This represents the first derivative of the unit quaternion with respect to time. For the spacecraft's angular velocity, For the spacecraft's rotational inertia, This represents the first derivative of the spacecraft's angular velocity with respect to time, with the superscript T indicating transpose. For the spacecraft's rotational inertia, This is the thrust torque. Wherein, Indicates rotation angle, , , These represent the directions of rotation about the X, Y, and Z axes, respectively. , , These represent the angular velocity components about the X, Y, and Z axes, respectively. , , These represent the magnitudes of the thrust torque along the X, Y, and Z axes, respectively. , , These represent the moments of inertia of the rigid body about the X, Y, and Z axes, respectively. Let t represent the system's state variables, and t represent time. The attitude dynamics equations are related to state, thrust torque, and time.

[0015] Linearizing the above dynamic equations using a first-order Taylor expansion yields a linear stochastic system:

[0016] (12);

[0017] in Given a state vector, the coefficient matrix is ​​defined as follows:

[0018] (13);

[0019] Put time Divide into N equal parts:

[0020] (14);

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

[0022] (15);

[0023] definition The discrete linear stochastic system is:

[0024] (16);

[0025] in, For system status, It is a system matrix, describing the relationships between state variables within the system. It is the input matrix, representing the effect of the input on each state variable. For thrust torque, Let k be the uncertainty matrix, where the index k represents the state at time k, and the index k+1 represents the state at time k+1. State feedback is used to control the spacecraft's attitude to reach the target state; the control law is:

[0026] (17);

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

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

[0029] (18);

[0030] in, Represents the expected probability. This represents the maximum thrust torque. Then, the probabilistic constraint is transformed into a more general constraint form:

[0031] (19);

[0032] (20);

[0033] in, It is the dimension of torque. This represents the coefficients related to the expected probability and the dimension of the variable.

[0034] Accordingly, the opportunity constraint of the pointing constraint takes the form of:

[0035] (twenty one);

[0036] in, , It is a mean matrix. The mean is 0 and the covariance is Gaussian distribution. Transform it into a general hard-constrained form:

[0037] (twenty two);

[0038] in, It is the inverse cumulative distribution function of the standard normal distribution. Vectorization is represented.

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

[0040] (twenty three);

[0041] in, This represents the mean torque at time k. express Reference value;

[0042] Using Schul's complement theorem, torque constraints can be equivalently expressed as:

[0043] (twenty four);

[0044] in, It is a non-negative auxiliary variable. The Schul complement of a block matrix;

[0045] Moment constraints can be written in the form of linear inequalities:

[0046] (25);

[0047] in, , This represents the reference torque. To obtain a convex problem, in... Perform a Taylor expansion at the reference value:

[0048] (26);

[0049] in, Indicates the relaxation term; This represents the average value of the reference torque.

[0050] Add penalty items In the objective function.

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

[0052] (27);

[0053] The second term of Schul complement theorem can be used to transform it into a second-order cone constraint:

[0054] (28);

[0055] Add penalty items In the objective function.

[0056] The advantages of this invention over the prior art are as follows: The lossless convexity method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment proposed in this invention makes up for the deficiency of not considering uncertainty in the traditional attitude alignment process; lossless convexity is achieved by using nonlinear stochastic systems and state feedback control for stochastic optimal control of gravitational wave spacecraft formation attitude alignment; the chance constraint is more consistent with the system in which the state or parameters have a probability distribution form, thus enhancing the flexibility of the system. Attached Figure Description

[0057] Figure 1 This is a schematic diagram of the method flow of the present invention. Detailed Implementation

[0058] To facilitate understanding by those skilled in the art, the present invention will be further described below with reference to examples and accompanying drawings. The content mentioned in the embodiments is not intended to limit the present invention.

[0059] Reference Figure 1 As shown, a lossless convexity transformation method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment is described, with the following steps:

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

[0061] Set the relevant parameters for the spacecraft and its mission scenario, specifically 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), and the target state of the spacecraft x f (Including quaternion q) f and angular velocity ω f (and parameters describing the pointing constraints, including the sensor's orientation vector in the body coordinate system). The direction vector of a bright celestial body in an inertial coordinate system The direction vector of the center of observation in the body coordinate system The corresponding sensor field of view half angle and .

[0062] 2) Establish a stochastic attitude dynamics model for the spacecraft.

[0063] A spacecraft attitude dynamics equation based on quaternions is established, and a stochastic term is added to obtain the corresponding stochastic attitude dynamics equation. The stochastic system is discretized, and the torque is controlled according to a zero-order hold law. The spacecraft attitude is controlled to reach the target state through state feedback control.

[0064] 3) Introduce opportunity constraints to reformulate torque constraints and orientation constraints.

[0065] To handle the uncertainties in the system, hard constraints are relaxed to probabilistic form. Chance constraints are transformed into equivalent, easily tractable general constraints.

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

[0067] By using the Schuler theorem and introducing dummy variables, the non-convex terms of the constraint terms in step 3 are made convex, resulting in a convex problem of the equivalent original problem.

[0068] The following example illustrates the spacecraft attitude alignment process during the LISA mission:

[0069] Step 1: First, in this problem, it is assumed that both the initial state of the spacecraft and the target state are known. The spacecraft state can be measured by the star sensor, so this assumption is reasonable. It is assumed that the spacecraft is equipped with a field-launched electric thruster (FEEP) capable of providing 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 initial quaternion of the spacecraft is q0 = [0.73224, 0.07897, 0.42632, 0.52521], and the initial angular velocity of the spacecraft is ω0 = [0.08606, 0.10219, 0.06650] rad / s. The target quaternion is q f =[0.74458,-0.068725,-0.428083,-0.507564], angular velocity ω of the spacecraft target. 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 for the pointing constraints are set as follows: the direction vector of taboo cone 1 is f1=[0.5000,-0.8660,0], the direction vector of taboo cone 2 is f2=[0.2432,0.9077,-0.3420], the direction vector of taboo cone 3 is f3=[0.4924,0.0868,-0.8660], and the half-angle of the field of view is 30 degrees deg. The direction vector of the forced cone is m=[-0.7660,0.6428,0], the half-angle of the field of view is 60 degrees deg, and the expected probability is 0.95.

[0070] Step 2: Establish stochastic attitude dynamics equations based on quaternions:

[0071] (29);

[0072] in For state variables, This 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 stochastic system is:

[0078] (33);

[0079] in, For system status, It is a system matrix, describing the relationships between state variables within the system. It is the input matrix, representing the effect of the input on each state variable. For thrust torque, This is an uncertainty matrix, where the subscript k represents the state at time k, and the subscript k+1 represents the state at time k+1. State feedback is used to control the spacecraft's attitude to reach the target state. The control law for using state feedback to control the spacecraft's attitude to reach the target state is:

[0080] (34);

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

[0082] Step 3: Introduce opportunity constraints. The moment constraint and the general form of the pointing constraint are as follows:

[0083] (35);

[0084] (36);

[0085] in, Represents the expected probability. Indicates the maximum thrust torque. , It is a mean matrix. The mean is 0 and the covariance is Gaussian distribution. Transform it into a general hard-constrained form:

[0086] (37);

[0087] (38);

[0088] in, It is the dimension of torque. This represents the coefficients related to the expected probability and the dimension of the variable.

[0089] (39);

[0090] in, It is the inverse cumulative distribution function of the standard normal distribution. Vectorization is represented.

[0091] Step 4: Convexification of Constraints. The pointing constraints and moment constraints converted to general hard constraints in Step 3 are convexified. The moment constraints are written in the form of linear inequalities:

[0092] (40);

[0093] in, Indicates the reference torque. This represents the mean torque at time k. express The reference value. To obtain the convex problem, in Perform a Taylor expansion at the reference value:

[0094] (41);

[0095] in, It is a non-negative auxiliary variable. , Indicates the relaxation term. This represents the average value of the reference torque;

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

[0097] (42);

[0098] The second term of Schul complement theorem can be used to transform it into a second-order cone constraint (SOC):

[0099] (43);

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

[0101] As can be seen from the above examples, this invention establishes a discrete stochastic attitude dynamics model for spacecraft; utilizes state feedback to control the uncertainties in the initial state; relaxes the constraints into probabilistic form through chance constraints; and uses the Schur complement theorem and the introduction of dummy variables to convexize the non-convex terms in the constraints to obtain a directly solvable convex problem.

[0102] This invention has many specific applications. The above description is only a preferred embodiment of this invention. It should be noted that for those skilled in the art, several improvements can be made without departing from the principle of this invention, and these improvements should also be considered within the scope of protection of this invention.

Claims

1. A lossless convexity transformation method for stochastic optimal control of attitude alignment in gravitational wave spacecraft formations, characterized in that, include: Step 1: Establish a spacecraft attitude dynamics model based on quaternions 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 constraints. Step 2: Add a noise term representing uncertainty to the spacecraft attitude dynamics model to establish a spacecraft stochastic attitude dynamics model; Step 3: Express the moment constraints and orientation constraints containing uncertainty through opportunity constraints; Step 4: Using the convex relaxation technique of Schuler's complement theorem and the method of introducing dummy variables, the non-convex constraint terms are made convex to obtain the equivalent convex problem of the original problem.

2. The lossless convexity reduction method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1, characterized in that, In step 1, the attitude dynamics equations based on quaternions are established as follows: (1)； in, It is a unit quaternion, and the superscript T indicates transpose. This represents the first derivative of the unit quaternion with respect to time. For the spacecraft's angular velocity, For the spacecraft's rotational inertia, This represents the first derivative of the spacecraft's angular velocity with respect to time. For thrust torque; ； in, Indicates rotation angle, , , These represent the directions of rotation about the X, Y, and Z axes, respectively. , , These represent the angular velocity components about the X, Y, and Z axes, respectively. , , These represent the magnitudes of the thrust torque along the X, Y, and Z axes, respectively. Define state vector The attitude dynamics equations are expressed as follows: ； in, , , Let X, Y, and Z represent the moments of inertia of the rigid body about the X, Y, and Z axes, respectively. Let t represent the system's state variables, and t represent time. The attitude dynamics equations are related to state, thrust torque, and time. The initial and final state constraints of the spacecraft are as follows: ； in, These represent the initial time and the terminal time, respectively. Pointing constraints include taboo constraints and mandatory constraints, which are represented as follows: ； in, This represents the direction vector of the sensor in the body coordinate system. This represents the direction vector of a bright celestial body in an inertial coordinate system. This represents the direction vector of the center of observation in the body coordinate system. and These represent the corresponding sensor field of view half-angles; they are then restated in a more general form: ； in, , These correspond to the real symmetric matrices in the quadratic form; ； in, The direction vector representing the taboo constraint. Represents the direction vector of the sensor. This represents the direction vector of the forced constraint. Represents a three-dimensional identity matrix.

3. The lossless convexity reduction method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1, characterized in that, In step 2, a stochastic attitude dynamics model of the spacecraft is established, including: multidimensional Brownian motion to describe uncertainties. and the initial state that follows a Gaussian distribution , including the mean Covariance .

4. The lossless convexity reduction method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1, characterized in that, In step 2, based on nonlinear stochastic systems The discrete-time linear stochastic system is obtained by first-order Taylor expansion and discretization: ； in, ； in, It is a system matrix that describes the relationships between state variables within the system. It is the input matrix, representing the effect of the input on each state variable. This represents the effect of random disturbances on state variables. This indicates a random perturbation. Indicates thrust torque. Indicates from arrive The state transition matrix is ​​given by the index k, which represents the state at time k, and the index k+1, which represents the state at time k+1.

5. The lossless convexity reduction method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1, characterized in that, In step 3, the chance constraint is relaxed to a probabilistic form, allowing for a small probability of violation; the probabilistic forms of the torque constraint and the pointing constraint are: ； Where P represents the probability of the event occurring. denoted by , p represents the maximum thrust torque, and p is the expected probability.

6. The lossless convexity reduction method for stochastic optimal control of gravitational wave spacecraft formation attitude alignment according to claim 1, characterized in that, In step 4, the chance constraint satisfying the Gaussian distribution is equivalent to a more conservative constraint form; the nonlinear constraint in the optimization problem is transformed into a linear matrix inequality using the Schur complement theorem, 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 magnitude constraint of the control torque is expanded at the reference torque value, and a penalty term is also added to the objective function to ensure that all constraint conditions are met.

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