A lossless convexification method for stochastic optimal control of six-degree-of-freedom spacecraft

Abstract

This invention discloses a lossless convexity transformation method for stochastic optimal control of a six-degree-of-freedom spacecraft, comprising the following steps: Under an uncertain environment, setting the initial and final state parameters of the six-degree-of-freedom spacecraft, including position, velocity, spin angular velocity, and modifying the Rodriguez parameter; the initial state contains uncertainty information, and the final state limits its uncertainty range; establishing a dynamic model of the six-degree-of-freedom spacecraft with uncertainty; establishing an open-loop optimal control problem and obtaining an initial reference solution; employing closed-loop control with state feedback, applying uncertain path constraints, viewpoint constraints, and control amplitude constraints, and using lossless convexity transformation techniques to transform them into convex constraints; obtaining the corresponding robust control strategy through continuous convex optimization methods; this invention uses deterministic transformation of chance constraints, auxiliary variables, and virtual control to perform lossless convexity transformation of uncertain constraints, and uses a simplified polyhedral gravitational field model to improve computational efficiency.

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 a six-degree-of-freedom spacecraft. Background Art

[0002] Due to the strong uncertainty of the space environment itself, and the possibility of certain failures and damage to the spacecraft during the mission, the asteroid exploration process must have a certain degree of uncertainty. The irregular shape of the asteroid and the small mass of the asteroid lead to an irregular weak gravitational field near it. When the spacecraft arrives near the asteroid, its motion environment is more complex and there are many unknown disturbance factors, which leads to dynamic uncertainty and uncertainty in the spacecraft state parameters. In addition, due to the irregular shape of the asteroid, the spacecraft is prone to collide with it during close-range detection. In addition, when approaching the surface of the asteroid, it is necessary to control the orientation of the sensors on the spacecraft to achieve a precise soft landing. The problem of random control of uncertain trajectories in different scenarios has become a hot area of ​​research.

[0003] The existing convexification methods need to meet certain prerequisites to be implemented, and their scene settings are relatively simple. In addition, in actual detection missions, there are certain requirements for the attitude of the spacecraft. Summary of the invention

[0004] In view of the deficiencies in the above-mentioned technologies, the present invention provides a lossless convexification method for random optimal control of a six-degree-of-freedom spacecraft, which can solve the trajectory optimization problem of a six-degree-of-freedom spacecraft under an uncertain environment.

[0005] To achieve the above object, the present invention provides a lossless convexification method for stochastic optimal control of a six-degree-of-freedom spacecraft, comprising the following steps:

[0006] Step 1, determining the state parameters of the six-degree-of-freedom spacecraft; wherein the state parameters include initial state parameters and terminal state parameters;

[0007] Step 2: Establish a six-degree-of-freedom spacecraft dynamics model with uncertainty;

[0008] Step 3: establishing an open-loop optimal control problem according to the state parameters and the six-degree-of-freedom spacecraft dynamics model to obtain a reference solution;

[0009] Step 4: Based on the reference solution obtained in step 3, a closed-loop control with state feedback is adopted to impose path constraints, viewing angle constraints, and control amplitude constraints with uncertainties and convert them into convex constraints using lossless convexification technology;

[0010] Step 5: Obtain the corresponding robust control strategy through continuous convex optimization method.

[0011] Optionally, in step 1, the initial state parameters include relevant parameters under an uncertain scenario, and the final state limits the uncertainty range of the relevant parameters.

[0012] Optionally, the step 1 includes:

[0013] Set relevant parameters in uncertain scenarios, including flight time, control threshold, disturbance parameter, confidence level, penalty coefficient, and convergence index;

[0014] Then, the initial and final state parameters of the spacecraft are set to have Gaussian distribution, expressed as and ,in, represents a Gaussian distribution, is the initial state parameter of the spacecraft, is the initial state expectation, is the initial state covariance; is the spacecraft terminal state parameter, is the terminal state expectation, is the terminal state covariance;

[0015] Finally, all variables are dimensionless, parameters are reasonably selected to keep the state quantity in uniform dimension, and the state covariance is additionally scaled using a scaling matrix.

[0016] Optionally, step 2 includes:

[0017] The orbital dynamics model is constructed using the four-particle gravitational field model, and the attitude dynamics model is constructed using the modified Rodriguez method. The two are combined to establish the six-degree-of-freedom spacecraft dynamics equations. ,in x For status, u For control, t is time; considering the disturbance in the dynamic process, the uncertain disturbance is modeled as a standard Brownian motion W (t) and apply the perturbation coefficient G d (t), add dynamic uncertainty to construct the stochastic dynamic equation:

[0018] (1);

[0019] The dynamic equation is transformed into a linear time-varying system through the first-order Taylor expansion.

[0020] (2);

[0021] in, A ( t ) is the Jacobian matrix of the dynamic equation with respect to the state, B( t ) is the Jacobian matrix of the dynamic equation with respect to control, c ( t ) is the higher-order term of Taylor expansion, using the time step Δ t Discretize the state into N intervals, and in the k Interval The upper integral establishes the dynamic equation of state, subscript k Indicates k Time interval

[0022] (3);

[0023] in,

[0024] , , , ;

[0025] in, is the state transfer function, represents the state transfer matrix, is the control coefficient matrix, is the integral term corresponding to the higher-order term, is the integral term corresponding to the disturbance; the state quantity has initial uncertainty, and the first two statistical moments of the state estimate, namely the expectation and covariance, are analyzed; the dynamic equation is analyzed using the covariance analysis method to obtain the dynamic equation of expectation and covariance:

[0026] (4);

[0027] (5);

[0028] In the formula, is the state expectation, To control the amount, is the state covariance, ,in, express The transpose of .

[0029] Optionally, step 3 includes: adding a control amplitude constraint to keep the control variable within a set threshold, and adding boundary conditions according to the set initial and final states; taking the sum of discrete values ​​of the control amplitude as the objective function, using the dynamic equation established in step 2, establishing a convex optimization model, and using the CVX toolbox in MATLAB to process the stochastic optimal control problem with open-loop control.

[0030] Optionally, step 4 includes:

[0031] Based on the reference solution, a closed-loop feedback control with state feedback is added, and the control variable is fed back according to the difference between the state estimate and the expected value, as shown in formula (6):

[0032] (6);

[0033] in, For control, To control expectations, For status, is the state expectation, K is the feedback matrix; Substituting equation (6) into equation (4) and equation (5) yields the expected and covariance dynamic equations with feedback control:

[0034] (7);

[0035] (8);

[0036] in, is the state expectation, is the state transfer matrix, is the control coefficient matrix, is the integral term corresponding to the higher-order term, is the state covariance, , K k is the feedback matrix;

[0037] Introducing new variables After relaxing the equation, the Schur complement theorem is used to decouple the feedback matrix and the covariance matrix in the covariance dynamics and convexify the equation (8) into

[0038] (9);

[0039] in, is the state covariance, is the state transfer matrix, is the control coefficient matrix, K k is the feedback matrix, is a new variable for decoupling; path constraints and view constraints are added; the path constraint consists of a no-entry zone and a slope constraint, the no-entry zone consists of three ball constraints, and the slope constraint is a second-order cone constraint; the view constraint has a second-order cone constraint form and is imposed in the landing phase; these constraints need to be converted into chance constraints due to the existence of uncertainty, which are in the following two forms:

[0040] (10);

[0041] (11);

[0042] in, represents the opportunity constraint, x, y are the variables that need to be constrained. a , b , ρ is the constant required by the constraint, represents the probability of violating the constraint; by using auxiliary variables and dummy controls, the chance constraint is convexified into a convex constraint; the second-order cone constraint has the following form

[0043] (12);

[0044] in, a, b, c, d are constraint coefficients; for the second-order cone constraint, by introducing the intermediate auxiliary variable m , split the second-order cone constraint in the form of chance constraint into two parts

[0045] (13);

[0046] Then, additional auxiliary variables and virtual controls are used to convexify them respectively, thereby constructing a stochastic convex optimization problem with closed-loop feedback.

[0047] Optionally, step 5 includes:

[0048] Use continuous convex optimization method to iteratively solve the problem model established in step 4, use CVX toolbox to solve it, and use the solution as a reference value;

[0049] Repeat step 4 for iterative solution until the difference between the two iterative solutions meets the convergence condition, thereby obtaining a robust control strategy under uncertain scenarios.

[0050] The accuracy and effectiveness of the results can be verified through Monte Carlo simulation of a large number of samples.

[0051] The beneficial effects of the present invention compared with the prior art are as follows: the lossless convexification method is suitable for detection trajectories in uncertain environments; the six-degree-of-freedom spacecraft random dynamics equation is adopted, and the state estimation value analysis is performed through the covariance analysis method; closed-loop control of state feedback is applied, and corresponding constraints are imposed according to the scene, and the constraints are convexified by the lossless convexification technology to establish a random convex optimization problem; the continuous convex optimization method is applied, and a robust control strategy is obtained through multiple solution iterations; Monte Carlo simulation verifies the effectiveness and applicability of the solution strategy, and the terminal state of the spacecraft under a large number of samples satisfies the terminal constraints. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 A flowchart of a lossless convexification method for random optimal control of a six-degree-of-freedom spacecraft provided in accordance with an embodiment of the present invention. DETAILED DESCRIPTION

[0053] In order to facilitate the understanding of those skilled in the art, the present invention is further described below in conjunction with embodiments and drawings. The contents mentioned in the implementation modes are not intended to limit the present invention.

[0054] Reference Figure 1 As shown in Figure 1, a lossless convexification method for stochastic optimal control of a six-degree-of-freedom spacecraft is presented. The steps are as follows:

[0055] 1) Set the initial and final state parameters of the six-degree-of-freedom spacecraft

[0056] Set the state parameters of the six-degree-of-freedom spacecraft, including position, velocity, spin angular velocity, and modify the Rodriguez parameters. The initial state of the spacecraft has a certain uncertainty and follows a Gaussian distribution. The setting parameters include the state expectation and state covariance matrix. Based on the initial state distribution of the spacecraft, the terminal state parameters of the spacecraft are given, and the distribution range of the terminal of the spacecraft is limited according to different scenarios.

[0057] 2) Establishing a six-degree-of-freedom spacecraft dynamics model with uncertainty

[0058] The four-particle gravitational field model is adopted. Based on the polyhedron gravitational field model, four particles are used to fit the gravitational field to construct the orbital dynamics model. Based on the classical quaternion method, the attitude dynamics model is constructed by modifying the Rodriguez parameters. This method can optimize the parameter dimension and avoid the singularity of dynamics, which reduces the amount of calculation to a certain extent. The two are combined into a six-degree-of-freedom spacecraft dynamics model. Due to the uncertainty of dynamics, uncertainty terms are added to the dynamics according to the stochastic optimal control method, and the first two statistical moments of the state estimate, namely the expectation and covariance, are analyzed to establish a six-degree-of-freedom spacecraft dynamics model with uncertainty.

[0059] 3) Establishing the open-loop optimal control problem

[0060] A simple open-loop control problem is established. According to the given spacecraft state parameters, an initial guess is arbitrarily generated, and the open-loop solution is obtained by performing 1-2 optimizations through the MATLAB convex optimization toolbox CVX.

[0061] 4) Establish a closed-loop optimal control problem and convexify it

[0062] The closed-loop control with state feedback is adopted, and the path constraint, view constraint, and control amplitude constraint with uncertainty are imposed to construct the closed-loop optimal control problem. The uncertainty constraint is transformed into a chance constraint with a high confidence level. The chance constraint is convexified by auxiliary variables and virtual control is introduced to ensure that the convexification process is lossless. Based on the open-loop solution obtained in step 3), the convex optimization toolbox is also used to solve it.

[0063] 5) Obtain the corresponding robust control strategy through continuous convex optimization method

[0064] The solution obtained in step 4) is used as a reference value, and step 4) is repeated until the difference between the two iterative solutions meets the convergence condition, thereby obtaining a robust control strategy under uncertain scenarios.

[0065] The following is an example of the hopping trajectory of asteroid 1996 HW1:

[0066] Step 1: First, in this problem, the state covariance matrix dimension is too different from the variable. First, scale the covariance matrix; use the scaling matrix to scale the initial and final state covariances according to the covariance size, so that the covariance matrix dimension is close to 1. Then make all variables dimensionless; the length of the longest axis of the asteroid divided by 2 is the unit distance, and the inverse of the asteroid's spin angular velocity is the unit time. Asteroid spin angular velocity ω ast =1.9929×10 -4 1 / s, maximum acceleration a max =5×10 -6 km / s 2 , maximum torque M max =2×10 -3 N·m 2 , flight time t f =3600 s, spacecraft inertia matrix J=diag([2014,1897,1357])kg·m 2 , take-off position path constraint angle θ i =30deg, landing position path constraint angle θ f =60deg, viewing angle constraint angle θ FOV = 25deg, sensor installation position d in the spacecraft system s =[0,0,-1] T m, the sensor orientation n in the spacecraft system s =[0,0,-1] T , confidence level 99.9%, convergence index ε=1×10 -6 , penalty coefficient η = 1 × 10 4 , the initial reference value of the auxiliary parameter τ=0.01.

[0067] Next, set the state parameters of the six-degree-of-freedom spacecraft. The state parameter x is composed of position r, velocity v, spacecraft spin angle ω, and modified Rodriguez parameter σ. The control variable u is composed of acceleration a and torque M. Given the initial state parameters: initial position r0=[-0.42772,0.24801,-0.12193] T km, initial position covariance P r0=diag([(1×10 -5 ) 2 , (1×10 -5 ) 2 , (1×10 -5 ) 2 ]) km 2 , initial velocity v0=[0,0,0] T km / s, initial velocity covariance P v0 =diag([(1×10 -6 ) 2 , (1×10 -6 ) 2 , (1×10 -6 ) 2 ]) (km / s) 2 , initial spin angular velocity ω0=[0,0,0] T 1 / s, initial spin angular velocity covariance P ω0 =diag([(1×10 -6 ) 2 , (1×10 -6 ) 2 , (1×10 -6 ) 2 ]) (1 / s) 2 , initial modified Rodriguez parameter σ0=[-0.40889,-0.37001,0] T , initial modified Rodriguez parameter covariance P σ0 =diag([(1×10 -3 ) 2 , (1×10 -3 ) 2 , (1×10 -3 ) 2 ]). Given the terminal state parameters, the terminal position r f =[0.71802,0.71131,0.22479] T km, terminal position covariance P rf =diag([(1×10 -5 ) 2 , (1×10 -5 ) 2 , (1×10 -5 ) 2 ]) km 2 , terminal speed v f =[0,0,0] T km / s, terminal velocity covariance P vf =diag([(1×10 -6 ) 2, (1×10 -6 ) 2 , (1×10 -6 ) 2 ])(km / s) 2 , the terminal spin angular velocity ω f =[0,0,0] T 1 / s, terminal spin angular velocity covariance P ωf =diag([(1×10 -6 ) 2 ,(1×10 -6 ) 2 , (1×10 -6 ) 2 ]) (1 / s) 2 , the terminal modified Rodrigues parameter σ f =[-0.30281,-0.06987,0] T , terminal modified Rodriguez parameter covariance P σf =diag([(1×10 -3 ) 2 , (1×10 -3 ) 2 , (1×10 -3 ) 2 ]).

[0068] Step 2: Establish the stochastic dynamics equation of the six-degree-of-freedom spacecraft, expressed as

[0069] (1)

[0070] in,

[0071] (2)

[0072] In the formula, Ω represents the vector expression of the asteroid's spin angular velocity in the asteroid system, I3 represents the three-dimensional unit matrix, represents the direction cosine matrix expressed in terms of modified Rodriguez parameters, is the acceleration generated by the asteroid’s gravitational field, which can be expressed as

[0073] (3)

[0074] in, is the gravitational constant corresponding to the ith particle, is the distance from the spacecraft to the i-th particle. The second term in the dynamic equation represents the disturbance, W represents the standard Brownian motion, G d is the disturbance coefficient, G d =diag([0 3×3, (1×10-7 )I 3, (1×10 -8 )I 3, 0 3×3 ]).

[0075] The six-degree-of-freedom spacecraft dynamic equations are transformed into a linear time-varying system through Taylor expansion and discretized with a uniform time step of 60 s. The expectation and covariance of the state are analyzed using the covariance analysis method to obtain the expectation and covariance dynamic equations of open-loop control.

[0076] When establishing the six-DOF spacecraft dynamics system, the attitude dynamics and four-particle gravitational field model are established by modifying the Rodriguez parameters, which simplifies the calculations.

[0077] Step 3: Constrain the initial and final values ​​of the state expectation and covariance, add the chance constraint of the control amplitude, and establish the open-loop optimal control problem with the discretized control amplitude and as the optimization objective function.

[0078] Step 4. A reference solution can be obtained from step 3, including reference values ​​such as state and control. Add closed-loop feedback control with state feedback, and the control variable is fed back according to the difference between the state estimate and the expected value. Introduce new variables to decouple and convexify the feedback matrix and covariance matrix in covariance dynamics. Then add path constraints and perspective constraints; the path constraint consists of a no-entry zone and a slope constraint, the no-entry zone consists of three ball constraints, and the slope constraint is a second-order cone constraint; the perspective constraint also has a second-order cone constraint form and is imposed in the landing section. By introducing intermediate auxiliary variables, the second-order cone constraint in the form of a chance constraint is split into two parts, and then convexified separately using additional auxiliary variables and virtual control. In this way, a stochastic convex optimization problem with closed-loop feedback is constructed.

[0079] Step 5: Use the result of step 4 as the new reference solution and repeat step 4) until the difference between the two iterative solutions meets the convergence condition ε, thereby obtaining a robust control strategy for uncertain scenarios. In this scenario, convergence occurs after 8 iterations, and the expected range of the state error is [10 -4 m, 10 -8 m / s,10 -7 1 / s,10 -8 ]. The effectiveness of the control strategy can be verified by Monte Carlo simulation method.

[0080] It can be seen from the above embodiments that the present invention adopts the six-degree-of-freedom spacecraft stochastic dynamics equation, and analyzes the state estimation value through the covariance analysis method; applies closed-loop control of state feedback, and imposes corresponding constraints according to the scenario, convexifies the constraints through lossless convexification technology, and establishes a random convex optimization problem; applies the continuous convex optimization method, and obtains a robust control strategy through multiple solution iterations; Monte Carlo simulation verifies the effectiveness and applicability of the solution strategy, and the terminal state of the spacecraft under a large number of samples satisfies the terminal constraints.

[0081] 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 principle of the present invention. These improvements should also be regarded as the protection scope of the present invention.

Claims

1. A lossless convexification method for stochastic optimal control of a six-degree-of-freedom spacecraft, characterized in that: include: Step 1, determining the state parameters of the six-degree-of-freedom spacecraft; The state parameters include initial state parameters and terminal state parameters; Step 2: Establish a six-degree-of-freedom spacecraft dynamics model with uncertainty; Step 3: establishing an open-loop optimal control problem according to the state parameters and the six-degree-of-freedom spacecraft dynamics model to obtain a reference solution; Step 4: Based on the reference solution obtained in step 3, a closed-loop control with state feedback is adopted to impose path constraints, viewing angle constraints, and control amplitude constraints with uncertainties and convert them into convex constraints using lossless convexification technology; Step 5: Obtain the corresponding robust control strategy through continuous convex optimization method.

2. The lossless convexification method according to claim 1, characterized in that: In step 1, the initial state parameters include relevant parameters in an uncertain scenario, and the final state limits the uncertainty range of the relevant parameters.

3. The lossless convexification method according to claim 2, characterized in that: The step 1 comprises: Set relevant parameters in uncertain scenarios, including flight time, control threshold, disturbance parameter, confidence level, penalty coefficient, and convergence index; Then, the initial and final state parameters of the spacecraft are set to have Gaussian distribution, expressed as and ,in, represents a Gaussian distribution, is the initial state parameter of the spacecraft, is the initial state expectation, is the initial state covariance; is the spacecraft terminal state parameter, is the terminal state expectation, is the terminal state covariance; Finally, all variables are dimensionless, parameters are reasonably selected to keep the state quantity in uniform dimension, and the state covariance is additionally scaled using a scaling matrix.

4. The lossless convexification method according to claim 1, characterized in that: Step 2 includes: The orbital dynamics model is constructed using the four-particle gravitational field model, and the attitude dynamics model is constructed using the modified Rodriguez method. The two are combined to establish the six-degree-of-freedom spacecraft dynamics equations. ,in x For status, u For control, t is time; considering the disturbance in the dynamic process, the uncertain disturbance is modeled as a standard Brownian motion W(t) And apply the perturbation coefficient G d (t) , add the kinetic uncertainty to construct the stochastic kinetic equation: （1）； The dynamic equation is transformed into a linear time-varying system through the first-order Taylor expansion. （2）； in, A ( t ) is the Jacobian matrix of the dynamic equation with respect to the state, B ( t ) is the Jacobian matrix of the dynamic equation with respect to control, c ( t ) is the higher-order term of Taylor expansion, using the time step Δ t Discretize the state into N intervals, and in the k Interval The upper integral establishes the dynamic equation of state, subscript k Indicates k Time interval （3）； in, , , , ; in, is the state transfer function, represents the state transfer matrix, is the control coefficient matrix, is the integral term corresponding to the higher-order term, is the integral term corresponding to the disturbance; the state quantity has initial uncertainty, and the first two statistical moments of the state estimate, namely the expectation and covariance, are analyzed; the dynamic equation is analyzed using the covariance analysis method to obtain the dynamic equation of expectation and covariance: （4）； （5）； In the formula, is the state expectation, To control the amount, is the state covariance, .

5. The lossless convexification method according to claim 4, characterized in that: The step 3 includes: adding a control amplitude constraint to keep the control variable within a set threshold, and adding boundary conditions according to the set initial and final states; taking the sum of discrete values ​​of the control amplitude as the objective function, using the dynamic equation established in step 2, establishing a convex optimization model, and using the CVX toolbox in MATLAB to process the stochastic optimal control problem with open-loop control.

6. The lossless convexification method according to claim 4, characterized in that: Step 4 includes: Based on the reference solution, a closed-loop feedback control with state feedback is added, and the control variable is fed back according to the difference between the state estimate and the expected value, as shown in formula (6): （6）； in, For control, To control expectations, For status, is the state expectation, K is the feedback matrix; Substituting equation (6) into equation (4) and equation (5) yields the expected and covariance dynamic equations with feedback control: （7）； （8）； in, is the state expectation, is the state transfer matrix, is the control coefficient matrix, is the integral term corresponding to the higher-order term, is the state covariance, , K k is the feedback matrix; Introducing new variables After relaxing the equation, the Schur complement theorem is used to decouple the feedback matrix and the covariance matrix in the covariance dynamics and convexify the equation (8) into （9）； in, is the state covariance, is the state transfer matrix, is the control coefficient matrix, K k is the feedback matrix, is a new variable for decoupling; path constraints and view constraints are added; the path constraint consists of a no-entry zone and a slope constraint, the no-entry zone consists of three ball constraints, and the slope constraint is a second-order cone constraint; the view constraint has a second-order cone constraint form and is imposed in the landing phase; these constraints need to be converted into chance constraints due to the existence of uncertainty, which are in the following two forms: （10）； （11）； in, represents the opportunity constraint, x, y are the variables that need to be constrained. a , b , ρ is the constant required by the constraint, represents the probability of violating the constraint; by using auxiliary variables and dummy controls, the chance constraint is convexified into a convex constraint; the second-order cone constraint has the following form （12）； in, a, b, c, d are constraint coefficients; for the second-order cone constraint, by introducing the intermediate auxiliary variable m , split the second-order cone constraint in the form of chance constraint into two parts （13）； Then, additional auxiliary variables and virtual controls are used to convexify them respectively, thereby constructing a stochastic convex optimization problem with closed-loop feedback.

7. The lossless convexification method according to claim 1, characterized in that: Step 5 includes: Use continuous convex optimization method to iteratively solve the problem model established in step 4, use CVX toolbox to solve it, and use the solution as a reference value; Repeat step 4 for iterative solution until the difference between the two iterative solutions meets the convergence condition, thereby obtaining a robust control strategy under uncertain scenarios.

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