A De-spinning Control Method for Underactuated Tethered Assemblies Based on Hierarchical Sliding Mode

Abstract

The present invention discloses an anti-rotation control method for an under-actuated tethered assembly based on hierarchical sliding mode, which is applicable to the attitude control of the assembly system formed after a flexible net captures space debris. Considering the uncertainties in the dynamic parameters of space debris, the errors in the measurement of the target initial attitude, and the existence of external disturbance forces, the present invention supplements and improves the dynamic model of the assembly, and proposes an attitude control method for the assembly based on hierarchical sliding mode. This method stabilizes the entire under-actuated tethered assembly system by applying torques to the actively controlled service satellite. At the same time, in order to improve the stability efficiency of the system, a radial propulsion force is applied to suppress the swing of the service satellite in the radial direction. The control method proposed by the present invention simplifies the control input parameters, and the space debris can be de-spun through the tether only by adjusting the attitude of the service satellite. Compared with the traditional proportional derivative controller, the control performance and robustness of the control method in the present invention are better.

Description

Technical Field

[0001] The present invention discloses a de-spinning control method for an under-actuated tethered assembly based on hierarchical sliding mode, which is applicable to the attitude control of an assembly system formed after a flexible net captures space debris. Background Art

[0002] Space debris poses a threat to satellites in normal service in orbit. Flexible nets are a very promising active space debris removal technology at present. After the net captures space debris, the service satellite, the space debris, and the net form a tethered assembly. The characteristics of the tethered assembly are as follows: 1) It includes a rigid service satellite, space debris, and a flexible net, and is a complex rigid-flexible coupled system; 2) The space debris is a non-cooperative target, and the dynamic model of the tethered assembly is accompanied by uncertainties; 3) The space debris loses power, and the tethered assembly only relies on the service satellite equipped with a control input device for motion control, and is a typical under-actuated system.

[0003] The dynamic model of the tethered assembly is affected by the uncertainties of the dynamic parameters of space debris, the initial attitude error of space debris spin, and external disturbances. On the basis of improving the description of the uncertainties of the dynamic model, the present invention adopts attitude control based on the hierarchical sliding mode method. Compared with the proportional derivative attitude control, the efficiency and accuracy of the de-spinning control are improved, and good robustness is shown. At the same time, an auxiliary radial propulsion force control is applied to suppress the radial swing of the service satellite and accelerate the de-spinning and stabilization control of the tethered assembly system. Summary of the Invention

[0004] The purpose of the present invention is to provide a de-spinning control method for an under-actuated tethered assembly based on hierarchical sliding mode. This method shows good robustness in dealing with the uncertainties of the dynamic parameters of space debris, the initial attitude error, and external disturbances on the basis of ensuring good control efficiency and control accuracy of the control system, and improves the de-spinning and stabilization ability.

[0005] A de-spinning control method for an under-actuated tethered assembly based on hierarchical sliding mode includes the following steps:

[0006] Step 1: Establish the dynamic model of the tethered assembly. Considering the uncertainties of the dynamic parameters of non-cooperative space debris, the initial attitude error of space debris spin, and external disturbances, supplement and improve the dynamic model. On this basis, express the dynamic model in the form of a state space equation;

[0007] Step 2: Based on the dynamic model established in Step 1, define a hierarchical sliding mode surface, design a sliding mode surface reaching law, deduce the control law of the sliding mode control method, and realize the de-spinning control of the tethered assembly.

[0008] Step 3: On the basis of the attitude control in Step 2, design a radial swing suppression controller, set the control objective, and enhance the anti-spin stabilization ability of the tethered assembly.

[0009] Advantages of the Invention

[0010] The present invention mainly relates to an anti-spin control method for an under-actuated tethered assembly based on hierarchical sliding mode, realizing the anti-spin stabilization control of the tethered assembly. Its advantages are as follows: First, considering the uncertain factors existing in the tethered assembly, including the uncertainty of space debris parameters, the initial spin attitude error of space debris, and external disturbances, the dynamic model of the assembly is improved; Second, the torque provided by the service satellite is the control input of the tethered assembly, which simplifies the hardware equipment of the system while ensuring the anti-spin effect; Third, compared with the proportional-derivative attitude controller, the hierarchical sliding mode attitude controller has improved control efficiency and control accuracy, and has strong robustness in dealing with the tethered assembly model containing uncertainties. Description of the Drawings

[0011] Figure 1 is the tethered assembly formed after capture

[0012] Figure 2 is the configuration of the tethered assembly considering the initial attitude error

[0013] Figure 3 is the anti-spin effect of space debris with accurate model in the embodiment

[0014] Figure 4 is the angular velocity distribution of space debris under parameter uncertainty in the embodiment

[0015] Figure 5 is the anti-spin effect of space debris under attitude error and external disturbances in the embodiment Detailed Embodiment

[0016] See Figures 1 - 5 , the technical solution adopted by the present invention includes the following content:

[0017] 1. Establish the dynamic model of the tethered assembly

[0018] 1.1 Dynamic model of the assembly

[0019] The tethered assembly formed after the net capture is as Figure 1 shown. Its dynamic model includes a service satellite, space debris, and a tether. The dynamic models of the service satellite and space debris are established using the Newton-Euler method, and the tether adopts a mass-spring model.

[0020] A. Dynamic model of the service satellite

[0021] F + T0 = m S aS (1)

[0022]

[0023] Among them, F = [F x F y is the propulsive force in the orbital motion direction provided by the thruster to ensure the tension of the tether. T0 = [T 0x T 0y is the tension vector on the main tether, a S = [a Sx a Sy is the linear acceleration of the service satellite. m S , J S are the mass and moment of inertia of the service satellite respectively. The connection point of the main tether on the service satellite is O1P0 = [-l S 0], where l S is half of the length of the service satellite. is the rotation matrix for converting the vector from the inertial coordinate system to the body coordinate system of the service satellite, where θ S is the rotation angle of the service satellite. M C is the torque input provided by devices such as thrusters and momentum wheels on the service satellite.

[0024] B. Space debris dynamics model

[0025] T1 + T2 = m T a T (3)

[0026]

[0027] Among them, T1 = [T 1x T 1y , T2 = [T 2x T 2y are the tension vectors on the two sub-tethers respectively. a T = [a Tx a Ty is the linear acceleration of the space debris. m T , J T are the mass and moment of inertia of the space debris respectively. The connection points of the tether net and the space debris are O2P1 = [l T w T , O2P2 = [l T -w T respectively, where l T , w T are half of the length and width of the space debris respectively. is the rotation matrix for converting the vector from the inertial coordinate system to the body coordinate system of the space debris, where θT is the rotation angle of the space debris.

[0028] C. Tether dynamics model

[0029]

[0030] where k is the stiffness of the tether, and l i is the length of each tether after deformation, and l i0 is the natural length of each tether before deformation, and Δl i = [Δl xi , Δl yi T is the elongation of each tether.

[0031] 1.2 Complementary model uncertainty

[0032] A. Considering the uncertainty of the space debris dynamic parameters, the improved dynamic equation is obtained as follows:

[0033]

[0034] where δ is the boundary value used to correct the nominal inertia coefficient.

[0035] B. Considering the initial attitude error of the space debris, the configuration of the combined body is as Figure 2 shown, and the initial attitude angle is defined as follows:

[0036] α T = π / 4·η (7)

[0037] where η is a random number between 0 and 1.

[0038] The change in the initial attitude angle causes a change in the connection point of the tether to the space debris. The connection point of the tether to the space debris in the inertial coordinate system is defined as:

[0039]

[0040] where, is the rotation matrix that transforms the connection point position vector from the space debris body coordinate system to the inertial coordinate system. Assuming the x - coordinate of the connection point in the inertial coordinate system is d, the position vector of the connection point in the body coordinate system is calculated as:

[0041] C. Considering the external disturbance D ∼ N(0, 0.0001 2 ) N·m received by the space debris, the improved dynamic equation is obtained as follows:

[0042]

[0043] 1.3 Express the dynamic model in the form of a state equation:

[0044]

[0045] Among them,

[0046] f1 = (O2P1 × (R -1 (θ T ) · T1) + O2P2 × (R -1 (θ T ) · T2)) / J T ,

[0047] f2 = (O1P1 × (R -1 (θ S ) · T0)) / J S ,

[0048] u1 = 0, u2 = M C , b2 = 1 / J s .

[0049] 2. Hierarchical sliding mode controller design

[0050] 2.1 Define the hierarchical sliding mode surface

[0051]

[0052] Among them, z1 and z2 are two sub - sliding mode surfaces, s is the overall sliding mode surface, c1, c2, and α are weight coefficients greater than zero, and the values of the desired state variables are set to x 1d = x 2d = x 3d = x 4d = 0.

[0053] 2.2 Design of the sliding mode switching law

[0054] To avoid the jitter problem caused by frequent switching, a saturation function is introduced to establish the switching law:

[0055] ds / dt = -ε · sat(s) + f(x1, x2, x3, x4) (12)

[0056] Among them, ε is the approaching rate, and the definitions of other functions are as follows:

[0057]

[0058] Among them, Δ is the sliding boundary value of the overall sliding mode surface.

[0059]

[0060] Among them, γ is the scaling coefficient of the control input, which is a decimal between 0 and 1.

[0061] 2.3 Derive the sliding mode control law as

[0062]

[0063] 3. Thruster Controller for Suppressing Radial Oscillation

[0064] 3.1 Define the position error and obtain the thruster control law

[0065]

[0066] where y S and are the radial position and velocity of the servicing satellite respectively, and y d and are the expected values of the radial position and velocity of the servicing satellite respectively.

[0067] Y C =-k3·e1 - k4·e2 (17)

[0068] where k3 and k4 are positive control gains, and Y C is the radial force input generated by the thruster.

[0069] 3.2 Set the control objective as:

[0070]

[0071] where y sb is the switching boundary and y pb is the convergence position boundary.

[0072] Thus, the despin control of the tethered assembly is achieved.

[0073] Embodiment

[0074] Based on the underactuated tethered assembly despin control method with hierarchical sliding mode provided by the present invention, verification is carried out with the tethered assembly system shown in Figure 1 . Its parameters are shown in Table 1. The initial conditions of the state variables of the tethered assembly are [x1, x2, x3, x4] = [0, 0.5, 0, 0], and the expected state quantity target is [x 1d , x 2d , x 3d , x 4d = [0, 0, 0, 0].[[]END]]

[0075] Table 1 Tethered Assembly Parameters

[0076]

[0077] The proposed control method of the present invention and the proportional derivative control method are respectively used to perform de-spin control on the tethered assembly. The de-spin effect of the space debris in the assembly is as Figure 3 shown. The results show that the proportional derivative control makes the angular velocity of the space debris enter the 5% error band at 78 s, while the control method proposed by the present invention makes the angular velocity of the space debris enter the 5% error band at 55 s, improving the efficiency of de-spinning the space debris. In terms of control accuracy, within the limited simulation time of 100 s, the accuracy achieved by the proportional derivative control is 1.2° / s, while the accuracy of the control method proposed by the present invention is higher, at 0.5° / s.

[0078] Considering the case where the dynamic parameters of the space debris are inaccurate, the de-spin effect of the space debris in the assembly is as Figure 4 shown. The results show that when the correction coefficient δ varies within ±0.3, within the limited simulation time of 100 s, the accuracy of the proportional derivative control fluctuates within a large range of 0.4° / s to 7° / s, while the control method proposed by the present invention has strong robustness, and the control accuracy fluctuates within a small range of 0.2° / s to 0.8° / s. Considering the case of the initial attitude error of the space debris and the external interference received, the de-spin effect of the space debris in the assembly is as Figure 5 shown. The control method proposed by the present invention has strong robustness and can make the angular velocity of the space debris enter the 5% error band more than 30 s earlier than the proportional derivative control.

Claims

1. A de-spinning control method for an under-actuated tethered combination based on hierarchical sliding mode, characterized in that The space debris can be despun by the tether only through the attitude adjustment of the servicing satellite, thereby realizing the stability of the combined body system. The specific steps are as follows: Step 1: Establish the dynamic model of the tethered combined body, supplement the uncertainty descriptions of multiple parameters involved in the model, and express the dynamic model in the state space, including the following sub-steps: Step 1.1: Establish the dynamic model of the tethered combined body A. Dynamic model of the servicing satellite F + T0 = m S a S (1) Among them, only considering the motion within the orbital plane, F = [F x F y is the propulsive force provided by the thruster in the direction of the motion of the service satellite within the orbital plane to ensure the tension of the tether. T0 = [T 0x T 0y is the tension vector on the main tether. a S = [a Sx a Sy is the linear acceleration of the service satellite. m S , J S are the mass and moment of inertia of the service satellite respectively. The connection point of the main tether on the service satellite is O1P0 = [-l S 0], where l S is half of the length of the service satellite. is the rotation matrix for converting the vector from the inertial coordinate system to the body coordinate system of the service satellite, where θ S is the rotation angle of the service satellite. M C is the torque input provided by devices such as thrusters and momentum wheels on the service satellite. B. Dynamic model of the space debris T1 + T2 = m T a T (3) where, T1 = [T 1x T 1y , T2 = [T 2x T 2y are the tension vectors on two sub-tethers respectively, a T = [a Tx a Ty is the linear acceleration of the space debris, m T , J T are the mass and moment of inertia of the space debris respectively. The connection points of the net and the space debris are O2P1 = [l T w T , O2P2 = [l T -w T , where l T , w T are respectively half of the length and width of the space debris, is the rotation matrix for transforming the vector from the inertial coordinate system to the body coordinate system of the space debris, where θ T is the rotation angle of the space debris; C. Dynamic model of the tether where k is the stiffness of the tether, l i is the length of each tether after deformation, l i0 is the natural length of each tether before deformation, and Δl i = [Δl xi , Δl yi T is the elongation of each tether;​ Step 1.2: Uncertainty description of the dynamic model A. Considering the uncertainty of the dynamic parameters of the space debris, the improved dynamic equation is obtained as follows: where δ is the boundary value used to correct the nominal inertia coefficient; B. Considering the initial attitude error of the space debris, the initial attitude angle is defined as follows: α T = π / 4·η (7) where η is a random number between 0 and 1; The change in the initial attitude angle causes a change in the connection point of the tether and the space debris. The connection point of the tether and the space debris in the inertial coordinate system is defined as: Among them, is the rotation matrix for converting the position vector of the connection point from the space debris body coordinate system to the inertial coordinate system. Assuming the x-direction coordinate of the connection point in the inertial coordinate system is d, the position vector of the connection point in the body coordinate system is calculated as follows: C. Considering the external interference on space debris, Gaussian noise D~N(0,0.0001 2 ) N·m is used to simulate the interference torque on space debris, and the improved dynamic equation is as follows: Step 1.3: The mathematical expression of the dynamic model in the state space is: where, f1 = (O2P1 × (R -1 (θ T ) · T1) + O2P2 × (R -1 (θ T ) · T2)) / J T , f2 = (O1P1 × (R -1 (θ S )) · T0)) / J S , u1 = 0, u2 = M C , b2 = 1 / J s ; Step 2: Design a hierarchical sliding mode control method, including the following sub-steps: Step 2.1 The hierarchical sliding mode surface is defined as: where z1 and z2 are two sub - sliding mode surfaces, s is the overall sliding mode surface, c1, c2, and α are weight coefficients greater than zero, and the desired state variable values are set to x 1d = x 2d = x 3d = x 4d = 0; Step 2.2 The design of the sliding mode reaching law is: ds / dt = -ε·sat(s) + f(x1,x2,x3,x4) (12) where ε is the reaching rate, and the definitions of other functions are as follows: where Δ is the sliding boundary value of the overall sliding mode surface; where γ is the scaling coefficient of the control input, 0 < γ < 1; Step 2.3 Derive the control law as: Step 3: Design a thruster controller to suppress the radial swing; including the following sub-steps: Step 3.1 Define the position error to obtain the thruster control law where y S and are the radial position and velocity of the servicing satellite respectively, and y d and are the expected values of the radial position and velocity of the servicing satellite respectively; Y C = -k3·e1 - k4·e2 (17) where k3 and k4 are positive control gains, and Y C is the radial force input generated by the thruster; Step 3.2 The thruster control objective is: where y sb is the switching boundary, and y pb is the convergence position boundary.

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