Three-axis control method for underactuated sail spacecraft based on planar control moment gyroscope

Abstract

The application discloses a sail spacecraft underactuated three-axis control method based on a planar control moment mechanism and belongs to the technical field of spacecraft attitude control and maintenance. The method is based on an attitude dynamics equation, a kinematics equation and a space geometric relation to establish a sail surface normal control law, a sail surface roll axis angular velocity control law and a sail surface roll angle control law, and then orderly combines the three control laws, so that the underactuated three-axis control is realized only by using the control moment on the sail surface. The application solves the problem that sail spacecraft represented by a solar sail and an atmospheric sail is difficult to autonomously generate a control moment in the roll axis direction to realize three-axis attitude control. The application can reduce the overall mass of a platform of the sail spacecraft due to the use of an additional roll axis moment generating mechanism, reduce the failure risk caused by the use of an additional control mechanism, is favorable for deep space exploration of the solar sail, and prolongs the service life of the sail spacecraft.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of spacecraft attitude control and maintenance, in particular to a sail spacecraft underactuated three-axis control method based on a planar control moment mechanism. BACKGROUND

[0002] A sail spacecraft is a new type of spacecraft that uses the area force widely existing in space to generate power after acting on an extremely thin sail. It includes solar sails, atmospheric sails, magnetic sails, etc. Although the solar radiation pressure, atmospheric resistance and other area forces in space are very small, this continuous thrust will continuously increase the impulse value of the sail spacecraft in the vacuum space environment. With the help of a special mechanism, the sail spacecraft can complete various space tasks that cannot be completed under normal circumstances due to the need for large propellant consumption.

[0003] Since the sail is subjected to a planar force acting on the centroid, it cannot generate a roll axis torque component in the normal direction of the sail, so the sail spacecraft often needs an additional control moment generating mechanism to achieve three-axis stable control. Flywheels and attitude control engines consume a large amount of fuel or power, and greatly increase the load, so they have been abandoned by the industry. The sail torsion mechanism represented by the RSB rod and small sub-sail has the characteristics of no fuel consumption and small power, but its structure is complex, and it seriously interferes with the dynamic characteristics of the sail, and also interferes with the unfolding and folding process of the sail, which also has practical application problems. SUMMARY

[0004] The technical problem solved by the present application is that existing sail spacecraft often need an additional control moment generating mechanism to achieve three-axis stable control.

[0005] To solve the above problems, the technical scheme of the present application is as follows:

[0006] The sail spacecraft underactuated three-axis control method based on a planar control moment mechanism comprises:

[0007] Modeling the attitude dynamics of the sail spacecraft to obtain the attitude dynamics equation of the sail spacecraft; based on the structure of the sail spacecraft, establishing a sail normal control law under the premise of considering the motion of the slider on the sail; under the premise that the sail spacecraft tracks the preset angular velocity path, establishing a sail roll angle control law; establishing a sail roll axis angular velocity control law; sequentially combining the sail normal control law, the sail roll angle control law, and the sail roll axis angular velocity control law, and using the control moment on the sail plane to complete the underactuated three-axis control of the sail spacecraft.

[0008] As another aspect of the present application, modeling the attitude dynamics of the sail spacecraft to obtain the attitude dynamics equation of the sail spacecraft comprises the following contents:

[0009] A principal axis coordinate system S is established based on the structure of the sail spacecraft b , an orbit coordinate system S l , a moment of inertia matrix in the principal axis coordinate system S b is expressed as a diagonal matrix: {J} b = diag(J x ,J y ,J z ), and an expression formula of an attitude dynamics equation in the principal axis coordinate system S b is derived from the angular momentum theorem as follows:

[0010]

[0011]

[0012]

[0013] In the above formula, J is the moment of inertia matrix, ω is the angular velocity of the sail spacecraft, is the derivative of the angular velocity of the sail spacecraft, M is the moment generated by the force received by the sail spacecraft due to the deviation of the center of mass from the centroid of the sail surface, F is the area force received by the sail spacecraft, [d x ,d y ,0] T is the position of the center of mass of the spacecraft in the principal axis coordinate system S b , d x is the component of the x-axis projection of the position of the center of mass of the spacecraft in the principal axis coordinate system S b , d y is the component of the y-axis projection of the position of the center of mass of the spacecraft in the principal axis coordinate system S b , F zb is the component of the z-axis projection of the area force F received by the spacecraft in the principal axis coordinate system S b , z b is the z-axis direction of the principal axis coordinate system S b , P is the area coefficient of the area force received by the sail surface, l b is the side length of the sail surface, and α is the included angle between the z-axis of the principal axis coordinate system S b and the z-axis of the orbit coordinate system S l .

[0014] As another aspect of the present application, the structure of the sail spacecraft includes a sail surface and a spacecraft fixed on the sail surface, the sail surface is provided with an X-shaped support rod, the intersection of the support rod is the fixed position of the spacecraft core, and a first sliding block, a second sliding block, a third sliding block and a fourth sliding block are respectively connected to the four sides of the intersection.

[0015] As another aspect of the present application, the expression of the sail surface normal control law is as follows:

[0016]

[0017] In the above formula, u is a control variable corresponding to the sliding surface, A is a system matrix, B is a control matrix, C is a designed sliding mode control coefficient matrix, x is a state variable, q is a fourth normal number, S1 is the first component of the sliding surface vector, and S2 is the second component of the sliding surface vector.

[0018] As another aspect of the present application, the establishment of the sail surface normal control law based on the sail spacecraft structure under the premise of considering the movement of the slider on the sail surface includes the following contents:

[0019] The state variable x describing the change state of the pitch angle and the yaw angle is defined as x = [α e , δ e , ω xb , ω yb ] T , where α e is the difference between the current value and the target value of the cone angle, δ e is the difference between the current value and the target value of the clock angle, ω xb is the component of the angular velocity ω of the spacecraft in the x-axis projection of the principal axis coordinate system S b , and ω yb is the component of the angular velocity ω of the spacecraft in the y-axis projection of the principal axis coordinate system S b .

[0020] Considering the change amount of J x and J y caused by the movement of the slider, J t = J x = J y is set, and ΔI = J z -J t is set, where J t is a first constant, J t takes the value of J x when all sliders are located at the center of mass of the sail spacecraft, and ΔI is the difference between J z and J t .

[0021] The state differential equation obtained from the attitude dynamics equation and the relationship between the angular velocity of the attitude angle and the angular velocity of the spacecraft is as follows:

[0022]

[0023]

[0024] In the above formula, x is a state variable, is the derivative of the state variable, is αe The derivative of It is δ e The derivative of It is ω xb The derivative of It is ω yb The derivative of M xb M is in the principal axis coordinate system S b The components of the mid-x-axis projection, M yb M is in the principal axis coordinate system S b The components of the y-axis projection are: A is the system matrix, corresponding to the corresponding part in the second equation above; B is the control matrix, corresponding to the corresponding part in the second equation above; and γ is the orbital coordinate system S. l The z-axis in the principal axis coordinate system S b The angle between the projection component on the xy plane and the x-axis.

[0025] The formulas for the sliding surface and the reaching law for the state differential equation are as follows:

[0026]

[0027]

[0028] In the above formula, s is the sliding surface vector. S1 is the derivative of the sliding surface vector s, C is the designed sliding mode control coefficient matrix, c1 is the first positive constant, c2 is the second positive constant, Δ is the third positive constant, q is the fourth positive constant, ε is the fifth positive constant, S1 is the first component of the sliding surface vector, and S2 is the second component of the sliding surface vector.

[0029] When the condition s = 0 for the sliding surface is satisfied, x will approach 0 according to the following rule:

[0030]

[0031]

[0032]

[0033] In the above formula, α e (t) is the difference between the current value and the target value of the cone angle at time t, ω xb (t) is the angular velocity ω of the spacecraft at time t in the principal axis coordinate system S. b The components of the x-axis projection, ω yb (t) is the angular velocity ω of the spacecraft at time t in the principal axis coordinate system S. b The component of the y-axis projection, δ e (t) is the difference between the current value and the target value of the clock angle at time t, α e(0) is the difference between the current value and the target value of the cone angle at t = 0, δ e (0) is the difference between the current value and the target value of the cone angle at t = 0, δ

[0034] The expression of the sail normal control law according to the control variable u corresponding to the sliding surface is as follows:

[0035]

[0036] In the above formula, u is the control variable corresponding to the sliding surface, A is the system matrix, B is the control matrix, C is the designed sliding mode control coefficient matrix, x is the state variable, q is the fourth normal number, S1 is the first component of the sliding surface vector, and S2 is the second component of the sliding surface vector.

[0037] As another aspect of the present application, the expression formula of the sail roll angle control law is as follows:

[0038]

[0039] In the above formula, Δψ is the roll angle difference value of the sail spacecraft, N roll is the first positive integer, A is the amplitude of the angular velocity of the sail spacecraft maintained in the x-y plane of the principal axis coordinate system S b , Ω n is the angular velocity of the sail spacecraft maintained in the x-y plane of the principal axis coordinate system S b , which rotates in the x-y plane of the principal axis coordinate system S b around the z axis of the principal axis coordinate system S b .

[0040] As another aspect of the present application, under the premise that the sail spacecraft tracks the preset angular velocity path, the sail roll angle control law is established and includes the following contents:

[0041] Suppose that the sail spacecraft tracks the designed angular velocity path, the angular velocity ω inplane of the sail spacecraft is maintained in the x-y plane of the principal axis coordinate system S b , and rotates in the x-y plane of the principal axis coordinate system S b around the z axis of the principal axis coordinate system S b at the angular velocity Ω n corresponding to the path, and the path expression is as follows:

[0042]

[0043]

[0044] In the above formula, ω inplane is the angular velocity of the sail spacecraft maintained in the x-y plane of the principal axis coordinate system Sb angular velocity in the xy plane, It is ω inplane The derivative of Ω n It is ω inplane Around the principal axis coordinate system S b The z-axis in the principal axis coordinate system S b The angular velocity of rotation in the xy plane, where A is ω inplane amplitude, It is ω inplane With principal axis coordinate system S b The initial angle between the x-axis and the x-axis, where t is time.

[0045] Establish an inertial coordinate system S in the initial state. os Inertial coordinate system S os The origin of the coordinate axes coincides with the centroid of the sail spacecraft, and the directions of its coordinate axes are relative to the principal axis coordinate system S at time t=0. b The coordinate axes of the initial state in the inertial coordinate system S are all in the same direction and remain unchanged in inertial space. os With principal axis coordinate system S b The Euler angle transformation formula is:

[0046]

[0047] In the above formula, L z (ψ) is the coordinate transformation matrix corresponding to the rotation of the coordinate system around the z-axis by an angle ψ, L y (θ) is the coordinate transformation matrix corresponding to the rotation of the coordinate system around the y-axis by an angle θ. It is a rotation of the coordinate system around the x-axis The coordinate transformation matrix corresponding to the angle.

[0048] The formula for the effect of the angular velocity of a sail spacecraft on each Euler angle after it completes one revolution along a given angular velocity path is as follows:

[0049]

[0050]

[0051] In the above formula, Ω is the angle of rotation of the coordinate system about the x-axis in the Euler transformation, ψ is the angle of rotation of the coordinate system about the z-axis in the Euler transformation, θ is the angle of rotation of the coordinate system about the y-axis in the Euler transformation, and Ω is the angle of rotation of the coordinate system about the y-axis in the Euler transformation. n It is ω inplane Around the principal axis coordinate system S b The z-axis in the principal axis coordinate system S b angular velocity of rotation in the xy plane,

[0052] The maximum change in the normal direction of the sail during the flight of a sail-type spacecraft, Δξmax and final value Δξ e The formula for calculating (the value corresponding to the spacecraft completing one full revolution along a given angular velocity path) is:

[0053]

[0054] In the above formula, Δξ max It is the maximum change in the normal direction of the sail during the flight of a sail-type spacecraft, Δξ. e It is the final value of the change in the normal direction of the sail during the flight of a sail-type spacecraft. Let θ(t) be the angle of rotation of the coordinate system about the x-axis in the Euler transformation corresponding to the attitude of the sail spacecraft at time t, and let A be the angle of rotation of the coordinate system about the y-axis in the Euler transformation corresponding to the attitude of the sail spacecraft at time t. inplane The amplitude, Ω n It is ω inplane Around the principal axis coordinate system S b The z-axis in the principal axis coordinate system S b angular velocity of rotation in the xy plane,

[0055] Based on the aforementioned formula for the effect of the angular velocity of a sail spacecraft on each Euler angle after one revolution along a given angular velocity path, the expression for the sail normal control law is as follows:

[0056]

[0057] In the above formula, Δψ is the roll angle difference of the sail spacecraft, and N roll It is the first positive integer, and A is the coordinate system in which the sail spacecraft remains in the principal axis coordinate system S. b The magnitude of the angular velocity in the xy plane, Ω n The sail-shaped spacecraft maintains its position in the principal axis coordinate system S. b The angular velocity in the xy plane about the principal axis coordinate system S b The z-axis in the principal axis coordinate system S b The angular velocity of rotation in the xy plane.

[0058] When it is necessary to control the roll angle of a sail spacecraft by changing the angle Δψ, first select roll Choose the desired control level after N laps. Select Ω based on the actual situation. n And A, so that they satisfy the relationship in the following equation ψ, then the spacecraft can travel along the selected parameter N. roll Ω n The angular velocity path corresponding to A will change the roll angle ψ by the angle Δψ after the motion.

[0059] As another aspect of the invention, the preset angular velocity path is obtained according to the space mission planning of the sail spacecraft.

[0060] As another aspect of the present application, the establishment of the sail roll axis angular velocity control law comprises the following contents:

[0061] Assume that the phase space PS=R 4 xb yb , and constitute, there are closed curves Curve1 and Curve2 in the phase space, according to the attitude dynamics equation, the sail spacecraft moves along Curve1 and Curve2 respectively, the control formula of the directional control roll axis angular velocity is:

[0062]

[0063]

[0064] In the above formula, m sail is the overall mass of the sail spacecraft, ω xb is the component of the angular velocity ω of the spacecraft in the x-axis projection of the principal axis coordinate system S b , ω yb is the component of the angular velocity ω of the spacecraft in the y-axis projection of the principal axis coordinate system S b , ω zb is the component of the angular velocity ω of the spacecraft in the z-axis projection of the principal axis coordinate system S b , is the derivative of ω xb , is the derivative of ω yb , J x is the moment of inertia matrix J corresponding to the x-axis of the principal axis coordinate system S b , J y is the moment of inertia matrix J corresponding to the y-axis of the principal axis coordinate system S b , J z is the moment of inertia matrix J corresponding to the z-axis of the principal axis coordinate system S b .

[0065] As another aspect of the present application, the ordered combination of the sail normal control law, the sail roll angle control law, and the sail roll axis angular velocity control law, and the use of the control torque on the sail plane to complete the underactuated three-axis control of the sail spacecraft comprises the following contents:

[0066] During the sailing of the sail spacecraft, the pitch angle, pitch angle velocity, yaw angle, and yaw angle velocity control of the sail of the sail spacecraft is realized through the sail normal control law based on the sailing situation, the roll angle control of the sail of the sail spacecraft is realized through the sail roll angle control law, and the roll angle angular velocity control is realized through the sail roll axis angular velocity control law. ​​

[0067] The beneficial effects of the present application are:

[0068] The present application considers to adopt a sail spacecraft configuration based on a planar control moment mechanism, such as a slider sail and a variable reflectivity sail, to realize three-axis control without adding an additional roll axis control moment generating mechanism, to greatly reduce the structural complexity and load of the sail spacecraft. BRIEF DESCRIPTION OF DRAWINGS

[0069] Figure 1 A configuration schematic diagram of a sail spacecraft based on a planar control moment mechanism is provided for an embodiment of the present application;

[0070] Figure 2 A related coordinate system schematic diagram of a sail spacecraft based on a planar control moment mechanism is provided for an embodiment of the present application;

[0071] Figure 3 A control schematic diagram of a sail surface roll angle control law is provided for an embodiment of the present application;

[0072] Figure 4 A schematic diagram of a roll axis angular velocity path of a roll axis angular velocity control law of a sail surface roll axis angular velocity control law is provided for an embodiment of the present application;

[0073] Figure 5 A schematic diagram of a roll axis angular velocity path of a roll axis angular velocity control law of a sail surface roll axis angular velocity control law is provided for an embodiment of the present application;

[0074] Figure 6 A control schematic diagram of a sail surface roll axis angular velocity control law is provided for an embodiment of the present application;

[0075] Figure 7 An attitude angle change schematic diagram under the control of an underactuated three-axis control method is provided for an embodiment of the present application;

[0076] Figure 8 An angular velocity change schematic diagram under the control of an underactuated three-axis control method is provided for an embodiment of the present application. DETAILED DESCRIPTION

[0077] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be further described in detail below with reference to the drawings. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative labor fall within the scope of protection of the present application.

[0078] The terminology used in the description of the application herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. As used in the description of the application and the appended claims, the singular forms "a", "an" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be understood that the terms "comprises" and / or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0079] It should be understood that, although the terms first, second, third, etc. can be used herein to describe various …, these … should not be limited to these terms. These terms are only used to distinguish one … from another. For example, a first … could also be termed a second …, and, similarly, a second … could also be termed a first …, without departing from the scope of the embodiments of the application.

[0080] In order to solve the problem in the prior art that the implementation of three-axis stabilization control of a sail spacecraft requires the use of additional roll-axis control torque generation mechanisms such as RSB rods, flywheels, small sail surfaces, attitude control engines, and the like, which increases the overall mass and structural complexity, the embodiments of the application propose a sail spacecraft underactuated three-axis control method based on a planar control torque mechanism. By designing a special angular velocity path in the phase space, the spacecraft can complete the angle and angular velocity control in the roll-axis direction without involving the control torque in the roll-axis direction when moving along the corresponding angular velocity path, and then combined with the traditional sail surface normal direction control, the underactuated three-axis control is realized. At the same time, a relatively simple configuration design is adopted, which reduces the overall mass and fuel and power consumption, reduces the structural complexity and the risk of accidents, and is more conducive to the space mission of the sail spacecraft and prolongs the service life of the sail spacecraft.

[0081] The sail spacecraft underactuated three-axis control method of the embodiments of the application comprises:

[0082] (1) modeling the attitude dynamics of the sail spacecraft to obtain the attitude dynamics equation of the sail spacecraft:

[0083] It can be understood that, in the embodiments of the application, in order to carry out the spacecraft attitude control task, the attitude dynamics of the spacecraft and the related parameter coordinate system need to be modeled first. In order to simplify the structural complexity of the sail spacecraft as much as possible, a slider is used as a planar control torque generation mechanism, and a configuration is adopted to generate a planar control torque by means of the deviation of the center of mass and the shape center of the sail surface. The structure of the sail spacecraft includes a sail surface and a spacecraft fixed on the sail surface. The sail surface is provided with an X-shaped support rod, and the intersection point of the support rod is the core of the spacecraft. Four edges of the intersection point are respectively connected with a first slider, a second slider, a third slider and a fourth slider.

[0084] And as shown in Figure 1 , a main axis coordinate system S b of the sail spacecraft is defined.b The origin is located at the centroid of the sail; the zb axis is along the normal direction of the sail; the xb axis is in the plane of the sail and points to the vertex corresponding to slider 1; the yb axis is determined by the right-hand rule.

[0085] like Figure 2 As shown, the orbital coordinate system S is defined. l S l The origin is located at the centroid of the sail; the direction of the zl axis is the straight line from the sun to the centroid of the sail; the xb axis is parallel to the ecliptic plane and perpendicular to zl; the yb axis is determined by the right-hand rule. The cone angle α, bell angle δ, and roll angle γ are defined. The cone angle α is the angle between the zl and zb axes; the bell angle δ is the angle between the projection components of the xl and zb axes onto the xl-yl plane; and the roll angle γ is the angle between the projection component of the zl axis onto the xb-yb plane and the xb axis.

[0086] Establish the principal axis coordinate system S based on the structure of the sail-shaped spacecraft. b Orbital coordinate system S l The moment of inertia matrix in the principal axis coordinate system S b The following is represented as a diagonal matrix: {J} b =diag(J x J y J z The formulas for calculating each component of the moment of inertia matrix are as follows:

[0087]

[0088] In the above formula, J x The moment of inertia matrix J corresponds to the principal axis coordinate system S. b The x-axis component, J y The moment of inertia matrix J corresponds to the principal axis coordinate system S. b The y-axis component, J z The moment of inertia matrix J corresponds to the principal axis coordinate system S. b In the z-axis component, m1 is the mass of the spacecraft core, m2 is the mass of each slider, m3 is the mass of the support rod, m4 is the mass of the sail, and l b 'a' is the side length of the sail, 'a' is the side length of the slider, and 'r' is the side length of the slider. pole is the radius of the support rod, s is the side length of the spacecraft core, and l1, l2, l3, and l4 are the distances between the first, second, third, and fourth sliders and the centroid of the sail spacecraft, respectively.

[0089] The angular momentum theorem is derived for the principal axis coordinate system S. b The expression for the attitude dynamics equations is as follows:

[0090]

[0091]

[0092]

[0093] In the above formula, J is the moment of inertia matrix, and ω is the angular velocity of the sail spacecraft. is the derivative of the angular velocity of the sail spacecraft, M is the torque generated by the force acting on the sail spacecraft due to the deviation between the center of mass and the centroid of the sail, F is the area force acting on the sail spacecraft, [d x ,d y ,0] T The spacecraft's center of mass is in the principal axis coordinate system S b The position in the middle, d x The position of the spacecraft's center of mass in the principal axis coordinate system S b The components of the x-axis projection, d y The position of the spacecraft's center of mass in the principal axis coordinate system S b The component of the y-axis projection, F zb The area force F acting on the spacecraft in the principal axis coordinate system S b The z-axis projection component, z b It is the principal coordinate system S b In the z-axis direction, P is the area coefficient of the area force on the sail surface, l b α is the side length of the sail, and α is the principal coordinate system S. b z-axis and orbital coordinate system S l The angle between the z-axis and the z-axis.

[0094] The formulas relating the angular velocities of the cone angle α, the bell angle δ, and the roll angle γ to the angular velocity of the spacecraft are shown below.

[0095]

[0096] Optionally, in this embodiment, the total mass and the weight of the slider are set to 200kg and 2kg, respectively. Based on actual engineering results, the initial value is 3.11g / m³. 2 The areal density is used as the areal density of the membrane used for the sail. Hollow carbon fiber tubes are used as support rods, with a linear density of 0.2 kg / m. The sail size is set to 100 m × 100 m, and the dimensions of the slider and spacecraft core are 0.1 m × 0.1 m × 0.1 m and 0.5 m × 0.5 m × 0.5 m, respectively.

[0097] (2) Considering the sliding motion on the sail surface, a normal control law for the sail surface is established based on the sail-type spacecraft structure:

[0098] Define a state variable x to describe the changes in pitch and yaw angles, and x = [α]. e ,δ e ,ω xb, ω yb ] T wherein α e is the difference between the current value and the target value of the cone angle, δ e is the difference between the current value and the target value of the clock angle, ω xb is the component of the angular velocity ω of the spacecraft in the x-axis projection in the principal axis coordinate system S b , ω yb is the component of the angular velocity ω of the spacecraft in the y-axis projection in the principal axis coordinate system S b ,

[0099] Considering the change amount of J x and J y due to the movement of the sliders, let J t = J x = J y , and let ΔI = J z - J t , wherein J t is a first constant value, taking the value of J x when all sliders are located at the center of mass of the sail spacecraft, and ΔI is the difference between J z and J t ,

[0100] The state differential equation obtained from the attitude dynamics equation and the relationship between the angular velocity of the attitude angle and the angular velocity of the spacecraft is as follows:

[0101]

[0102]

[0103] In the above formula, x is a state variable, is the derivative of the state variable, is the derivative of α e , is the derivative of δ e , is the derivative of ω xb , is the derivative of ω yb , M xb is the component of M in the x-axis projection in the principal axis coordinate system S b , M yb is the component of M in the y-axis projection in the principal axis coordinate system S b , A is a system matrix corresponding to the corresponding part in the second formula of the above formula, B is a control matrix corresponding to the corresponding part in the second formula of the above formula, and γ is the angle between the projection component of the z-axis of the orbital coordinate system S l in the x-y plane of the principal axis coordinate system S b and the x-axis.

[0104] The formulas for the sliding surface and the reaching law for the state differential equation are as follows:

[0105]

[0106]

[0107] In the above formula, s is the sliding surface vector. S1 is the derivative of the sliding surface vector s, C is the designed sliding mode control coefficient matrix, c1 is the first positive constant, c2 is the second positive constant, Δ is the third positive constant, q is the fourth positive constant, ε is the fifth positive constant, S1 is the first component of the sliding surface vector, and S2 is the second component of the sliding surface vector.

[0108] When the condition s = 0 for the sliding surface is satisfied, x will approach 0 according to the following rule:

[0109]

[0110]

[0111]

[0112] In the above formula, α e (t) is the difference between the current value and the target value of the cone angle at time t, ω xb (t) is the angular velocity ω of the spacecraft at time t in the principal axis coordinate system S. b The components of the x-axis projection, ω yb (t) is the angular velocity ω of the spacecraft at time t in the principal axis coordinate system S. b The component of the y-axis projection, δ e (t) is the difference between the current value and the target value of the clock angle at time t, α e (0) is the difference between the current value and the target value of the cone angle at time t=0, δ e (0) is the difference between the current value and the target value of the clock angle at t=0.

[0113] Based on the control variable u corresponding to the sliding surface, the expression for the sail normal control law is as follows:

[0114]

[0115] In the above formula, u is the control variable corresponding to the sliding surface, A is the system matrix, B is the control matrix, C is the designed sliding control coefficient matrix, x is the state variable, q is the fourth positive constant, S1 is the first component of the sliding surface vector, and S2 is the second component of the sliding surface vector.

[0116] Let the Lyapunov function be V = s Ts / 2, then from the above equation, the derivative of the Lyapunov function is less than zero:

[0117]

[0118] According to the Lyapunov second method, since the derivative of the Lyapunov function of the controlled system under the sliding mode control is less than zero, the system is stable.

[0119] (3) Under the premise that the sail spacecraft tracks the preset angular velocity path, the sail surface roll angle control law is established:

[0120] Since the current sail spacecraft configuration can only generate control moments in the xb and yb directions, it is assumed that the sail spacecraft tracks the designed angular velocity path, and the angular velocity ω inplane of the sail spacecraft is kept in the x-y plane of the principal axis coordinate system S b , and rotates around the z axis of the principal axis coordinate system S n at an angular velocity of Ω b in the x-y plane of the principal axis coordinate system S b , and the corresponding path is expressed as:

[0121]

[0122]

[0123] In the above equation, ω inplane is the angular velocity of the sail spacecraft kept in the x-y plane of the principal axis coordinate system S b , ω inplane is the derivative of ω n , Ω inplane is the angular velocity of ω b rotating around the z axis of the principal axis coordinate system S b in the x-y plane of the principal axis coordinate system S inplane , A is the amplitude of ω inplane , θ b is the initial angle between ω os and the x axis of the principal axis coordinate system S b , t is time,

[0124] An initial state inertial coordinate system S os is established, whose coordinate axis origin coincides with the center of mass of the sail spacecraft at the moment, and its coordinate axis direction is the same as that of the principal axis coordinate system S b at the moment t = 0 and remains unchanged in the inertial space. The Euler angle transformation formula between the initial state inertial coordinate system S z and the principal axis coordinate system S y is:

[0125]

[0126] In the above formula, L z (ψ) is the coordinate transformation matrix corresponding to the rotation of the coordinate system around the z axis by an angle ψ, L y (θ) is the coordinate transformation matrix corresponding to the rotation of the coordinate system around the y axis by an angle θ, is the coordinate transformation matrix corresponding to the rotation of the coordinate system around the x axis by an angle ,

[0127] The conversion relationship obtained from the Euler angle transformation formula is as follows,

[0128]

[0129]

[0130] Using the small angle assumption and integrating the rate of change of each Euler angle of the spacecraft moving along a given angular velocity path, the change formula of the Euler angle is as follows,

[0131]

[0132]

[0133]

[0134] Considering the boundary conditions then Therefore, the formula for the effect of the angular velocity of the sail spacecraft on each Euler angle after one cycle according to the given angular velocity path is:

[0135]

[0136]

[0137] In the above formula, is the angle of rotation of the coordinate system around the x axis in the Euler transformation, ψ is the angle of rotation of the coordinate system around the z axis in the Euler transformation, θ is the angle of rotation of the coordinate system around the y axis in the Euler transformation, Ω n is ω inplane rotating around the z axis of the principal axis coordinate system S b in the x-y plane of the principal axis coordinate system S b ,

[0138] The calculation formula of the maximum value Δξ max of the change of the sail surface normal direction of the sail spacecraft during the voyage and the final value (the value corresponding to the complete one cycle of the spacecraft along the given angular velocity path) Δξ e is as follows:

[0139]

[0140] In the above formula, Δξ max It is the maximum change in the normal direction of the sail during the flight of a sail-type spacecraft, Δξ. e It is the final value of the change in the normal direction of the sail during the flight of a sail-type spacecraft. Let θ(t) be the angle of rotation of the coordinate system about the x-axis in the "3-2-1" Euler transformation corresponding to the attitude of the sail spacecraft at time t, and let A be the angle of rotation of the coordinate system about the y-axis in the "3-2-1" Euler transformation corresponding to the attitude of the sail spacecraft at time t. inplane The amplitude, Ω n It is ω inplane Around the principal axis coordinate system S b The z-axis in the principal axis coordinate system S b The angular velocity of rotation in the xy plane.

[0141] Based on the aforementioned formula for the effect of the angular velocity of a sail spacecraft on each Euler angle after one revolution along a given angular velocity path, the expression for the sail normal control law is as follows:

[0142]

[0143] In the above formula, Δψ is the roll angle difference of the sail spacecraft, and N roll It is the first positive integer, and A is the coordinate system in which the sail spacecraft remains in the principal axis coordinate system S. b The magnitude of the angular velocity in the xy plane, Ω n The sail-shaped spacecraft maintains its position in the principal axis coordinate system S. b The angular velocity in the xy plane about the principal axis coordinate system S b The z-axis in the principal axis coordinate system S b The angular velocity of rotation in the xy plane.

[0144] When it is necessary to control the roll angle of a sail spacecraft by changing the angle Δψ, first select roll Choose the desired control level after N laps. Select Ω based on the actual situation. n And A, so that they satisfy the relationship in the following equation ψ, then the spacecraft can travel along the selected parameter N. roll Ω n The angular velocity path corresponding to A will change the roll angle ψ by the angle Δψ after the motion.

[0145] like Figure 3 As shown, the roll angle control law achieves roll angle control without significantly interfering with pitch and yaw angles.

[0146] (4) Establish the angular velocity control law for the sail's roll axis:

[0147] Similarly, in order to realize the control of the roll axis angular velocity only by the control moment within the sail surface, here is still to make the spacecraft move along a given angular velocity path, and realize the control of the roll axis angular velocity without great influence on other attitude angles. Assuming that the phase space PS=R 4 xb yb , and there are closed curves Curve1 and Curve2 in the phase space, then according to the attitude dynamics equation, the sail spacecraft moves along Curve1 and Curve2 respectively, and the control formula for directional control of the roll axis angular velocity is:

[0148]

[0149]

[0150] In the above formula, m sail is the overall mass of the sail spacecraft, ω xb is the component of the angular velocity ω of the spacecraft projected on the x-axis in the principal axis coordinate system S b , ω yb is the component of the angular velocity ω of the spacecraft projected on the y-axis in the principal axis coordinate system S b , ω zb is the component of the angular velocity ω of the spacecraft projected on the z-axis in the principal axis coordinate system S b , is the derivative of ω xb , is the derivative of ω yb , J x is the component of the moment of inertia matrix J corresponding to the x-axis in the principal axis coordinate system S b , J y is the component of the moment of inertia matrix J corresponding to the y-axis in the principal axis coordinate system S b , and J z is the component of the moment of inertia matrix J corresponding to the z-axis in the principal axis coordinate system S b .

[0151] Here, we find that when the angular velocity path satisfies the following two types, that is, the coordinate system corresponding to the wx, wy angular velocity plane and the positive proportional straight line corresponding to the positive and negative 45-degree angle. Then the rate of change of the roll axis angular velocity is 0 when the spacecraft runs on the corresponding angular velocity path:

[0152]

[0153]

[0154] Considering F zb ​​The following relationship exists between the rate of change of angular velocity and the inertia matrix:

[0155]

[0156]

[0157] Here we consider along ω xb -ω yb In the plane with Ω n The angular velocity of rotation about the zb axis is ω inplane The corresponding path is expressed as:

[0158]

[0159]

[0160] Where A is ω inplane amplitude, It is ω inplane The initial angle between the x and b axes. Then, as the spacecraft travels along this path, the following relationship holds:

[0161]

[0162]

[0163]

[0164]

[0165] Obviously, when hour It will be less than 0 when hour It will be greater than 0. Therefore, by concatenating the above three types of angular velocity paths, a path that satisfies the entire process can be obtained. A closed angular velocity path with a value greater than zero or less than zero. When a spacecraft travels along the corresponding angular velocity path, its roll axis angular velocity will monotonically increase or monotonically decrease with time. Figure 4 and Figure 5 The diagrams show the angular velocity paths for monotonically decreasing and monotonically increasing angular velocities of the roll axis, respectively. Figure 6 The paper presents the roll axis angular velocity control results corresponding to these four angular velocity paths. Furthermore, based on the formula for the rate of change of roll axis angular velocity, it can be seen that when the intensity of the area force in space decreases, the control speed of the sail on the roll axis angular velocity actually increases, which will greatly enhance the applicability of this control method.

[0166] Understandably, in this embodiment, the preset angular velocity path is obtained based on the space mission planning of the sail spacecraft.

[0167] (5) The sail normal control law, the sail roll angle control law, and the sail roll axis angular velocity control law are sequentially combined to complete the underactuated three-axis control of the sail spacecraft using the control moment on the sail plane:

[0168] During the sailing of the sail spacecraft, the pitch angle, the pitch angular velocity, the yaw angle, and the yaw angular velocity of the sail of the sail spacecraft are controlled by the sail normal control law based on the sailing condition, the roll angle of the sail of the sail spacecraft is controlled by the sail roll angle control law, and the roll angle angular velocity is controlled by the sail roll axis angular velocity control law.

[0169] It can be understood that the three-axis control of the spacecraft requires the pitch angle, the pitch angular velocity, the yaw angle, the yaw angular velocity, the roll angle, and the roll axis angular velocity to be controlled, and the conventional method can only control the former four, and an additional mechanism is required to control the latter two. The method provided in the embodiment can control all the six through different control laws.

[0170] At present, the sail normal control law, the sail roll angle control law, and the sail roll axis angular velocity control law have been implemented. As shown in Figure 1 , they can control the attitude in space. By sequentially combining them, the underactuated three-axis control using only the control moment on the sail plane can be achieved.

[0171] The parameters involved in the sail normal control law, the sail roll angle control law, and the sail roll axis angular velocity control law described above are set as follows: c1=0.002, c2=0.002, A(sail roll angle control law)=0.0002 rad / s, A(sail roll axis angular velocity control law)=0.0012 rad / s, Ω n (sail roll angle control law)=0.0001 rad / s, Ω n (sail roll axis angular velocity control law)=0.004 rad / s, and kn=0.001. The motion constraints of the slider are set as follows: the upper limit of the distance dmax=70.71 m, the upper limit of the speed 2 m / s, and the upper limit of the acceleration 0.5 m / s. The upper limit of the cone angle is set to 50 degrees.

[0172] Considering that the sail spacecraft is a solar sail and operates near the Earth, the initial state is set as The target state is set as The solar radiation pressure P can be set to 4.6 μPa. The underactuated three-axis control method formed by the combination of the sail normal control law, the sail roll angle control law, and the sail roll axis angular velocity control law has the control result as Figure 7 , Figure 8The cone angle a is kept below 50 degrees, and the angle difference between the measured value and the target value of the attitude angle converges to zero after 240000s. The roll axis angular velocity is reduced to zero at about 80000s, and the components of the angular velocity are all less than 1.2x10 -3 rad / s. The above results show that the underactuated three-axis control method is effective, and can meet the cone angle constraint during operation.

[0173] The underactuated three-axis control method for the sail spacecraft based on the planar control moment mechanism provided by the embodiment of the present application has the following beneficial effects:

[0174] The sail spacecraft involved in the present application has a simple configuration, does not need to install an additional roll axis torque generating mechanism, reduces the overall mass and fuel and power consumption, reduces the structural complexity and accident risk, and prolongs the service life of the sail spacecraft. In addition, the present application proposes an underactuated three-axis control method considering engineering constraints, which designs a special angular velocity path in the phase space, so that the spacecraft can complete the angle and angular velocity control in the roll axis direction without involving the control torque in the roll axis direction when moving along the corresponding angular velocity path. And meet the cone angle constraint and the position, velocity and acceleration constraints of the slider. At the same time, the underactuated three-axis control method can also work normally under the condition of small area force intensity in space, which is very beneficial to deep space exploration and other scenes, and is more beneficial to the sail spacecraft to perform space tasks.

[0175] The specific embodiments described in the present application can enable those skilled in the art to have a more comprehensive understanding of the present application, but in no way limit the present application. Therefore, those skilled in the art should understand that modifications or equivalent replacements to the present application can still be made; and all technical solutions and improvements that do not deviate from the spirit and technical essence of the present application should be covered in the protection scope of the present application patent.

Claims

1. A method for underactuated three-axis control of a sail-type spacecraft based on a planar control torque mechanism, characterized in that, include: The attitude dynamics of the sail spacecraft are modeled to obtain the attitude dynamics equations of the sail spacecraft; considering the motion of the slider on the sail surface, a normal control law for the sail surface is established based on the structure of the sail spacecraft; under the premise that the sail spacecraft tracks according to a preset angular velocity path, a roll angle control law for the sail surface is established; and a roll axis angular velocity control law for the sail surface is established. The preset angular velocity path is obtained based on the space mission planning of the sail-shaped spacecraft; the sail-shaped spacecraft tracks according to the preset angular velocity path, the expression of which is: In the above formula, ω inplane The sail-shaped spacecraft maintains its position in the principal axis coordinate system S. b angular velocity in the xy plane, It is ω inplane The derivative of Ω n It is ω inplane Around the principal axis coordinate system S b The z-axis in the principal axis coordinate system S b The angular velocity of rotation in the xy plane, where A is ω inplane amplitude, It is ω inplane With principal axis coordinate system S b The initial angle between the x-axis and the x-axis, where t is time; By combining the sail normal control law, sail roll angle control law, and sail roll axis angular velocity control law in an orderly manner, the underactuated three-axis control of the sail spacecraft is achieved by utilizing the control torque on the sail plane.

2. The underactuated three-axis control method for a sail-type spacecraft as described in claim 1, characterized in that, The attitude dynamics modeling of the sail-shaped spacecraft yields the following attitude dynamics equations: Establish the principal axis coordinate system S based on the structure of the sail-shaped spacecraft. b Orbital coordinate system S l The moment of inertia matrix in the principal axis coordinate system S b The following is represented as a diagonal matrix: {J} b =diag(J x J y J z The angular momentum theorem is derived in the principal axis coordinate system S. b The expression for the attitude dynamics equations is as follows: {M} b =[d x ,d y ,0] T ×{F} b =[-F zb d y ,-F zb d x ,0] T In the above formula, J is the moment of inertia matrix, and ω is the angular velocity of the sail spacecraft. is the derivative of the angular velocity of the sail spacecraft, M is the torque generated by the force acting on the sail spacecraft due to the deviation between the center of mass and the centroid of the sail, F is the area force acting on the sail spacecraft, [d x ,d y ,0] T The spacecraft's center of mass is in the principal axis coordinate system S b The position in the middle, d x The position of the spacecraft's center of mass in the principal axis coordinate system S b The components of the x-axis projection, d y The position of the spacecraft's center of mass in the principal axis coordinate system S b The component of the y-axis projection, F zb The area force F acting on the spacecraft in the principal axis coordinate system S b The z-axis projection component, z b It is the principal coordinate system S b In the z-axis direction, P is the area coefficient of the area force on the sail surface, l b α is the side length of the sail, and α is the principal coordinate system S. b z-axis and orbital coordinate system S l The angle between the z-axis and the z-axis.

3. The underactuated three-axis control method for a sail-type spacecraft as described in claim 1, characterized in that, The structure of the sail-type spacecraft includes a sail and a spacecraft fixed on the sail. The sail is provided with X-shaped support rods. The intersection of the support rods is the fixed position of the spacecraft core. A first slider, a second slider, a third slider, and a fourth slider are slidably connected to the four sides of the intersection.

4. The underactuated three-axis control method for a sail-type spacecraft as described in claim 1, characterized in that, The expression for the sail normal control law is as follows: In the above formula, u is the control variable corresponding to the sliding surface, A is the system matrix, B is the control matrix, C is the designed sliding control coefficient matrix, x is the state variable, q is the fourth positive constant, S1 is the first component of the sliding surface vector, and S2 is the second component of the sliding surface vector.

5. The underactuated three-axis control method for a sail-type spacecraft as described in claim 4, characterized in that, The formula for the sail roll angle control law is as follows: In the above formula, Δψ is the roll angle difference of the sail spacecraft, and N roll It is the first positive integer, and A is the coordinate system in which the sail spacecraft remains in the principal axis coordinate system S. b The magnitude of the angular velocity in the xy plane, Ω n The sail-shaped spacecraft maintains its position in the principal axis coordinate system S. b The angular velocity in the xy plane about the principal axis coordinate system S b The z-axis in the principal axis coordinate system S b The angular velocity of rotation in the xy plane.

6. The underactuated three-axis control method for a sail-type spacecraft as described in claim 1, characterized in that, The establishment of the sail roll axis angular velocity control law includes the following: Assume phase space PS = R 4 By ω xb ω yb , and Given that there are closed curves Curve1 and Curve2 in phase space, according to the attitude dynamics equations, the sail spacecraft moves along Curve1 and Curve2 respectively. The control formula for directional control of the roll axis angular velocity is as follows: In the above formula, m sail It is the overall mass of the sail-type spacecraft, ω xb The angular velocity ω of the spacecraft in the principal axis coordinate system S b The components of the x-axis projection, ω yb The angular velocity ω of the spacecraft in the principal axis coordinate system S b The components of the y-axis projection, ω zb The angular velocity ω of the spacecraft in the principal axis coordinate system S b The components of the z-axis projection, It is ω xb The derivative of It is ω yb The derivative of J x The moment of inertia matrix J corresponds to the principal axis coordinate system S. b The x-axis component, J y The moment of inertia matrix J corresponds to the principal axis coordinate system S. b The y-axis component, J z The moment of inertia matrix J corresponds to the principal axis coordinate system S. b The z-axis component.

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