CN103412573A

CN103412573A - Elliptical orbit spacecraft relative position regressing control method based on cascade connection equation

Info

Publication number: CN103412573A
Application number: CN2013103085670A
Authority: CN
Inventors: 李鹏; 岳晓奎; 袁建平
Original assignee: Northwestern Polytechnical University
Current assignee: Northwestern Polytechnical University
Priority date: 2013-07-22
Filing date: 2013-07-22
Publication date: 2013-11-27

Abstract

The invention provides a relative position regression control method of an elliptical orbit spacecraft based on cascading equations. Firstly, a relative dynamics model of the elliptical orbit in the form of cascading equations is established, and then a relative position regression controller design based on the cascading equations is carried out. The present invention has the advantages of structure and systemization, which makes the design process more flexible, not only can design the subsystems of each order of the system separately, but also consider the special problems existing in the subsystems separately, so that the system can obtain a good overall or Local stability, following properties and parameter robustness.

Description

Elliptical orbit spacecraft relative position room for manoeuvre control method based on the cascade equation

Technical field

The present invention relates to a kind of elliptical orbit spacecraft control method that closely regresses.

Background technology

Along with the deep development of space technology, the spacecraft that 26S Proteasome Structure and Function becomes increasingly complex is sent constantly into space, spacecraft and robot for space independently in-orbit operation be subject to extensive concern.Yet a lot of inert satellites that still stop have in-orbit formed increasing space trash, even spacecraft is subject to the atmospherical drag impact and causes orbit altitude constantly to reduce and the situation of finally falling also happens occasionally.These still safety of ground staff of carrying out for space tasks have all formed very important threat.Space junk cleaning and that inert satellite is caught to the technology that reclaims or repair is urgently to be resolved hurrily, and the closely relative orbit control technology of spacecraft is the basis of implementing above-mentioned task.Therefore be necessary to set up the relative position relation of Servicing spacecraft and passive space vehicle, design relative orbit control algorithm closely to realize that pursuit spacecraft is to the position control in the passive space vehicle approximate procedure.

The relative position that current spacecraft relative orbit control mainly concentrates on circular orbit is controlled, still not for operation task in-orbit, and the two spacecraft relative position room for manoeuvre controller design methods closely that move on elliptical orbit.

While for meeting, operating in-orbit, spacecraft relative position control accuracy is high, the requirement that departure is little, to guarantee the safety and reliability of operation task, existing method for optimally controlling often the resolving time long, operand is large, in the situation that the spaceborne computer arithmetic capability is limited, be difficult to realize the spacecraft relative orbit real-time from main control, thereby limited its application in space tasks.Generally speaking, currently also fail to set up a kind of mathematical model that is applicable to the spacecraft relative orbit cascade equation form of calculating fast, also fail to propose a kind of effective high-accuracy control method in real time for the relative position in spacecraft terminal approximate procedure.

Summary of the invention

In order to overcome the deficiencies in the prior art, the invention provides closely relative orbit describing method of a kind of spacecraft based on the cascade equation form, and, based on this mathematical model, adopt the room for manoeuvre control theory to design the room for manoeuvre controller.

The technical solution adopted for the present invention to solve the technical problems comprises the following steps:

Step 1, set up the elliptical orbit Relative dynamic equation of cascade equation form

(1) geocentric inertial coordinate system o _ix _iy _iz _i(s _i);

(2) orbital coordinate system oxyz (s _o): initial point o is positioned at the passive space vehicle barycenter, and the x axle points to initial point by the earth's core, and the y axle points to direction of motion in orbital plane, and the z axle meets the right-hand rule;

(3) body coordinate system o _bx _by _bz _b(S _b): initial point is spacecraft barycenter o _b, x _b, y _b, z _bConsistent with the principal axis of inertia respectively;

Under the passive space vehicle orbital coordinate system, select nonlinear T-H equation to describe the relative motion of two spacecrafts:

\{\begin{matrix} \overset{\cdot \cdot}{x} = {\overset{\cdot}{θ}}^{2} x + \overset{\cdot \cdot}{θ} y + 2 \overset{\cdot}{θ} \overset{\cdot}{y} + \frac{μ}{r_{t}^{2}} - \frac{μ (r_{t} + x)}{r_{c}^{3}} + f_{dx} + f_{cx} \\ \overset{\cdot \cdot}{y} = {- \overset{\cdot \cdot}{θ}}^{2} x + {\overset{\cdot}{θ}}^{2} y - 2 \overset{\cdot}{θ} \overset{\cdot}{x} - \frac{μy}{r_{c}^{3}} + f_{dy} + f_{cy} \\ \overset{\cdot \cdot}{z} = - \frac{μz}{r_{c}^{3}} + f_{dz} + f_{cz} \end{matrix} - - - (1)

Wherein, pursuit spacecraft and passive space vehicle mean with subscript c and t respectively, r _cAnd r _tFor the position vector of the earth's core to the spacecraft barycenter, μ is Gravitational coefficient of the Earth, and the pursuit spacecraft quality is m _cF _d=[f _Dxf _Dyf _Dz] ^TBe the poor of the perturbation acceleration that is subject to of two spacecrafts, F _c=[f _Cxf _Cyf _Cz] ^TFor the pursuit spacecraft track is controlled the acceleration that thrust produces; θ is the true anomaly of passive space vehicle track,

With

For orbit angular velocity and the angular acceleration of passive space vehicle, (1) formula is turned to the kinetics equation of following cascade form:

\overset{\cdot}{ρ} = v - - - (2)

\overset{\cdot}{v} = - C_{1} \overset{\cdot}{ρ} - D_{1} ρ - N_{1} (ρ) + F_{c} + F_{d} - - - (3)

In formula, ρ=[x y z] ^TFor from passive space vehicle, pointing to the relative position vector of pursuit spacecraft,

V=[v _xv _yv _z] ^TFor the relative velocity vector; Nonlinear terms in formula specifically are expressed as follows

N_{1} (ρ) = {[\frac{μ (r_{t} + x)}{r_{c}^{3}} - \frac{μ}{r_{t}^{2}} + \frac{2 μx}{r_{t}^{3}}, \frac{μy}{r_{c}^{3}} - \frac{μy}{r_{t}^{3}}, \frac{μz}{r_{c}^{3}} - \frac{μz}{r_{t}^{3}}]}^{T},

C_{1} = [\begin{matrix} 0 & - 2 \overset{\cdot}{θ} & 0 \\ 2 \overset{\cdot}{θ} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}],

D_{1} = [\begin{matrix} - {\overset{\cdot}{θ}}^{2} - \frac{2 μ}{r_{t}^{3}} & - \overset{\cdot \cdot}{θ} & 0 \\ \overset{\cdot \cdot}{θ} & - {\overset{\cdot}{θ}}^{2} + \frac{μ}{r_{t}^{3}} & 0 \\ 0 & 0 & \frac{μ}{r_{t}^{3}} \end{matrix}];

On the basis of equation (2), (3), choose state variable x ₁=ρ ^T, x ₂=v ^T, the six degree of freedom the coupled dynamical equation that obtains the relative orbit attitude is:

{\overset{\cdot}{x}}_{1} = A x_{2} - - - (4)

M {\overset{\cdot}{x}}_{2} = - {Cx}_{2} - {Dx}_{1} - N + U + P - - - (5)

Wherein, E ₃Be three rank unit matrix, A=E ₃, M=E ₃, C=C ₁, D=D ₁, N=[N ₁(ρ)] ^T,

P = F_{d}^{T};

Step 2, definition expectation state variable

x_{d 1} = ρ_{d}^{T} - - - (6)

x_{d 2} = v_{d}^{T} - - - (7)

Wherein, ρ _d, v _dBe respectively the expectation value of relative position and speed;

The definition error variance

e_{1} = x_{1} - x_{d 1} = {\tilde{ρ}}^{T} - - - (8)

Have

{\overset{\cdot}{x}}_{d 1} = {Ax}_{d 2} - - - (10)

{\overset{\cdot}{e}}_{1} = {Ae}_{2} - - - (11)

The definition status variable

z ₁＝e ₁ (12)

z ₂＝e ₂-α ₁ (13)

The definition stability function

α ₁＝-A ^Tz ₁ (14)

Have

{\overset{\cdot}{z}}_{1} = A (α_{1} + z_{2}) - - - (15)

{\overset{\cdot}{z}}_{2} = {\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{d 2} - {\overset{\cdot}{α}}_{1} - - - (16)

The Lyapunov function of definition one, second-order system is

V_{1} (z_{1}) = \frac{1}{2} z_{1}^{T} z_{1} - - - (17)

V_{2} (z_{1}, z_{2}) = V_{1} + \frac{1}{2} z_{2}^{T} {Mz}_{2} - - - (18)

To V ₁Differentiate, and by α ₁Substitution obtains

{\overset{\cdot}{V}}_{1} = - z_{1}^{T} {AA}^{T} z_{1} + z_{1}^{T} {Az}_{2} - - - (19)

By formula (19) substitution

Can obtain

M {\overset{\cdot}{z}}_{2} = M {\overset{\cdot}{x}}_{2} - M {\overset{\cdot}{x}}_{d 2} - M {\overset{\cdot}{α}}_{1}

(20)

= - {Cx}_{2} - {Dx}_{1} - N + U + P - M {\overset{\cdot}{x}}_{d 2} - M {\overset{\cdot}{α}}_{1}

To V ₂(z ₁, z ₂) differentiate by the following formula substitution, can obtain

{\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{1} + z_{2}^{T} [- {Cx}_{2} - {Dx}_{1} - N + U + P - M {\overset{\cdot}{x}}_{d 2} - M {\overset{\cdot}{α}}_{1}] - - - (21)

Due to C ₁, C ₂Skew-symmetry, Matrix C is also antisymmetric matrix, has

So design control law

U = C (x_{d 2} + α_{1}) + {Dx}_{1} + N - P + M {\overset{\cdot}{x}}_{d 2} + M {\overset{\cdot}{α}}_{1} - A^{T} z_{1} - K_{d} z_{2} - - - (22) .

The invention has the beneficial effects as follows: regressing and controlling (Backstepping Control) is a kind of recursive design method of controller based on Lyapunov stability theory, by the higher order term in nonlinear model, carry out the multistep Recursive Design, draw systematized Feedback Control Laws and corresponding Lyapunov function, make system obtain the good overall situation or local stability, following feature and parameter robustness.For describing, the relative appearance rail coupling model state of rotation noncooperative target approximate procedure nearly tens is tieed up, even up to tens dimensions, comparatively complicated based on the controller design of non-linear dynamic model while considering flexible appendage.The Backstepping Control Based method has structuring, systematized advantage, make design process more flexible, not only can design respectively each rank subsystem of system, and can consider separately the existing specific question of subsystem, this paper just is being based on the method and is designing closely relative orbit room for manoeuvre controller of spacecraft.

The accompanying drawing explanation

Fig. 1 is relative position curve synoptic diagram under the passive space vehicle orbital coordinate system.

Fig. 2 is relative velocity curve synoptic diagram under the passive space vehicle orbital coordinate system.

Fig. 3 is passive space vehicle orbital coordinate system lower railway control curve synoptic diagram.

Embodiment

The present invention is further described below in conjunction with drawings and Examples, the present invention includes but be not limited only to following embodiment.

A kind of spacecraft based on the cascade equation form is relative orbit room for manoeuvre control method closely, and its concrete steps comprise:

(1) geocentric inertial coordinate system o _ix _iy _iz _i(s _i); (2) orbital coordinate system oxyz (s _o): initial point o is positioned at the passive space vehicle barycenter, and the x axle points to initial point by the earth's core, and the y axle points to direction of motion in orbital plane, and the z axle meets the right-hand rule; (3) body coordinate system o _bx _by _bz _b(S _b): initial point is spacecraft barycenter o _b, x _b, y _b, z _bConsistent with the principal axis of inertia respectively.

\{\begin{matrix} \overset{\cdot \cdot}{x} = {\overset{\cdot}{θ}}^{2} x + \overset{\cdot \cdot}{θ} y + 2 \overset{\cdot}{θ} \overset{\cdot}{y} + \frac{μ}{r_{t}^{2}} - \frac{μ (r_{t} + x)}{r_{c}^{3}} + f_{dx} + f_{cx} \\ \overset{\cdot \cdot}{y} = {- \overset{\cdot \cdot}{θ}}^{2} x + {\overset{\cdot}{θ}}^{2} y - 2 \overset{\cdot}{θ} \overset{\cdot}{x} - \frac{μy}{r_{c}^{3}} + f_{dy} + f_{cy} \\ \overset{\cdot \cdot}{z} = - \frac{μz}{r_{c}^{3}} + f_{dz} + f_{cz} \end{matrix} - - - (1)

Wherein, pursuit spacecraft and passive space vehicle mean with subscript c and t respectively, r _cAnd r _tFor the position vector of the earth's core to the spacecraft barycenter, μ is Gravitational coefficient of the Earth, and the pursuit spacecraft quality is m _c.F _d=[f _Dxf _Dyf _Dz] ^TBe the poor of the perturbation acceleration that is subject to of two spacecrafts, F _c=[f _Cxf _Cyf _Cz] ^TFor the pursuit spacecraft track is controlled the acceleration that thrust produces.θ is the true anomaly of passive space vehicle track,

With For orbit angular velocity and the angular acceleration of passive space vehicle,

Through deriving, (1) formula is turned to following cascade

The kinetics equation of form:

\overset{\cdot}{ρ} = v - - - (2)

\overset{\cdot}{v} = - C_{1} \overset{\cdot}{ρ} - D_{1} ρ - N_{1} (ρ) + F_{c} + F_{d} - - - (3)

V=[v _xv _yv _z] ^TFor the relative velocity vector.Nonlinear terms in formula specifically are expressed as follows

N_{1} (ρ) = {[\frac{μ (r_{t} + x)}{r_{c}^{3}} - \frac{μ}{r_{t}^{2}} + \frac{2 μx}{r_{t}^{3}}, \frac{μy}{r_{c}^{3}} - \frac{μy}{r_{t}^{3}}, \frac{μz}{r_{c}^{3}} - \frac{μz}{r_{t}^{3}}]}^{T},

C_{1} = [\begin{matrix} 0 & - 2 \overset{\cdot}{θ} & 0 \\ 2 \overset{\cdot}{θ} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}],

D_{1} = [\begin{matrix} - {\overset{\cdot}{θ}}^{2} - \frac{2 μ}{r_{t}^{3}} & - \overset{\cdot \cdot}{θ} & 0 \\ \overset{\cdot \cdot}{θ} & - {\overset{\cdot}{θ}}^{2} + \frac{μ}{r_{t}^{3}} & 0 \\ 0 & 0 & \frac{μ}{r_{t}^{3}} \end{matrix}] .

On the basis of equation (2), (3), choose state variable x ₁=ρ ^T, x ₂=v ^T, the six degree of freedom the coupled dynamical equation that obtains the relative orbit attitude of deriving is:

{\overset{\cdot}{x}}_{1} = A x_{2} - - - (4)

M {\overset{\cdot}{x}}_{2} = - {Cx}_{2} - {Dx}_{1} - N + U + P - - - (5)

P = F_{d}^{T};

Step 2, based on the design of the relative position room for manoeuvre controller of cascade equation form

Definition expectation state variable

x_{d 1} = ρ_{d}^{T} - - - (6)

x_{d 2} = v_{d}^{T} - - - (7)

Wherein, ρ _d, v _dBe respectively the expectation value of relative position and speed.

The definition error variance

e_{1} = x_{1} - x_{d 1} = {\tilde{ρ}}^{T} - - - (8)

Have

{\overset{\cdot}{x}}_{d 1} = {Ax}_{d 2} - - - (10)

{\overset{\cdot}{e}}_{1} = {Ae}_{2} - - - (11)

The definition status variable

z ₁＝e ₁ (12)

z ₂＝e ₂-α ₁ (13)

The definition stability function

α ₁＝-A ^Tz ₁ (14)

Have

{\overset{\cdot}{z}}_{1} = A (α_{1} + z_{2}) - - - (15)

{\overset{\cdot}{z}}_{2} = {\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{d 2} - {\overset{\cdot}{α}}_{1} - - - (16)

The Lyapunov function of definition one, second-order system is

V_{1} (z_{1}) = \frac{1}{2} z_{1}^{T} z_{1} - - - (17)

V_{2} (z_{1}, z_{2}) = V_{1} + \frac{1}{2} z_{2}^{T} {Mz}_{2} - - - (18)

To V ₁Differentiate, and by α ₁Substitution can obtain

{\overset{\cdot}{V}}_{1} = - z_{1}^{T} {AA}^{T} z_{1} + z_{1}^{T} {Az}_{2} - - - (19)

By formula (19) substitution

Can obtain

M {\overset{\cdot}{z}}_{2} = M {\overset{\cdot}{x}}_{2} - M {\overset{\cdot}{x}}_{d 2} - M {\overset{\cdot}{α}}_{1}

(20)

= - {Cx}_{2} - {Dx}_{1} - N + U + P - M {\overset{\cdot}{x}}_{d 2} - M {\overset{\cdot}{α}}_{1}

{\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{1} + z_{2}^{T} [- {Cx}_{2} - {Dx}_{1} - N + U + P - M {\overset{\cdot}{x}}_{d 2} - M {\overset{\cdot}{α}}_{1}] - - - (21)

Due to C ₁, C ₂Skew-symmetry, Matrix C is also antisymmetric matrix, has

So design control law

U = C (x_{d 2} + α_{1}) + {Dx}_{1} + N - P + M {\overset{\cdot}{x}}_{d 2} + M {\overset{\cdot}{α}}_{1} - A^{T} z_{1} - K_{d} z_{2} - - - (22) .

Substitution formula (5), derive

{\overset{\cdot}{V}}_{2} (z_{1}, z_{2}) = - z_{1}^{T} {AA}^{T} z_{1} - z_{2}^{T} K_{d} z_{2} < 0 - - - (23)

Work as K _dDuring for positive definite matrix, V ₂(z ₁, z ₂)>0,

According to Lyapunov stability theory, the equilibrium point (z of closed-loop system ₁, z ₂)=(0,0) consistent progressive stable, during t → ∞, e ₁→ 0, e ₂→ 0, be also x ₁→ x _D1, x ₂→ x _D2.

The case verification of the inventive method:

1) passive space vehicle track six key elements are respectively

2) initial relative position and relative velocity are respectively ρ=[15-10 30] ^TM,

\overset{\cdot}{ρ} = {[\begin{matrix} 0.6 & - 0.2 & 0.3 \end{matrix}]}^{T} m / s,

Phase

Hope relative position vector relative velocity ρ _d=[0 0 5] ^TM,

3) Servicing spacecraft quality 100kg;

4) simulation time is 100s, step-length 0.1s.

By Fig. 1, Fig. 2 can see, x during 50s, and the relative distance basic controlling of y direction is to zero, and Servicing spacecraft moves to 15m place directly over target, and relative distance and relative velocity all significantly reduce, and hover in this position.When 150s-200s, pursuit spacecraft moves to assigned address gradually directly over target, with slow speed, further close on stop to passive space vehicle, during 200s, substantially reach desired location, show that two spacecrafts have reached docking location, relative velocity also is reduced to zero simultaneously, two spacecrafts reach and keep relatively static, for the tasks such as operation in-orbit that next stage will carry out ready.

Fig. 3 is track control curve synoptic diagram; As shown in Figure 3, when 50s-150s, for guaranteeing to approach safety, Servicing spacecraft first is controlled to passive space vehicle top certain distance, and now two spacecraft relative velocities are zero substantially, reach relative static conditions.During 150s, the rail control engine applies velocity pulse again makes pursuit spacecraft slowly move to target, along with distance approach and arrive assigned address, control decays to zero gradually.In whole control procedure, the controlled quentity controlled variable line smoothing, the track control is in the normal output area of topworks.

Claims

1. a kind of elliptical orbit spacecraft relative position regression control method based on cascade equation, it is characterized in that comprising the steps:

Step 1. Establish the relative dynamics model of the elliptical orbit in the form of cascade equations

(1) Geocentric inertial coordinate system o _i x _i y _i z _i (s _i );

(2) Orbital coordinate system oxyz(s _o ): the origin o is located at the center of mass of the target spacecraft, the x-axis points from the center of the earth to the origin, the y-axis points to the direction of motion in the orbital plane, and the z-axis satisfies the right-hand rule;

(3) Body coordinate system o _b x _b y _b z _b (S _b ): the origin is the center of mass o _b of the spacecraft, and x _b , y _b , and z _b are respectively consistent with the main axes of inertia;

In the orbital coordinate system of the target spacecraft, the nonlinear T-H equation is used to describe the relative motion of the two spacecraft:

\{\begin{matrix} \overset{\cdot \cdot \cdot &Center Dot;}{x x} = = {\overset{\cdot \cdot}{θ θ}}^{22} x x + + \overset{\cdot \cdot \cdot &Center Dot;}{θ θ} y the y + + 22 \overset{\cdot &Center Dot;}{θ θ} \overset{\cdot &Center Dot;}{y the y} + + \frac{μ μ}{{r r}_{t t}^{22}} - - \frac{μ μ (({r r}_{t t} + + x x))}{{r r}_{c c}^{33}} + + {f f}_{dx dx} + + {f f}_{cx cx} \\ \overset{\cdot &Center Dot; \cdot &Center Dot;}{y the y} = = {- - \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ}}^{22} x x + + {\overset{\cdot &Center Dot;}{θ θ}}^{22} y the y - - 22 \overset{\cdot &Center Dot;}{θ θ} \overset{\cdot &Center Dot;}{x x} - - \frac{μy μy}{{r r}_{c c}^{33}} + + {f f}_{dy dy} + + {f f}_{cy cy} \\ \overset{\cdot &Center Dot; \cdot &Center Dot;}{z z} = = - - \frac{μz μz}{{r r}_{c c}^{33}} + + {f f}_{dz dz} + + {f f}_{cz cz} \end{matrix} - - - - - - ((11))

Among them, the tracking spacecraft and the target spacecraft are denoted by the subscripts c and t respectively, r _c and r _t are the position vectors from the center of the earth to the center of mass of the spacecraft, μ is the gravitational constant of the earth, and the mass of the tracking spacecraft is m _c ; F _d ＝[f _dx f _dy f _dz ] ^T is the difference between the perturbed accelerations received by the two spacecraft, F _c ＝[f _cx f _cy f _cz ] ^T is the acceleration produced by tracking the orbital control thrust of the spacecraft; θ is the target spaceflight the true anomaly of the orbit,

and

For the orbital angular velocity and angular acceleration of the target spacecraft, formula (1) is transformed into the dynamic equation in the following concatenated form:

\overset{\cdot &Center Dot;}{ρ ρ} = = v v - - - - - - ((22))

\overset{\cdot &Center Dot;}{v v} = = - - {C C}_{11} \overset{\cdot &Center Dot;}{ρ ρ} - - {D D.}_{11} ρ ρ - - {N N}_{11} ((ρ ρ)) + + {F f}_{c c} + + {F f}_{d d} - - - - - - ((33))

In the formula, ρ=[x y z] ^T is the relative position vector from the target spacecraft to the tracking spacecraft,

v=[v _x v _y v _z ] ^T is the relative velocity vector; the nonlinear term in the formula is specifically expressed as follows

{N N}_{11} ((ρ ρ)) = = {[[\frac{μ μ (({r r}_{t t} + + x x))}{{r r}_{c c}^{33}} - - \frac{μ μ}{{r r}_{t t}^{22}} + + \frac{22 μx μx}{{r r}_{t t}^{33}},, \frac{μy μy}{{r r}_{c c}^{33}} - - \frac{μy μy}{{r r}_{t t}^{33}},, \frac{μz μz}{{r r}_{c c}^{33}} - - \frac{μz μz}{{r r}_{t t}^{33}}]]}^{T T},,

{C C}_{11} = = [\begin{matrix} 00 & - - 22 \overset{\cdot \cdot}{θ θ} & 00 \\ 22 \overset{\cdot \cdot}{θ θ} & 00 & 00 \\ 00 & 00 & 00 \end{matrix}],,

{D D.}_{11} = = [\begin{matrix} - - {\overset{\cdot &Center Dot;}{θ θ}}^{22} - - \frac{22 μ μ}{{r r}_{t t}^{33}} & - - \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} & 00 \\ \overset{\cdot &Center Dot; \cdot \cdot}{θ θ} & - - {\overset{\cdot \cdot}{θ θ}}^{22} + + \frac{μ μ}{{r r}_{t t}^{33}} & 00 \\ 00 & 00 & \frac{μ μ}{{r r}_{t t}^{33}} \end{matrix}];;

On the basis of equations (2) and (3), select the state variables x ₁ =ρ ^T , x ₂ =v ^T , and obtain the six-degree-of-freedom coupled dynamic equation of the relative orbital attitude as:

{\overset{\cdot \cdot}{x x}}_{11} = = A A {x x}_{22} - - - - - - ((44))

M m {\overset{\cdot \cdot}{x x}}_{22} = = - - {Cx Cx}_{22} - - {Dx Dx}_{11} - - N N + + U u + + P P - - - - - - ((55))

Wherein, E ₃ is a third-order unit matrix, A=E ₃ , M=E ₃ , C=C ₁ , D=D ₁ , N=[N ₁ (ρ)] ^T ,

P = f_{d}^{T};

Step 2. Define the desired state variables

{x x}_{d d 11} = = {ρ ρ}_{d d}^{T T} - - - - - - ((66))

{x x}_{d d 22} = = {v v}_{d d}^{T T} - - - - - - ((77))

Among them, ρ _d , v _d are the expected values of relative position and velocity respectively;

Define the error variable

{e e}_{11} = = {x x}_{11} - - {x x}_{d d 11} = = {\overset{~ ~}{ρ ρ}}^{T T} - - - - - - ((88))

then there is

{\overset{\cdot &Center Dot;}{x x}}_{d d 11} = = {Ax Ax}_{d d 22} - - - - - - ((1010))

{\overset{\cdot &Center Dot;}{e e}}_{11} = = {Ae Ae}_{22} - - - - - - ((1111))

define state variables

z ₁ =e ₁ (12)

z ₂ = e ₂ -α ₁ (13)

Define a stable function

α ₁ = ^-AT z ₁ (14)

then there is

{\overset{\cdot &Center Dot;}{z z}}_{11} = = A A (({α α}_{11} + + {z z}_{22})) - - - - - - ((1515))

{\overset{\cdot &Center Dot;}{z z}}_{22} = = {\overset{\cdot &Center Dot;}{x x}}_{22} - - {\overset{\cdot \cdot}{x x}}_{d d 22} - - {\overset{\cdot &Center Dot;}{α α}}_{11} - - - - - - ((1616))

Define the Lyapunov functions of the first-order and second-order systems as

{V V}_{11} (({z z}_{11})) = = \frac{11}{22} {z z}_{11}^{T T} {z z}_{11} - - - - - - ((1717))

{V V}_{22} (({z z}_{11},, {z z}_{22})) = = {V V}_{11} + + \frac{11}{22} {z z}_{22}^{T T} {Mz Mz}_{22} - - - - - - ((1818))

Deriving V ₁ and substituting α ₁ into

{\overset{\cdot \cdot}{V V}}_{11} = = - - {z z}_{11}^{T T} {AA AAA}^{T T} {z z}_{11} + + {z z}_{11}^{T T} {Az Az}_{22} - - - - - - ((1919))

Substitute (19) into

Available

m {\overset{&Center Dot;}{z}}_{2} = m {\overset{\cdot}{x}}_{2} - m {\overset{\cdot}{x}}_{d 2} - m {\overset{\cdot}{α}}_{1}

(20)

= = - - {Cx Cx}_{22} - - {Dx Dx}_{11} - - N N + + U u + + P P - - M m {\overset{\cdot \cdot}{x x}}_{d d 22} - - M m {\overset{\cdot &Center Dot;}{α α}}_{11}

Taking the derivative of V ₂ (z ₁ , z ₂ ) and substituting the above formula, we can get

{\overset{\cdot &Center Dot;}{V V}}_{22} = = {\overset{\cdot &Center Dot;}{V V}}_{11} + + {z z}_{22}^{T T} [[- - {Cx Cx}_{22} - - {Dx Dx}_{11} - - N N + + U u + + P P - - M m {\overset{\cdot &Center Dot;}{x x}}_{d d 22} - - M m {\overset{\cdot &Center Dot;}{α α}}_{11}]] - - - - - - ((21 twenty one))

Due to the anti-symmetry of C ₁ and C ₂ , the matrix C is also an anti-symmetric matrix, then

Therefore, the design control law

U u = = C C (({x x}_{d d 22} + + {α α}_{11})) + + {Dx Dx}_{11} + + N N - - P P + + M m {\overset{\cdot \cdot}{x x}}_{d d 22} + + M m {\overset{\cdot \cdot}{α α}}_{11} - - {A A}^{T T} {z z}_{11} - - {K K}_{d d} {z z}_{22} - - - - - - ((22 twenty two)) . .