CN105955028A - On-orbit guidance avoidance control integrated algorithm for spacecraft - Google Patents

Abstract

The invention discloses an on-orbit guidance avoidance control integrated algorithm for a spacecraft. The on-orbit guidance avoidance control integrated algorithm includes the steps: establishing a relative movement model; designing a potential function and a sliding-mode control guidance control integrated algorithm; analyzing the robustness of the algorithm, and providing a safety orbit space; and finally verifying the effectiveness of the on-orbit guidance avoidance control integrated algorithm for a spacecraft through a simulation example. The on-orbit guidance avoidance control integrated algorithm for a spacecraft analyzes the safety section of the safety orbit under a disturbed condition, can quickly perform on-orbit calculation and control the spacecraft to perform real-time obstacle avoidance, can guarantee that the spacecraft can avoid the obstacle in the safety range, and is conductive to a spacecraft to more reliably perform an on-orbit flight task in the future.

Description

A kind of spacecraft evades Guidance and control Integrated Algorithm in-orbit

[technical field]

The invention belongs to technical field of spacecraft control, relate to a kind of spacecraft and evade Guidance and control integration in-orbit Algorithm.

[background technology]

Along with improving constantly space research, exploitation with application power, various countries in succession develop and transmit in a large number Towards the spacecraft of various mission requirementses, by being initially entered into space and utilizing space to answer to spatial operation and space With.And the autonomy of spatial operation, safety etc. are the most gradually paid close attention to by people.

In recent years, artificial potential function method in the guidance of spacecraft safe trajectory (Artificial potential function, APF) more and more used.It is initially at " Real-Time Obstacle Avoidance for by Khatib Manipulator and Mobile Robots " middle proposition, it is used for solving the mechanical arm path planning in working place Problem, its basic thought is structure target gravitational field and barrier repulsion field, makes they effect search gesture letters jointly Number potential energy descent direction is found without touching path.McInnes is at " Autonomous Rendezvous Using Artificial Potential Function Guidance " in propose applied and in spacecraft field be up till now Only having had 20 years, the document of related fields has a lot, develops relative maturity.Such as Ender St.John is at " Ankersen F.Safety-critical autonomous spacecraft proximity operations Via potential function guidance " in propose apply it to autonomous transfer vehicle ATV, HTV With in the spacecrafts rendezvous of ISS, result is satisfactory.Zhang Dawei et al. is at " Safe Guidance for Autonomous Rendezvous and Docking with a Non-Cooperative Target " in propose by people Work potential function is applied in the spacecrafts rendezvous task of noncooperative target, and combines with fuzzy control method, for Dynamic barrier hide and dock safe trajectory constraint carried out research and policy.Document " Swarm Aggregations using artificial potentials and sliding mode control " in propose by artificial gesture letter Number and sliding formwork control to combine and are applied to satellite gathering and form into columns.Document " Autonomous Distributed Control Algorithm for Multiple Spacecraft in Close Proximity Operations " middle proposition will Artificial potential function method and LQR method combine and are applied in multiobject short-range operation.Artificial potential function method tool The remarkable advantage such as have stability easily to judge, computational efficiency is high, is a kind of intersection compared and be suitably applied in autonomous type Artificial intelligence approach in docking mission.And sliding formwork controls to be applicable to linearity and non-linearity system, continuously with discrete System, determining and uncertain system etc., this control method makes system mode along cunning by the switching of controlled quentity controlled variable Die face is slided, and makes system have invariance when by Parameter Perturbation and external interference.Experience years development, existing Become and automatically controlled a kind of universal method for designing.But there is certain buffeting problem.

By potential function method advantage in the evading of barrier, and sliding formwork control for nonlinear Control good Good performance, two kinds of methods are combined by the present invention, and devise one by robust control theory and can deposit The Guidance and control Integrated Algorithm of Obstacle avoidance, Ke Yishi is carried out in the case of Position disturbance and measurement error Hiding of the trajectory planning of existing optional position and static or dynamic barrier.

[summary of the invention]

Present invention aims to effective avoiding barrier during spacecraft flight in-orbit, improve flight safety Demand, it is provided that a kind of spacecraft evades Guidance and control Integrated Algorithm in-orbit, and this algorithm can improve the reality of avoidance Time property and safety.

For reaching above-mentioned purpose, the present invention is achieved by the following technical solutions:

A kind of spacecraft evades Guidance and control Integrated Algorithm in-orbit, comprises the following steps:

1) Spacecraft Relative Motion equation is set up

With target track coordinate system as reference frame, set up Spacecraft Relative Motion kinetics equation:

$\stackrel{\·\·}{x}=A\stackrel{\·}{x}+Dx+u+d---\left(1\right)$

Matrix A and D are described as:

$A=\left[\begin{array}{ccc}0& 2{\mathrm{\ω}}_k& -2{\mathrm{\ω}}_j\\ -2{\mathrm{\ω}}_k& 0& 2{\mathrm{\ω}}_i\\ 2{\mathrm{\ω}}_j& -2{\mathrm{\ω}}_i& 0\end{array}\right]$

$D=\left[\begin{array}{ccc}{{\mathrm{\ω}}_j}^2+{{\mathrm{\ω}}_k}^2& -{\mathrm{\ω}}_i{\mathrm{\ω}}_j+{\stackrel{\·}{\mathrm{\ω}}}_k& -{\mathrm{\ω}}_i{\mathrm{\ω}}_k-{\stackrel{\·}{\mathrm{\ω}}}_j\\ -{\mathrm{\ω}}_i{\mathrm{\ω}}_j-{\stackrel{\·}{\mathrm{\ω}}}_k& {{\mathrm{\ω}}_i}^2+{{\mathrm{\ω}}_k}^2& -{\mathrm{\ω}}_j{\mathrm{\ω}}_k+{\stackrel{\·}{\mathrm{\ω}}}_i\\ -{\mathrm{\ω}}_i{\mathrm{\ω}}_k+{\stackrel{\·}{\mathrm{\ω}}}_j& -{\mathrm{\ω}}_j{\mathrm{\ω}}_k-{\stackrel{\·}{\mathrm{\ω}}}_i& {{\mathrm{\ω}}_i}^2+{{\mathrm{\ω}}_j}^2\end{array}\right]$

Wherein, ω is the orbit angular velocity of passive space vehicle, and u is the control output of pursuit spacecraft, and d is not for build Mould error, including the Gradient of Gravitation error, various perturbations, target the unknown control power, or other immeasurabilities and The unknown disturbances estimated；

Kinetics equation therein can be replaced by any Spacecraft Relative Motion kinetic model；

2) design potential function and sliding formwork control integration guidance control algolithm

The potential function of spacecraft is defined in state space, if state variable is x, it is desirable to state be x_d；Draw Power potential function promotes pursuit spacecraft to arrive the dbjective state preset, and therefore gravitational potential function is designed as:

Gravitational potential function contains position and velocity information simultaneously, describes position and its motion that expectation is followed the tracks of Trend；Wherein P_p、P_vFor positive definite matrix；

Repulsion gesture selection Gaussian function:

Wherein, ψ represents the height of repulsion gesture, and σ represents the width of repulsion gesture, and matrix N is relevant with barrier shape Profile matrix；Subscript i represents multiple different barrier；Total potential energy represents that spacecraft exists barrier In the case of potential energy be gravitational potential and repulsion gesture and；The sliding-mode surface that the gradient offer sliding formwork of potential function controls:

The gradient of artificial energy field representated by potential function is used for representing the direction that its energy declines, and energy is high Place is replaced by repulsion gesture virtual produced by barrier, and low potential field is replaced by the desired trajectory of planning, because of This is the secure path meeting mission planning along the path of gradient direction；

The convergence that sliding formwork controls must is fulfilled for:

$s^T\stackrel{\&CenterDot;}{s}<0---\left(5\right)$

Hamilton-Jacobi Inequality theory is used to be controlled device design；Its robustness uses L₂Gain Form be described:

$J=\underset{\left|\right|d\left|\right|\&NotEqual;0}{sup}\frac{\left|\right|z|{|}_2}{\left|\right|d|{|}_2}\&le;\mathrm{\&gamma;}---\left(6\right)$

Wherein assume that z is sliding-mode surface；Definition lyapunov function is:

$V=\frac{1}{2}{s}^T{P}_v^{-1}s---\left(7\right)$

Because control stability need to meetTherefore design controller is:

$u=(A-P_v^{-1}{P}_p)\stackrel{\&CenterDot;}{e}+De-\left(\frac{1+{\mathrm{\&gamma;}}^2}{2{\mathrm{\&gamma;}}^2}\right)s---\left(8\right)$

Obtained by checking Hamilton function:

$\begin{array}{c}H=\stackrel{\&CenterDot;}{V}-\frac{1}{2}{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2+\frac{1}{2}|\left|z\right|{|}^2\\ =s^T(-\frac{1}{2{\mathrm{\&gamma;}}^2}s-\frac{1}{2}s-d)-\frac{1}{2}{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2+\frac{1}{2}|\left|z\right|{|}^2\\ =(-\frac{1}{2{\mathrm{\&gamma;}}^2}{s}^{T}s-\frac{1}{2}{\mathrm{\&gamma;}}^2|\left|d\right|{|}^2-s^{T}d)+\left(\frac{1}{2}\right|\left|z\right|{|}^2-\frac{1}{2}{s}^{T}s)=-\frac{1}{2}\left|\right|\frac{1}{\mathrm{\&gamma;}}s+\mathrm{\&gamma;}d|{|}^2\&le;0\end{array}---\left(9\right)$

Hence it is demonstrated that control system is the most stable；

3) robust analysis

Control power to be expressed as when considering measurement error:

$u=(A-P_v^{-1}{P}_p)\stackrel{\&CenterDot;}{\tilde{e}}+D\tilde{e}-(\frac{1}{2{\mathrm{\&gamma;}}^2}\tilde{s}-\frac{1}{2}\tilde{s})---\left(10\right)$

Definitionλ(A) andIt is respectively the minimum and maximum eigenvalue of matrix A；F_tFor kinetics equation linearisation Coefficient matrix, H_tFor calculation matrix；For simplicity, orderAnd define:

$\begin{array}{c}a:=\sup \stackrel{\&OverBar;}{\mathrm{\&lambda;}}\left(F\right(\tilde{x}\left)\right),b:=\underset{t}{\sup }\stackrel{\&OverBar;}{\mathrm{\&lambda;}}\left(Q_t\right),\\ c:=\inf \underset{\&OverBar;}{\mathrm{\&lambda;}}\left(H_t{\left(f\left(\tilde{x},u\right)\right)}^{\&prime;}{R}_t^{-1}{H}_t\right(f\left(\tilde{x},u\right)\left)\right)\\ d:=bc/a+1\end{array}---\left(11\right)$

For sensor, measurement data employing rate is p, has a following expression in each moment:

$\stackrel{\&OverBar;}{E\&lsqb;l_t\&rsqb;}=(a\stackrel{\&OverBar;}{E\&lsqb;l_{t-1}\&rsqb;}+b)\left(\right(1-p)+\frac{p}{c\stackrel{\&OverBar;}{E\&lsqb;l_{t-1}\&rsqb;}+d})---\left(12\right)$

In the range of assuming that the track of pursuit spacecraft has the probability of 99.74% to be in safe trajectory, i.e. at state value InInterval, whereinTherefore the limits of error of sliding-mode surface s is obtained by mean value theorem:

$\left|\right|s-\tilde{s}\left|\right|\&le;M_s\left|\right|x-\tilde{x}\left|\right|=M_s\mathrm{\&eta;}---\left(13\right)$

Wherein

Now system is stable relevant with the certainty of measurement of sensor, through being derived by the measurement noise of sensor Need to meet in the range of stablizing such as lower inequality time control guarantee and be in above-mentioned safe trajectory:

$\mathrm{\&eta;}\left|\right|s\left|\right|\mathrm{\&phi;}+{\mathrm{\&eta;}}^2{M}_s^2+{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2\&le;\frac{1}{{\mathrm{\&gamma;}}^2}|\left|s\right|{|}^2---\left(14\right)$

Wherein,

Compared with prior art, the method have the advantages that

The present invention proposes the Guidance and control Integrated Algorithm of a kind of spacecraft avoidance in-orbit, and it is dry to analyze existence The security interval of safe trajectory in the case of disturbing.This algorithm can calculate and control spacecraft the most in-orbit to be carried out in real time Avoidance, and guarantee spacecraft avoiding barrier in safety range, it is of value to Future Spacecraft and enters more reliably Row aerial mission in-orbit.

[accompanying drawing explanation]

Fig. 1 is relative movement orbit figure during static-obstacle thing avoidance；

Fig. 2 is Monte-Carlo method figure；

Fig. 3 is relative movement orbit figure during barrier motion.

[detailed description of the invention]

Below in conjunction with the accompanying drawings the present invention is described in further detail:

Seeing Fig. 1-Fig. 3, spacecraft of the present invention is evaded Guidance and control Integrated Algorithm in-orbit, is comprised the following steps:

1) Spacecraft Relative Motion equation is set up

With target track coordinate system as reference frame, set up Spacecraft Relative Motion kinetics equation:

$\stackrel{\&CenterDot;\&CenterDot;}{x}=A\stackrel{\&CenterDot;}{x}+Dx+u+d---\left(1\right)$

Matrix A and D are described as:

$A=\left[\begin{array}{ccc}0& 2{\mathrm{\&omega;}}_k& -2{\mathrm{\&omega;}}_j\\ -2{\mathrm{\&omega;}}_k& 0& 2{\mathrm{\&omega;}}_i\\ 2{\mathrm{\&omega;}}_j& -2{\mathrm{\&omega;}}_i& 0\end{array}\right]$

$D=\left[\begin{array}{ccc}{{\mathrm{\&omega;}}_j}^2+{{\mathrm{\&omega;}}_k}^2& -{\mathrm{\&omega;}}_i{\mathrm{\&omega;}}_j+{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_k& -{\mathrm{\&omega;}}_i{\mathrm{\&omega;}}_k-{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_j\\ -{\mathrm{\&omega;}}_i{\mathrm{\&omega;}}_j-{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_k& {{\mathrm{\&omega;}}_i}^2+{{\mathrm{\&omega;}}_k}^2& -{\mathrm{\&omega;}}_j{\mathrm{\&omega;}}_k+{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_i\\ -{\mathrm{\&omega;}}_i{\mathrm{\&omega;}}_k+{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_j& -{\mathrm{\&omega;}}_j{\mathrm{\&omega;}}_k-{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_i& {{\mathrm{\&omega;}}_i}^2+{{\mathrm{\&omega;}}_j}^2\end{array}\right]$

Wherein, ω is the orbit angular velocity of passive space vehicle, and u is the control output of pursuit spacecraft, and d is not for build Mould error, including the Gradient of Gravitation error, various perturbations, target the unknown control power, or other immeasurabilities and The unknown disturbances estimated；

Kinetics equation therein can be replaced by any Spacecraft Relative Motion kinetic model；

2) design potential function and sliding formwork control integration guidance control algolithm

The potential function of spacecraft is defined in state space, if state variable is x, it is desirable to state be x_d；Draw Power potential function promotes pursuit spacecraft to arrive the dbjective state preset, and therefore gravitational potential function is designed as:

Gravitational potential function contains position and velocity information simultaneously, describes position and its motion that expectation is followed the tracks of Trend；Wherein P_p、P_vFor positive definite matrix；

Repulsion gesture selection Gaussian function:

Wherein, ψ represents the height of repulsion gesture, and σ represents the width of repulsion gesture, and matrix N is relevant with barrier shape Profile matrix；Subscript i represents multiple different barrier；Total potential energy represents that spacecraft exists barrier In the case of potential energy be gravitational potential and repulsion gesture and；The sliding-mode surface that the gradient offer sliding formwork of potential function controls:

The gradient of artificial energy field representated by potential function is used for representing the direction that its energy declines, and energy is high Place is replaced by repulsion gesture virtual produced by barrier, and low potential field is replaced by the desired trajectory of planning, because of This is the secure path meeting mission planning along the path of gradient direction；

The convergence that sliding formwork controls must is fulfilled for:

$s^T\stackrel{\&CenterDot;}{s}<0---\left(5\right)$

Hamilton-Jacobi Inequality theory is used to be controlled device design；Its robustness uses L₂Gain Form be described:

$J=\underset{\left|\right|d\left|\right|\&NotEqual;0}{sup}\frac{\left|\right|z|{|}_2}{\left|\right|d|{|}_2}\&le;\mathrm{\&gamma;}---\left(6\right)$

Wherein assume that z is sliding-mode surface；Definition lyapunov function is:

$V=\frac{1}{2}{s}^T{P}_v^{-1}s---\left(7\right)$

Because control stability need to meetTherefore design controller is:

$u=(A-P_v^{-1}{P}_p)\stackrel{\&CenterDot;}{e}+De-\left(\frac{1+{\mathrm{\&gamma;}}^2}{2{\mathrm{\&gamma;}}^2}\right)s---\left(8\right)$

Obtained by checking Hamilton function:

$\begin{array}{c}H=\stackrel{\&CenterDot;}{V}-\frac{1}{2}{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2+\frac{1}{2}|\left|z\right|{|}^2\\ =s^T(-\frac{1}{2{\mathrm{\&gamma;}}^2}s-\frac{1}{2}s-d)-\frac{1}{2}{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2+\frac{1}{2}|\left|z\right|{|}^2\\ =(-\frac{1}{2{\mathrm{\&gamma;}}^2}{s}^{T}s-\frac{1}{2}{\mathrm{\&gamma;}}^2|\left|d\right|{|}^2-s^{T}d)+\left(\frac{1}{2}\right|\left|z\right|{|}^2-\frac{1}{2}{s}^{T}s)=-\frac{1}{2}\left|\right|\frac{1}{\mathrm{\&gamma;}}s+\mathrm{\&gamma;}d|{|}^2\&le;0\end{array}---\left(9\right)$

Hence it is demonstrated that control system is the most stable；

3) robust analysis

Control power to be expressed as when considering measurement error:

$u=(A-P_v^{-1}{P}_p)\stackrel{\&CenterDot;}{\tilde{e}}+D\tilde{e}-(\frac{1}{2{\mathrm{\&gamma;}}^2}\tilde{s}-\frac{1}{2}\tilde{s})---\left(10\right)$

Definitionλ(A) andIt is respectively the minimum and maximum eigenvalue of matrix A；F_tFor kinetics equation linearisation Coefficient matrix, H_tFor calculation matrix；For simplicity, orderAnd define:

$\begin{array}{c}a:=\sup \stackrel{\&OverBar;}{\mathrm{\&lambda;}}\left(F\right(\tilde{x}\left)\right),b:=\underset{t}{\sup }\stackrel{\&OverBar;}{\mathrm{\&lambda;}}\left(Q_t\right),\\ c:=\inf \underset{\&OverBar;}{\mathrm{\&lambda;}}\left(H_t{\left(f\left(\tilde{x},u\right)\right)}^{\&prime;}{R}_t^{-1}{H}_t\right(f\left(\tilde{x},u\right)\left)\right)\\ d:=bc/a+1\end{array}---\left(11\right)$

For sensor, measurement data employing rate is p, has a following expression in each moment:

$\stackrel{\&OverBar;}{E\&lsqb;l_t\&rsqb;}=(a\stackrel{\&OverBar;}{E\&lsqb;l_{t-1}\&rsqb;}+b)\left(\right(1-p)+\frac{p}{c\stackrel{\&OverBar;}{E\&lsqb;l_{t-1}\&rsqb;}+d})---\left(12\right)$

In the range of assuming that the track of pursuit spacecraft has the probability of 99.74% to be in safe trajectory, i.e. at state value InInterval, whereinTherefore the limits of error of sliding-mode surface s is obtained by mean value theorem:

$\left|\right|s-\tilde{s}\left|\right|\&le;M_s\left|\right|x-\tilde{x}\left|\right|=M_s\mathrm{\&eta;}---\left(13\right)$

Wherein

Now system is stable relevant with the certainty of measurement of sensor, through being derived by the measurement noise of sensor Need to meet in the range of stablizing such as lower inequality time control guarantee and be in above-mentioned safe trajectory:

$\mathrm{\&eta;}\left|\right|s\left|\right|\mathrm{\&phi;}+{\mathrm{\&eta;}}^2{M}_s^2+{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2\&le;\frac{1}{{\mathrm{\&gamma;}}^2}|\left|s\right|{|}^2---\left(14\right)$

Wherein,

Initial phase is adjusted the distance: [-1,000 0 0], target relative distance: [-10 0 0]^T, the unknown disturbance upper limit: 2m/s², sample frequency: 10HZ, site error: σ_x=σ_y=σ_z=0.3m, velocity error: Measure angular error: σ_α=σ_β=0.06 °, measure range error: σ_ρ=0.5m, the thrust upper limit: 200N.

With flight intersection task avoidance in-orbit as example, Guidance and control Integrated Algorithm of the present invention and place of safety are described Between calculate effectiveness.In tracing process, if sensor can not provide correct measurement signal, posteriority is estimated Meter variance can increased, so that secure border amplifies, if secure border and barrier have coincidence, and must Must more emat sensor or change control parameter.As it is shown in figure 1, real movement locus is in tubular space. In order to further illustrate the reliability of safe trajectory and Robust Analysis, in the feelings not changing target preliminary orbit parameter Monte-Carlo method experiment has been carried out, as in figure 2 it is shown, all of track is in the tubular space calculated under condition In, and keep safe distance with barrier, as can be seen from the figure real track distance barrier nearest away from From also above 20m.

And demonstrating this algorithm effectiveness in dynamic barrier is evaded, design and simulation is as it is shown on figure 3, navigate It device is respectively at 70s, and 75s, 80s are shown, pursuit spacecraft can success avoiding dynamic barrier, and Zhongdao reaches target location.

Above content is only the technological thought that the present invention is described, it is impossible to limit protection scope of the present invention with this, all It is the technological thought proposed according to the present invention, any change done on the basis of technical scheme, each fall within this Within the protection domain of bright claims.

Claims (1)

1. a spacecraft evades Guidance and control Integrated Algorithm in-orbit, it is characterised in that comprise the following steps:

1) Spacecraft Relative Motion equation is set up

With target track coordinate system as reference frame, set up Spacecraft Relative Motion kinetics equation:

$\stackrel{\&CenterDot;\&CenterDot;}{x}=A\stackrel{\&CenterDot;}{x}+Dx+u+d---\left(1\right)$

Matrix A and D are described as:

$A=\left[\begin{array}{ccc}0& 2{\mathrm{\&omega;}}_k& -2{\mathrm{\&omega;}}_j\\ -2{\mathrm{\&omega;}}_k& 0& 2{\mathrm{\&omega;}}_i\\ 2{\mathrm{\&omega;}}_j& -2{\mathrm{\&omega;}}_i& 0\end{array}\right]$

$D=\left[\begin{array}{ccc}{{\mathrm{\&omega;}}_j}^2+{{\mathrm{\&omega;}}_k}^2& -{\mathrm{\&omega;}}_i{\mathrm{\&omega;}}_j+{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_k& -{\mathrm{\&omega;}}_i{\mathrm{\&omega;}}_k-{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_j\\ -{\mathrm{\&omega;}}_i{\mathrm{\&omega;}}_j-{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_k& {{\mathrm{\&omega;}}_i}^2+{{\mathrm{\&omega;}}_k}^2& -{\mathrm{\&omega;}}_j{\mathrm{\&omega;}}_k+{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_i\\ -{\mathrm{\&omega;}}_i{\mathrm{\&omega;}}_k+{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_j& -{\mathrm{\&omega;}}_j{\mathrm{\&omega;}}_k-{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_i& {{\mathrm{\&omega;}}_i}^2+{{\mathrm{\&omega;}}_j}^2\end{array}\right]$

Wherein, ω is the orbit angular velocity of passive space vehicle, and u is the control output of pursuit spacecraft, and d is not for build Mould error, including the Gradient of Gravitation error, various perturbations, target the unknown control power, or other immeasurabilities and The unknown disturbances estimated；

Kinetics equation therein can be replaced by any Spacecraft Relative Motion kinetic model；

2) design potential function and sliding formwork control integration guidance control algolithm

The potential function of spacecraft is defined in state space, if state variable is x, it is desirable to state be x_d；Draw Power potential function promotes pursuit spacecraft to arrive the dbjective state preset, and therefore gravitational potential function is designed as:

Gravitational potential function contains position and velocity information simultaneously, describes position and its motion that expectation is followed the tracks of Trend；Wherein P_p、P_vFor positive definite matrix；

Repulsion gesture selection Gaussian function:

Wherein, ψ represents the height of repulsion gesture, and σ represents the width of repulsion gesture, and matrix N is relevant with barrier shape Profile matrix；Subscript i represents multiple different barrier；Total potential energy represents that spacecraft exists barrier In the case of potential energy be gravitational potential and repulsion gesture and；The sliding-mode surface that the gradient offer sliding formwork of potential function controls:

The gradient of artificial energy field representated by potential function is used for representing the direction that its energy declines, and energy is high Place is replaced by repulsion gesture virtual produced by barrier, and low potential field is replaced by the desired trajectory of planning, because of This is the secure path meeting mission planning along the path of gradient direction；

The convergence that sliding formwork controls must is fulfilled for:

$s^T\stackrel{\&CenterDot;}{s}<0---\left(5\right)$

Hamilton-Jacobi Inequality theory is used to be controlled device design；Its robustness uses L₂Gain Form be described:

$J=\underset{\left|\right|d\left|\right|\&NotEqual;0}{sup}\frac{\left|\right|z|{|}_2}{\left|\right|d|{|}_2}\&le;\mathrm{\&gamma;}---\left(6\right)$

Wherein assume that z is sliding-mode surface；Definition lyapunov function is:

$V=\frac{1}{2}{s}^T{P}_v^{-1}s---\left(7\right)$

Because control stability need to meetTherefore design controller is:

$u=(A-P_v^{-1}{P}_p)\stackrel{\&CenterDot;}{e}+De-\left(\frac{1+{\mathrm{\&gamma;}}^2}{2{\mathrm{\&gamma;}}^2}\right)s---\left(8\right)$

Obtained by checking Hamilton function:

$\begin{array}{c}H=\stackrel{\&CenterDot;}{V}-\frac{1}{2}{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2+\frac{1}{2}|\left|z\right|{|}^2\\ =s^T(-\frac{1}{2{\mathrm{\&gamma;}}^2}s-\frac{1}{2}s-d)-\frac{1}{2}{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2+\frac{1}{2}|\left|z\right|{|}^2\\ =(-\frac{1}{2{\mathrm{\&gamma;}}^2}{s}^{T}s-\frac{1}{2}{\mathrm{\&gamma;}}^2|\left|d\right|{|}^2-s^{T}d)+\left(\frac{1}{2}\right|\left|z\right|{|}^2-\frac{1}{2}{s}^{T}s)=-\frac{1}{2}\left|\right|\frac{1}{\mathrm{\&gamma;}}s+\mathrm{\&gamma;}d|{|}^2\&le;0\end{array}---\left(9\right)$

Hence it is demonstrated that control system is the most stable；

3) robust analysis

Control power to be expressed as when considering measurement error:

$u=(A-P_v^{-1}{P}_p)\stackrel{\&CenterDot;}{\tilde{e}}+D\tilde{e}-(\frac{1}{2{\mathrm{\&gamma;}}^2}\tilde{s}-\frac{1}{2}\tilde{s})---\left(10\right)$

Definitionλ(A) andIt is respectively the minimum and maximum eigenvalue of matrix A；F_tFor kinetics equation linearisation Coefficient matrix, H_tFor calculation matrix；For simplicity, orderAnd define:

$a:=\sup \stackrel{\&OverBar;}{\mathrm{\&lambda;}}\left(F\right(\tilde{x}\left)\right),b:=\underset{t}{\sup }\stackrel{\&OverBar;}{\mathrm{\&lambda;}}\left(Q_t\right),$

$c:=\inf \underset{\&OverBar;}{\mathrm{\&lambda;}}\left(H_t{\left(f\left(\tilde{x},u\right)\right)}^{\&prime;}{R}_t^{-1}{H}_t\right(f\left(\tilde{x},u\right)\left)\right)---\left(11\right)$

D:=bc/a+1

For sensor, measurement data employing rate is p, has a following expression in each moment:

In the range of assuming that the track of pursuit spacecraft has the probability of 99.74% to be in safe trajectory, i.e. at state value InInterval, whereinTherefore the limits of error of sliding-mode surface s is obtained by mean value theorem:

$\left|\right|s-\tilde{s}\left|\right|\&le;M_s\left|\right|x-\tilde{x}\left|\right|=M_s\mathrm{\&eta;}---\left(13\right)$

Wherein

Now system is stable relevant with the certainty of measurement of sensor, through being derived by the measurement noise of sensor Need to meet in the range of stablizing such as lower inequality time control guarantee and be in above-mentioned safe trajectory:

$\mathrm{\&eta;}\left|\right|s\left|\right|\mathrm{\&phi;}+{\mathrm{\&eta;}}^2{M}_s^2+{\mathrm{\&gamma;}}^2\left|\right|d|{|}^2\&le;\frac{1}{{\mathrm{\&gamma;}}^2}|\left|s\right|{|}^2---\left(14\right)$

Wherein,