CN106647786A - Fractional order differential-based staged space tethered capture device attitude stabilization control method - Google Patents

Abstract

The present invention relates to a fractional order differential-based staged space tethered capture device attitude stabilization control method. According to the method, the attitude stabilization controller of the recovery stage of a space tethered capture device is designed in a staged manner according to the features of a tether recovery stage, the design of the controller is more fuel-efficient and can avoid the collision of a space platform and a tail end capture device at the later period of the recovery stage; the fractional order differential controller is faster in input response and is small in overshoot compared with an integer-order controller; and when fractional order differential calculation is performed, a state at a previous time point is memorized, and therefore, with the controller provided by the invention adopted, a tether can be recovered more stably at the later period of the recovery stage with a recovery speed and a capture device attitude angle ensured.

Description

Using the system of the space rope stage by stage catching device pose stabilization control method of fractional order differential

Technical field

The invention belongs to Spacecraft Control technical field of research, is related to a kind of space rope stage by stage of utilization fractional order differential It is catching device pose stabilization control method

Background technology

Space rope system catching device is that a kind of structure is " space platform+connection tether+flexible tether+autonomous motorised units " New spacecraft, the features such as it has flexible, safe, can be used for the space tasks such as track garbage-cleaning, inert satellite capture, behaviour Tens meters are can reach to several kms, there is extensive effect in On-orbit servicing application as scope.

According to the flow of task of space rope system catching device, its task can be divided into following five stages：Discharge, approach mesh Mark, target winding arrest, arrest after it is stable and reclaim.Wherein recovery stage is one of research emphasis of space rope system catching device.

Because space rope system catching device dynamics is complicated, its control is the difficult problem of a comparison.Control common at present Method mostly is the control method for coordinating using tether pulling force with thruster, due to fast due to reclaiming at recovery stage initial stage and later stage Degree difference is larger, and demand for control is different, therefore it is steady to design a space rope system catching device recovery stage attitude stage by stage Determine controller very necessary.

The content of the invention

The technical problem to be solved

In order to avoid the deficiencies in the prior art part, the present invention proposes a kind of space rope stage by stage of utilization fractional order differential It is catching device pose stabilization control method, solves the problems, such as space rope system catching device recovery stage attitude stabilization, is space rope system The research of catching device is laid a good foundation.

Technical scheme

A kind of system of the space rope stage by stage catching device pose stabilization control method of utilization fractional order differential, it is characterised in that step It is rapid as follows：

Step 1, set up space rope system catching device recovery stage kinetic model：

Wherein：

Wherein：K=1,2,3,4, work as k=1, q=2,3,4；Work as k=2, q=1,3,；4 work as k=3, q=1,2,；4 work as k= 4, q=1,2,；3 Λ, lw be nondimensionalization after connection tether length and flexible strand length, θ,Respectively space platform is bowed Face upward, roll angle,Respectively space platform pitching, angular velocity in roll, α, β respectively connect tether face interior angle, face exterior angle,Angular speed outside spatial tether face interior angle, face, α_k、β_k(k=1,2,3,4) are respectively flexible strand face interior angle, face exterior angle,Respectively flexible strand face interior angle, angular speed, I outside face_ozzFor space platform around z-axis rotary inertia, Q_θ,Q_ψQ_α,Q_βQ_l,The respectively control moment of the space platform angle of pitch, the control moment of roll angle, platform and end Connect control moment, the control moment of face interior angle, control moment, the end of connection tether length of tether face interior angle between catching device The control moment of catching device flexible strand face interior angle, the control moment at end catching device flexible strand face exterior angle, μ is that terrestrial gravitation is normal Number, R is distance of the earth's core to system barycenter, and Ω is orbit angular velocity, m₀,m_t, m be respectively with space platform quality, be connected tether The quality of quality, end catching device

The PD control device of step 2, design recovery stage based on tether recovery rate：

Wherein, k_pΛ,k_dΛ,k_cΛRespectively controller regulation parameter,And Λ^*,Point Wei not expect to connect tether length, expect connection tether length rate of change；

Step 3, design recovery stage are based on tether recovery rate and the PD of tether face interior angle^μController：

Wherein, D^λy₂It is the fractional order differential of new definition, is defined as：

Wherein, λ is real number, and n is the integer of a determination, and Γ () is defined asΓ equations；

Wherein, k_pα,k_dα,k_cα；k_pΛ,k_dΛ,k_cΛ；Respectively controller adjusts ginseng Number,And And α^*,Respectively expect connection tether face interior angle, expect angular speed in connection tether face, expect end Flexible strand face interior angle, expect end flexible strand face in angular speed, expect end flexible strand face exterior angle, expect end flexible strand face outside Angular speed；

Step 4：At the recovery stage initial stage, it is controlled with the PD control device based on tether recovery rate；Returning the rank stage Later stage and when connecting tether length and being less than total length 10%, with the PD based on tether recovery rate and tether face interior angle^μController It is controlled.

Beneficial effect

A kind of system of the space rope stage by stage catching device pose stabilization control method of utilization fractional order differential proposed by the present invention, According to the characteristics of tether recovery stage, the staged design pose stabilization control device of space rope system catching device recovery stage, institute The controller of design more saves fuel, and can avoid reclaiming the collision of later stage space platform and end catching device；Secondly, Using fractional order differential controller compared to integer rank controller to the response that is input into faster and overshoot is little；Finally, due to Fractional order differential has memory to previous moment state when calculating, therefore controller proposed by the invention can in the recovery stage later stage In the case where ensureing to reclaim speed and catching device attitude angle, more smoothly to reclaim tether.

The present invention is the new space flight that a kind of structure is " space platform+connection tether+flexible tether+autonomous motorised units " Device, the features such as it has flexible, safe, can be used for the space tasks such as track garbage-cleaning, inert satellite capture, and opereating specification can Tens meters are reached to several kms, there is extensive effect in On-orbit servicing application.

Description of the drawings

Fig. 1 is the structural representation of space rope system catching device；Wherein 1 is space platform, and 2 are connection tether, and 3 are end rope It is catching device.

Specific embodiment

In conjunction with embodiment, accompanying drawing, the invention will be further described：

The technical solution adopted in the present invention is comprised the following steps：

1) space rope system catching device recovery stage kinetic model is set up；

2) PD control device of the recovery stage based on tether recovery rate is designed；

3) recovery stage is designed based on tether recovery rate and the PD of tether face interior angle^μController.

Described step 1) in, space rope system catching device recovery stage kinetics equation is：

Wherein, in formula (6), (7), k=1,2,3,4.Work as k=1, q=2,3,4；Work as k=2, q=1,3,4；Work as k=3, q =1,2,4；Work as k=4, q=1,2,3.Λ, lw be nondimensionalization after connection tether length and flexible strand length, θ,Respectively For space platform pitching, roll angle,Respectively space platform pitching, angular velocity in roll, α, β respectively connect tether face Interior angle, face exterior angle,Angular speed outside spatial tether face interior angle, face, α_k、β_k(k=1,2,3,4) it is respectively in flexible strand face Angle, face exterior angle,Respectively flexible strand face interior angle, angular speed, I outside face_ozzIt is space platform around z-axis Rotary inertia, Q_θ,Q_ψQ_α,Q_βQ_l,The respectively control moment of the space platform angle of pitch, control moment of roll angle, flat Connect control moment, the control moment of face interior angle, the control of connection tether length of tether face interior angle between platform and end catching device Torque, the control moment of end catching device flexible strand face interior angle, the control moment at end catching device flexible strand face exterior angle, μ is ground Ball gravitational constant, R is distance of the earth's core to system barycenter, and Ω is orbit angular velocity, m₀,m_t, m be respectively with space platform quality, Connection tether quality, the quality of end catching device.

Described step 2) in, PD control device design of the recovery stage based on tether recovery rate is as follows：

Hypothesis below is made first：First, in recovery stage, space platform is considered as into attitude and keeps constant；Second, reference Lot of documents learns that face exterior angle amplitude of fluctuation is much smaller than face interior angle, therefore ignores connection tether face exterior angle only to all face interior angles And flexible strand face exterior angle is controlled.

Due at the recovery stage initial stage, reclaiming speed, and the attitude of end catching device need not be maintained at expectation Attitude, therefore only PD control device of the design based on tether recovery rate.

PD control device design based on tether recovery rate is as follows：

Wherein, k_pΛ,k_dΛ,k_cΛRespectively controller regulation parameter,And Λ^*,Point Wei not expect to connect tether length, expect connection tether length rate of change.

Described step 3) in, recovery stage is based on tether recovery rate and tether face interior angle, the PD at face exterior angle^λController Design process is as follows：

The later stage in rank stage is being returned, that is, is connecting (such as 20m) when tether length is less than a certain numerical value, to avoid end rope system from arresting Device is collided with space platform, need to reduce recovery speed, and needs the attitude for ensureing end catching device to be maintained at expectation attitude, because This need to design more smoothly controller.

Recovery stage is based on tether recovery rate and tether face interior angle, the PD at face exterior angle^λController design is as follows：

Wherein, D^λy₂It is the fractional order differential of new definition.It is defined as follows：

Wherein, λ is real number, and n is the integer of a determination, and Γ () is defined asΓ equations.

Wherein, k_pα,k_dα,k_cα；k_pΛ,k_dΛ,k_cΛ；Respectively controller adjusts ginseng Number, e_α=α^*-α,AndRespectively expect connection tether face interior angle, expect angular speed in connection tether face, expect that end is soft Property rope face interior angle, expect angular speed in the flexible strand face of end, expect end flexible strand face exterior angle, expect end flexible strand face exterior angle Speed.

Claims (1)

1. a kind of system of the space rope stage by stage catching device pose stabilization control method of utilization fractional order differential, it is characterised in that step It is as follows：

Step 1, set up space rope system catching device recovery stage kinetic model：

$\begin{array}{c}{\mathrm{\θ}}^{\′\′}\[M_A{\cos }^2\mathrm{\ψ}+Ioxx\·{\sin }^2\mathrm{\ψ}+Iozz\·{\cos }^2\mathrm{\ψ}\]+{\mathrm{\α}}^{\′\′}{M}_D\mathrm{\Λ}Lr\cos \mathrm{\ψ}\cos \mathrm{\β}\\ +\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[{{\mathrm{\α}}_k}^{\′\′}{M}_E{\mathrm{lwcos\ψcos\β}}_k\]\\ -{\mathrm{\ψ}}^{\′}\left[\begin{array}{c}2M_A\left(1+{\mathrm{\θ}}^{\′}\right)\cos \mathrm{\ψ}\sin \mathrm{\ψ}+M_D\mathrm{\Λ}Lr\left(1+{\mathrm{\α}}^{\′}\right)\sin \mathrm{\ψ}\cos \mathrm{\β}\\ +\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{E}lw\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{sin\ψcos\β}}_k\]+2Ioxx\left(1+{\mathrm{\θ}}^{\′}\right)\sin \mathrm{\ψ}\cos \mathrm{\ψ}-2Ioxx\left(1+{\mathrm{\θ}}^{\′}\right)\sin \mathrm{\ψ}\cos \mathrm{\ψ}\end{array}\right]\\ -{\mathrm{\β}}^{\′}{M}_D\mathrm{\Λ}Lr\left(1+{\mathrm{\α}}^{\′}\right)\cos \mathrm{\ψ}\sin \mathrm{\β}-\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[{{\mathrm{\β}}_k}^{\′}{M}_{E}lw\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{cos\ψsin\β}}_k\]\\ +{\mathrm{\Λ}}^{\′}{M}_{D}Lr\left(1+{\mathrm{\α}}^{\′}\right)\cos \mathrm{\ψ}\sin \mathrm{\β}+3M_A{\cos }^2\mathrm{\ψ}\cos \mathrm{\θ}\sin \mathrm{\θ}\\ +M_D\mathrm{\Λ}Lr\left(2\sin \mathrm{\θ}\cos \mathrm{\ψ}\cos \mathrm{\α}\cos \mathrm{\β}+\cos \mathrm{\θ}\cos \mathrm{\ψ}\sin \mathrm{\α}\cos \mathrm{\β}\right)\\ +\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{E}lw\left(2{\mathrm{sin\θcos\ψcos\α}}_k{\mathrm{cos\β}}_k+{\mathrm{cos\θcos\ψsin\α}}_k{\mathrm{cos\β}}_k\right)\]\\ -Q_{\mathrm{\θ}}/{\mathrm{\Ω}}^2=0\end{array}$

$\begin{array}{c}{\mathrm{\ψ}}^{\′\′}\[M_A+I_{oyy}\]+{\mathrm{\β}}^{\′\′}{M}_D\mathrm{\Λ}Lr+\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[{\mathrm{\β}}_k^{\′\′}{M}_{E}lw\]+M_D{\mathrm{Lr\Λ}}^{\′}{\mathrm{\β}}^{\′}+M_A{\left(1+{\mathrm{\θ}}^{\′}\right)}^2\cos \mathrm{\ψ}\sin \mathrm{\ψ}\\ +M_D\mathrm{\Λ}Lr\left(1+{\mathrm{\θ}}^{\′}\right)\left(1+{\mathrm{\α}}^{\′}\right)\cos \mathrm{\β}\sin \mathrm{\ψ}+\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{E}lw\left(1+{\mathrm{\θ}}^{\′}\right)\left(1+{\mathrm{\α}}_k^{\′}\right){\mathrm{cos\β}}_k\sin \mathrm{\ψ}\]\\ +3M_A{\cos }^2\mathrm{\θ}\cos \mathrm{\ψ}\sin \mathrm{\ψ}+M_D\mathrm{\Λ}Lr\left(2\sin \mathrm{\ψ}\cos \mathrm{\θ}\cos \mathrm{\α}\cos \mathrm{\β}-\sin \mathrm{\ψ}\sin \mathrm{\θ}\sin \mathrm{\α}\cos \mathrm{\β}+\cos \mathrm{\ψ}\sin \mathrm{\β}\right)\\ +\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{E}lw\left(2{\mathrm{sin\ψcos\θcos\α}}_k{\mathrm{cos\β}}_k-{\mathrm{sin\ψsin\θsin\α}}_k{\mathrm{cos\β}}_k+{\mathrm{cos\ψsin\β}}_k\right)\]\\ -I_{oxx}{\left(1+{\mathrm{\θ}}^{\′}\right)}^2\sin \mathrm{\ψ}\cos \mathrm{\ψ}+I_{ozz}{\left(1+{\mathrm{\θ}}^{\′}\right)}^2\sin \mathrm{\ψ}\cos \mathrm{\ψ}\\ -Q_{\mathrm{\ψ}}/{\mathrm{\Ω}}^2=0\end{array}$

$\begin{array}{c}{\mathrm{\α}}^{\′\′}{M}_B{\cos }^2{\mathrm{\β\Λ}}^2{\mathrm{Lr}}^2+{\mathrm{\θ}}^{\′\′}{M}_D\mathrm{\Λ}Lr\cos \mathrm{\ψ}\cos \mathrm{\β}+\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[{{\mathrm{\α}}_k}^{\′\′}{M}_{F}lw\mathrm{\Λ}Lr{\mathrm{cos\βcos\β}}_k\]\\ -{\mathrm{\β}}^{\′}\[2M_B{\mathrm{\Λ}}^2{\mathrm{Lr}}^2\cos \mathrm{\β}\sin \mathrm{\β}\·\left(1+{\mathrm{\α}}^{\′}\right)+M_D\mathrm{\Λ}Lr\left(1+{\mathrm{\θ}}^{\′}\right)\cos \mathrm{\ψ}\sin \mathrm{\β}+\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{F}lw\mathrm{\Λ}Lr\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{sin\βcos\β}}_k\]\]\\ -{\mathrm{\ψ}}^{\′}{M}_D\mathrm{\Λ}Lr\left(1+{\mathrm{\θ}}^{\′}\right)\sin \mathrm{\ψ}\cos \mathrm{\β}-\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[{{\mathrm{\β}}_k}^{\′}{M}_{F}lw\mathrm{\Λ}Lr\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{cos\βsin\β}}_k\]\\ +{\mathrm{Lr\Λ}}^{\′}\[2M_B\mathrm{\Λ}Lr\left(1+{\mathrm{\α}}^{\′}\right){\cos }^2\mathrm{\β}+M_D\left(1+{\mathrm{\θ}}^{\′}\right)\cos \mathrm{\ψ}\cos \mathrm{\β}+\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{F}lw\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{cos\βcos\β}}_k\]\]\\ +3M_B{\mathrm{\Λ}}^2{\mathrm{Lr}}^2{\cos }^2\mathrm{\β}\cos \mathrm{\α}\sin \mathrm{\α}+M_D\mathrm{\Λ}Lr\left(2\sin \mathrm{\α}\cos \mathrm{\θ}\cos \mathrm{\ψ}\cos \mathrm{\β}+\cos \mathrm{\α}\sin \mathrm{\θ}\cos \mathrm{\ψ}\cos \mathrm{\β}\right)\\ +\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{F}lw\mathrm{\Λ}Lr\left(2{\mathrm{sin\αcos\βcos\α}}_k{\mathrm{cos\β}}_k+{\mathrm{cos\αcos\βsin\α}}_k{\mathrm{cos\β}}_k\right)\]\\ -Q_{\mathrm{\α}}/{\mathrm{\Ω}}^2=0\end{array}$

$\begin{array}{c}{\mathrm{\β}}^{\′\′}{M}_B{\mathrm{\Λ}}^2{\mathrm{Lr}}^2+{\mathrm{\ψ}}^{\′\′}{M}_D\mathrm{\Λ}Lr+\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[{{\mathrm{\β}}_k}^{\′\′}{M}_{F}lw\mathrm{\Λ}Lr\]+{\mathrm{\Λ}}^{\′}Lr\[M_D{\mathrm{\ψ}}^{\′}+\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_F{{\mathrm{lw\β}}_k}^{\′}\]+2M_B{\mathrm{\β}}^{\′}\mathrm{\Λ}Lr\]\\ +M_B{\mathrm{\Λ}}^2{\mathrm{Lr}}^2{\left(1+{\mathrm{\α}}^{\′}\right)}^2\cos \mathrm{\β}\sin \mathrm{\β}+M_D\mathrm{\Λ}Lr\left(1+{\mathrm{\θ}}^{\′}\right)\left(1+{\mathrm{\α}}^{\′}\right)\cos \mathrm{\ψ}\sin \mathrm{\β}\\ +\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{F}lw\mathrm{\Λ}Lr\left(1+{\mathrm{\α}}^{\′}\right)\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{cos\β}}_k\sin \mathrm{\β}\]+3M_B{\mathrm{\Λ}}^2{\mathrm{Lr}}^2{\cos }^2\mathrm{\α}\cos \mathrm{\β}\sin \mathrm{\β}\\ +M_D\mathrm{\Λ}Lr\left(2\sin \mathrm{\β}\cos \mathrm{\θ}\cos \mathrm{\ψ}\cos \mathrm{\α}-\sin \mathrm{\β}\sin \mathrm{\θ}\cos \mathrm{\ψ}\sin \mathrm{\α}+\sin \mathrm{\ψ}\cos \mathrm{\β}\right)\\ +\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{F}lw\mathrm{\Λ}Lr\left(2{\mathrm{sin\βcos\αcos\α}}_k{\mathrm{cos\β}}_k-{\mathrm{sin\βsin\αsin\α}}_k{\mathrm{cos\β}}_k+{\mathrm{cos\βsin\β}}_k\right)\]\\ -Q_{\mathrm{\β}}/{\mathrm{\Ω}}^2=0\end{array}$

$\begin{array}{c}{\mathrm{\Λ}}^{\′\′}{\mathrm{LrM}}_B-M_B\mathrm{\Λ}Lr\[{\left(1+{\mathrm{\α}}^{\′}\right)}^2{\cos }^2\mathrm{\β}+{\mathrm{\β}}^{\′\′}\]-M_D\[\left(1+{\mathrm{\θ}}^{\′}\right)\left(1+{\mathrm{\α}}^{\′}\right)\cos \mathrm{\ψ}\cos \mathrm{\β}+{\mathrm{\ψ}}^{\′}{\mathrm{\β}}^{\′}\]\\ -\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{F}lw\[\left(1+{\mathrm{\α}}^{\′}\right)\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{cos\βcos\β}}_k+{\mathrm{\β}}^{\′}{{\mathrm{\β}}_k}^{\′}\]\]-3M_B\mathrm{\Λ}Lr{\cos }^2{\mathrm{\βcos}}^2\mathrm{\α}\\ +M_D\[-2\cos \mathrm{\θ}\cos \mathrm{\ψ}\cos \mathrm{\α}\cos \mathrm{\β}+\sin \mathrm{\θ}\cos \mathrm{\ψ}\sin \mathrm{\α}\cos \mathrm{\β}+\sin \mathrm{\ψ}\sin \mathrm{\β}\]\\ +\underset{k=1}{\overset{n=4}{\mathrm{\Σ}}}\[M_{F}lw\[-2{\mathrm{cos\αcos\βcos\α}}_k{\mathrm{cos\β}}_k+{\mathrm{sin\αcos\βsin\α}}_k{\mathrm{cos\β}}_k+{\mathrm{sin\βsin\β}}_k\]\]\\ -Q_{\mathrm{\Λ}}/{\mathrm{\Ω}}^2=0\end{array}$

$\begin{array}{c}{{\mathrm{\α}}_k}^{\′\′}{M}_C{\mathrm{lw}}^2{\cos }^2{\mathrm{\β}}_k+{\mathrm{\θ}}^{\′\′}{M}_{E}lw{\mathrm{cos\ψcos\β}}_k+{\mathrm{\α}}^{\′\′}{M}_{F}lw\mathrm{\Λ}Lr{\mathrm{cos\βcos\β}}_k\\ +\underset{q}{\mathrm{\Σ}}\[M_G{\mathrm{lw}}^2{\mathrm{cos\β}}_k{\mathrm{cos\β}}_q\·{{\mathrm{\α}}_q}^{\′\′}\]\\ +{{\mathrm{\β}}_k}^{\′}\left[\begin{array}{c}{M}_C{\mathrm{lw}}^2\left(1+{{\mathrm{\α}}_k}^{\′}\right)2{\mathrm{cos\β}}_k\left(-{\mathrm{sin\β}}_k\right)+M_E\·lw\left(1+{\mathrm{\θ}}^{\′}\right)\cos \mathrm{\ψ}\left(-{\mathrm{sin\β}}_k\right)\\ +M_{F}lw\mathrm{\Λ}Lr\left(1+{\mathrm{\α}}^{\′}\right)\cos \mathrm{\β}\left(-{\mathrm{sin\β}}_k\right)+\underset{q}{\mathrm{\Σ}}\[M_G{\mathrm{lw}}^2\left(1+{{\mathrm{\α}}_q}^{\′}\right)\left(-{\mathrm{sin\β}}_k\right){\mathrm{cos\β}}_q\]\end{array}\right]\\ +{\mathrm{\ψ}}^{\′}{M}_{E}lw\left(1+{\mathrm{\θ}}^{\′}\right)\left(-\sin \mathrm{\ψ}\right){\mathrm{cos\β}}_k+{\mathrm{\Λ}}^{\′}{\mathrm{LrM}}_{F}lw\left(1+{\mathrm{\α}}^{\′}\right){\mathrm{cos\βcos\β}}_k\\ +{\mathrm{\β}}^{\′}{M}_{F}lw\mathrm{\Λ}Lr\left(1+{\mathrm{\α}}^{\′}\right)\left(-\sin \mathrm{\β}\right){\mathrm{cos\β}}_k+\underset{q}{\mathrm{\Σ}}\[M_G{\mathrm{lw}}^2\left(1+{{\mathrm{\α}}_q}^{\′}\right){\mathrm{cos\β}}_k\left(-{\mathrm{sin\β}}_q\right){{\mathrm{\β}}_q}^{\′}\]\\ +3M_C{\mathrm{lw}}^2{\cos }^2{\mathrm{\β}}_k{\mathrm{cos\α}}_k{\mathrm{sin\α}}_k+M_{E}lw\left(2{\mathrm{sin\α}}_k{\mathrm{cos\θcos\ψcos\β}}_k+{\mathrm{cos\α}}_k{\mathrm{sin\θcos\ψcos\β}}_k\right)\\ +M_{F}lw\mathrm{\Λ}Lr\left(2{\mathrm{sin\α}}_k{\mathrm{cos\αcos\βcos\β}}_k+{\mathrm{cos\α}}_k{\mathrm{sin\αcos\βcos\β}}_k\right)\\ +\underset{q}{\mathrm{\Σ}}\[M_G{\mathrm{lw}}^2\left(2{\mathrm{sin\α}}_k{\mathrm{cos\β}}_k{\mathrm{cos\α}}_q{\mathrm{cos\β}}_q+{\mathrm{cos\α}}_k{\mathrm{cos\β}}_k{\mathrm{sin\α}}_q{\mathrm{cos\β}}_q\right)\]\\ -Q_{{\mathrm{\α}}_k}/{\mathrm{\Ω}}^2=0\end{array}$

$\begin{array}{c}{{\mathrm{\β}}_k}^{\′\′}{M}_C{\mathrm{lw}}^2+{\mathrm{\ψ}}^{\′\′}{M}_{E}lw+{\mathrm{\β}}^{\′\′}{M}_{F}lw\mathrm{\Λ}Lr+\underset{q}{\mathrm{\Σ}}\[M_G{\mathrm{lw}}^2{{\mathrm{\β}}_q}^{\′\′}\]+M_C{\mathrm{lw}}^2{\left(1+{{\mathrm{\α}}_k}^{\′}\right)}^2{\mathrm{cos\β}}_k{\mathrm{sin\β}}_k\\ +M_{E}lw\left(1+{\mathrm{\θ}}^{\′}\right)\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{cos\ψsin\β}}_k+M_F{\mathrm{lw\Λ}}^{\′}{\mathrm{Lr\β}}^{\′}+M_{F}lw\mathrm{\Λ}Lr\left(1+{\mathrm{\α}}^{\′}\right)\left(1+{{\mathrm{\α}}_k}^{\′}\right){\mathrm{cos\βsin\β}}_k\\ +\underset{q}{\mathrm{\Σ}}\[M_G{\mathrm{lw}}^2\left(1+{{\mathrm{\α}}_k}^{\′}\right)\left(1+{{\mathrm{\α}}_q}^{\′}\right){\mathrm{cos\β}}_q{\mathrm{sin\β}}_k\]+3M_C{\mathrm{lw}}^2{\cos }^2{\mathrm{\α}}_k{\mathrm{cos\β}}_k{\mathrm{sin\β}}_k\\ +M_{E}lw\left(2{\mathrm{sin\β}}_k{\mathrm{cos\θcos\ψcos\α}}_k-{\mathrm{sin\β}}_k{\mathrm{sin\θcos\ψsin\α}}_k+{\mathrm{sin\ψcos\β}}_k\right)\\ +M_{F}lw\mathrm{\Λ}Lr\left(2{\mathrm{sin\β}}_k{\mathrm{cos\αcos\βcos\α}}_k-{\mathrm{sin\β}}_k{\mathrm{sin\αcos\βsin\α}}_k+{\mathrm{sin\βcos\β}}_k\right)\\ +\underset{q}{\mathrm{\Σ}}\[M_G{\mathrm{lw}}^2\left(2{\mathrm{sin\β}}_k{\mathrm{cos\α}}_k{\mathrm{cos\α}}_q{\mathrm{cos\β}}_q-{\mathrm{sin\β}}_k{\mathrm{sin\α}}_k{\mathrm{sin\α}}_q{\mathrm{cos\β}}_q+{\mathrm{cos\β}}_k{\mathrm{sin\β}}_q\right)\]\\ -Q_{{\mathrm{\β}}_k}/{\mathrm{\Ω}}^2=0\end{array}$

Wherein：

Wherein：K=1,2,3,4, work as k=1, q=2,3,4；Work as k=2, q=1,3；Work as k=3, q=1,2；Work as k=4, q=1, 2；Λ, lw be nondimensionalization after connection tether length and flexible strand length, θ,Respectively space platform pitching, roll angle,Respectively space platform pitching, angular velocity in roll, α, β respectively connect tether face interior angle, face exterior angle,Space is Angular speed outside rope face interior angle, face, α_k、β_k(k=1,2,3,4) are respectively flexible strand face interior angle, face exterior angle,Respectively flexible strand face interior angle, angular speed, I outside face_ozzFor space platform around z-axis rotary inertia,Respectively the control moment of the space platform angle of pitch, the control moment of roll angle, platform with end Connect control moment, the control moment of face interior angle, control moment, the end of connection tether length of tether face interior angle between the catching device of end Control moment, the control moment at end catching device flexible strand face exterior angle of end catching device flexible strand face interior angle, μ is that terrestrial gravitation is normal Number, R is distance of the earth's core to system barycenter, and Ω is orbit angular velocity, m₀,m_t, m be respectively with space platform quality, be connected tether The quality of quality, end catching device

The PD control device of step 2, design recovery stage based on tether recovery rate：

$Q_{\mathrm{\Λ}}=k_{p\mathrm{\Λ}}{e}_{\mathrm{\Λ}}+k_{d\mathrm{\Λ}}{\stackrel{\·}{e}}_{\mathrm{\Λ}}+k_{c\mathrm{\Λ}}$

Wherein, k_pΛ,k_dΛ,k_cΛRespectively controller regulation parameter, e_Λ=Λ^*-Λ,And Λ^*,Respectively Expect to connect tether length, expect connection tether length rate of change；

Step 3, design recovery stage are based on tether recovery rate and the PD of tether face interior angle^μController：

$\left\{\begin{array}{c}{Q}_{\mathrm{\α}}=k_{p\mathrm{\α}}{e}_{\mathrm{\α}}+k_{d\mathrm{\α}}{D}^{\mathrm{\λ}}{\stackrel{\·}{e}}_{\mathrm{\α}}+k_{c\mathrm{\α}}\\ Q_{\mathrm{\Λ}}=k_{p\mathrm{\Λ}}{e}_{\mathrm{\Λ}}+k_{d\mathrm{\Λ}}{D}^{\mathrm{\λ}}{\stackrel{\·}{e}}_{\mathrm{\Λ}}+k_{c\mathrm{\Λ}}\\ Q_{{\mathrm{\α}}_k}=k_{{\mathrm{p\α}}_k}{e}_{{\mathrm{\α}}_k}+k_{{\mathrm{d\α}}_k}{D}^{\mathrm{\λ}}{\stackrel{\·}{e}}_{{\mathrm{\α}}_k}+k_{{\mathrm{c\α}}_k}\\ Q_{{\mathrm{\β}}_k}=k_{{\mathrm{p\β}}_k}{e}_{{\mathrm{\β}}_k}+k_{{\mathrm{d\β}}_k}{D}^{\mathrm{\λ}}{\stackrel{\·}{e}}_{{\mathrm{\β}}_k}+k_{{\mathrm{c\β}}_k}\end{array}\right.$

Wherein, D^λy₂It is the fractional order differential of new definition, is defined as：

$D_t^{\mathrm{\&lambda;}}f\left(t\right)=\frac{1}{\mathrm{\&Gamma;}(n-\mathrm{\&lambda;})}{\&Integral;}_0^t\frac{f^{\left(n\right)}\left(\mathrm{\&tau;}\right)}{{(t-\mathrm{\&tau;})}^{\mathrm{\&lambda;}-n+1}}d\mathrm{\&tau;},n-1<\mathrm{\&lambda;}<n$

Wherein, λ is real number, and n is the integer of a determination, and Γ () is defined asΓ equations；

Wherein,Respectively controller adjusts ginseng Number, e_α=α^*-α,AndRespectively expect connection tether face interior angle, expect angular speed in connection tether face, expect that end is soft Property rope face interior angle, expect angular speed in the flexible strand face of end, expect end flexible strand face exterior angle, expect end flexible strand face exterior angle Speed；

Step 4：At the recovery stage initial stage, it is controlled with the PD control device based on tether recovery rate；Returning the later stage in rank stage And when connecting tether length less than total length 10%, with the PD based on tether recovery rate and tether face interior angle^μController is carried out Control.