Research Center of Satellite Technology,Harbin Institute of Technology,Harbin 150080,China

Modi fi ed super twisting controller for servicing to uncontrolled spacecraft

Binglong Chen and YunhaiGeng*

Research Center of Satellite Technology,Harbin Institute of Technology,Harbin 150080,China

A relative position and attitude coupled sliding mode controller is proposed by combining the standard super twisting (ST)controland basic linearalgorithm forautonomous rendezvous and docking.Itis schemed for on-orbitservicing to a tumbling noncooperative target spacecraft subjected to external disturbances. A coupled dynamic modelis established including both kinematical and dynamic coupled effect of relative rotation on relative translation,which illustrates the relative movement between the docking port located in target spacecraft and another in service spacecraft.The modi fi ed super twisting(MST)control algorithm containing linear compensation items is schemed to manipulate the relative position and attitude synchronously.The correction provides more robustness and convergence velocity for dealing with linearly growing perturbations than the ST control algorithm. Moreover,the stability characteristic ofclosed-loop system is analyzed by Lyapunov method.Numerical simulations are adopted to verify the analysis with the comparison between MST and ST control algorithms.Simulation results demonstrate that the proposed MST controller is characterized by high precision,strong robustness and fast convergence velocity to attenuate the linearly increasing perturbations.

autonomous rendezvous and docking,coupled dynamic model,modi fi ed super twisting,Lyapunov method.

1.Introduction

The ability to perform routine autonomousrendezvous and docking(ARD)is needed in future space missions including assembly of internationalspace station(ISS),autonomous deployment,manipulation and repair[1].The collision probability increases with the decreasing distance between two spacecraft,especially docking with a tumbling non-cooperative targetspacecraft[2,3].Therefore,it is important to establish the full dynamics for ARD and controllers are designed to guarantee the reliability and success rate.Numbers of control strategies have been adopted for either orbital maneuvering or attitude tracking,such as adaptive control[4,5],optimal control[6,7] and sliding modelcontrol[8,9].

In prophase researches,models for relative translation and relative rotation are established separately,which restrictdevelopments ofits applications.Pan and Kapila[10] addressed a nonlineartracking controlproblem with adaptive feedback control to deal with unknown mass and inertia matrix of spacecraft.They took into accountthe dynamic coupled effectcaused by the gravity gradienttorque on relative translation and the globalasymptoticalstability of tracking errors is proved by the Lyapunov framework. However,the relative translation model is on the basis of point-mass model and the controller is proposed for the open-loop system.Shay and Pinisolved the errors resulted from the point-mass modelin distributed spacecraftformation fl ying[11],and developed a kinematicalcoupled relative translation modelbetween any arbitrary feature points on spacecraft.They consider the kinematical coupled effectofrelative rotation on relative translation derived from relative angular velocity,but they neglect the kinematic coupled effect caused by absolute angular velocity of the leader spacecraft and another effect introduced by disturbances.Environmental disturbance torque is inevitable existence,and therefore we consider both kinematicaland dynamical coupled effects of relative rotation on relative translation.

As is known to all,sliding mode control(SMC)is used widely because ofthe fi nite time convergence property and robustness for system uncertainties.Its capability to suppress disturbances is independent of dynamic model instead ofmodeling with uncertain states as the system function in robustcontrol[12,13].However,the standard SMC is based on 1-sliding mode[14]and itinduces controlchattering phenomenon caused by high frequency switching of control.Therefore,high order sliding mode(HOSM)technique is invented to eliminate the chattering phenomenon [15]by acting on the higher order time derivatives of thesystem deviation from the constraint.Consequently,there are increasing information demands in implementation of HOSMand the arbitrary order sliding mode controllaw is mostly stilltheoretically studied.However,2-sliding mode (SOSM)algorithms,such as twisting and super twisting, have already been used successfully in realproblems[16]. Super twisting sliding mode(ST)is one of widely used SOSM control algorithms[17,18],which can suppress bounded disturbances and does notneed to use the derivative ofthe switching function.In contrastto the linearalgorithm,the main disadvantage of SOSM algorithm is thatit cannotendure the linearly growing perturbation.However, the linear algorithm is not able to support strong disturbance near the equilibrium point,which is one of advantages of the SOSMalgorithm[19].

Inspired by the aforementioned issues,a modi fi ed super twisting sliding mode(MST)controlalgorithm is proposed by adding linear correction terms to the basic ST to obtain both excellent properties of them.It is applied in this study to design a relative position and attitude coupled SOSMcontroller.We take accountof the bounded linearly increasing perturbations,the limited disturbance torques, modeluncertainties and the actuatoroutputsaturation.The paper is organized as follows.In Section 2,a coupled dynamic model is established between the docking port located in targetspacecraftwith respectto anotherin service spacecraft including both kinematical and dynamic coupled effects of relative rotation on relative translation[20]. In Section 3,ST and MST are schemed to generate control operation ofservice spacecraftformaking ARDwith target spacecraftand the second method of Lyapunov is used to analyze the stability characteristic of the closed-loop system.In Section 4,numericalsimulations are performed to verify the performance of MST by comparing with basic ST method for ARD withoutcollision.Finally,the conclusions are represented in Section 5.

2.Coupled dynamic relative model

A coupled relative motion model is derived from the traditional point-mass model for relative motion between center-of-masses(CMs)of the target spacecraft and the service spacecraft.We take into consideration the kinematical coupled effect caused by relative attitude angular velocity,relative attitude quaternion and absolute attitude angular velocity of the service spacecraftand the dynamic coupled effectderived from externaldisturbance torques.

2.1 Coordinate systems

We de fi ne some coordinate systems to illustrate the relative motion between the two docking ports,so that the origins of coupled effects are distinct.The usefulcoordinate systems are shown in Fig.1.

Fig.1 Coordinate systems

The earth-centered inertial coordinate system(Fi): OXiYiZiis fi xed in an inertial space.It is a right-handed system with the origin atthe earth center O.Xiaxis points the vernal equinox direction,Ziaxis is along the North Pole and Yiaxis completes the setup to yield a Cartesian righthand system.

Euler-Hillreference frame(Fo):OsXoYoZois fi xed to the CM of the service spacecraft with the origin Os.Xoaxis is directing from the radially outward,Zoaxis is normal to the orbital plane,Yoaxis is pointing to the velocity direction of the service spacecraft in the orbital plane and perpendicular with OsXo.This frame is used to describe the attitude of the service spacecraftand the relative motion of the targetspacecraft with respect to the service spacecraft.

Orbit coordinate system of the target spacecraft(Ft): OtXtYtZtis fi xed to the CMof the targetspacecraftwith the origin Ot.Ztaxis is pointing to the earth center O,Ytaxis is along the opposite direction of orbit angular rate and Xtis along the velocity direction of the target spacecraftcompleting a righthand system.This frame is used to describe the attitude of the targetspacecraft.

Body coordinate system(Fb):It is a Cartesian righthand reference frame fi xed on the spacecraftand originates atthe spacecraft’s CM.The body coordinate systems ofthe service spacecraftand the targetspacecraftare denoted Fbsand Fbtrespectively.Itis assumed that Fbsand Fbtcoincide with Foand Ftseparately atthe initialtime.

Therefore,euler angles and attitude angular velocities of the service spacecraft and the target spacecraft are defi ned respectively by relative rotationalmotion of Fbwith respectto Foand Ft.Then de fi nition of absolute attitude angular velocities,severally noted byωbsandωbt,are rotational velocity of Fbsand Fbtrelative to Fi.Similarly, relative angular velocityωris rotational velocity of Fbtwith respectto Fbs.Therefor,ωrcan be expressed as

Then,the attitude can be parameterized by quaternion:

where =[q1,q2,q3]is the vector part and q4is thescalar part.It is subjected to the constraint that=1.qsand qtare denotations for attitude of the service spacecraftand the targetspacecraft.qrdenotes relative quaternion of Fbtwith respect to Fbs.Then,the rotation matrix can be expressed as

where[·×]denotes the cross productmatrix.

2.2 Relative rotation

Let a be an arbitrary vector measured with respect to the origin of Fiand˙a|Fdenotes time derivative of a measured in the reference frame F and(a)Fis the expression in F. According to(1),ωbtcan be rewritten in Fbtas follows:

Due to the angularmomentum theorem,the time derivative of(4)can be rewritten as

where J is the inertia matrix and T consists of controltorque Tc,gravity gradient torque Tgand disturbance torque Tdas follows:

whereω?is the magnitude of the orbit angular velocity, μis the earth gravitationalconstant,r is the magnitude of the radius vector from the CM of spacecraftto the earth’s center and Z0is the unit radius vector of r.Then,time derivative of the quaternion kinematical equation can be expressed as

Then the relative rotational model can be obtained by substituting(5)into(6).Express the vectorpartand itprovides us with the following expression:

and components of?2can be limited by

where i=1,2,3;λmax(·)is the maximum eigenvalue of a matrix and max(·)represents the maximum elementofa vector.

2.3 Relative translation

Consider two docking ports located in the target spacecraftand the service spacecraftseparately,as illustrated in Fig.2.denotes a vectordirected from the origin of Fbtto the docking portandis directed from the origin of Fbsto the docking portBy observing,we can obtain the relative position vectorρijas

where rt,rsare CM position vectors of the target spacecraft and the service spacecraft respectively.Then the second-order derivative ofρijwith respect to time in Focan be calculated as

whereω?is the orbitalangular velocity,ascis the control acceleration ofthe service spacecraft,asd=(1+kdt)asd0, atd=(1+kdt)atd0where kdis the increasing rate.

Fig.2 Relative translation of docking ports

Moreover,the relative translational model can be expressed as

where

3.Controllaw design

where i=x,y,z.

Let us de fi ne the desired control objective xd=,the state error e=x-xdand˙e=˙x-˙xd.

The controlgoalis to enforce the sliding mode on the manifold s=˙e+λe.

3.1 Standard ST design

According to the inequalities(9)and(13),we can suppose that the i th component ofδis bounded by positive constants expressingThus,the ST controlleris designed as follows:

where Mj(j=1,2)is the main diagonalmatrix with diagonal elements Mji＞0.The function sign(s)1/2is defi ned as follows:

Therefore,the system can converge to zero in fi nite time when the inequalities are satis fi ed.

3.2 MST controller design

In this section,MST is designed based on standard ST algorithm and basic linear algorithm.These modi fi cation allows the controlsystem to have both exponentialand fi nite time convergence properties.According to the components of?,itcan be divided into two parts as follows:

and they are assumed to be bounded by some positive constantsδji(j=1,2,...,4)as follows:

whereδ1i=|ζ1i|max,δ2i=|ζ2i|max,δ3i＞0,δ4i＞0; x1iis the i th componentof switching function x1de fi ned in the following section.The proposed MST controller is designed as follows:

and Kj(j=1,...,4)is the main diagonal matrix with kji＞0 as the diagonalelements.

Let us de fi ne vector x=[x1,x2]where x1=s,Thus the dynamic functions of the closed-loop system can be expressed as

In what follows,the proof of fi nite time convergence to equilibrium point in MST law is given by the second method of Lyapunov.

3.3 MST stability analysis

The Lyapunov function forsystem(21)is de fi ned with perturbations as follows:

Note that Viis continuous but it is not differentiable at x1i=0 and it is positive de fi nite and radially unbounded if kji＞0.

whereλmin(P)andλmax(P)denote the minimum and maximum eigenvalues of the matrix P.Then the derivative of Viwith respectto time can be expressed as

can be established if x1ix2i＞0,where

Thus,(24)can be rewritten as˙Vi

Then(24)can be rewritten as

Thus,we can deriveV˙＜0 if kjiare appropriately chosen to make the matrices Qjibe positive de fi nite.Under this condition,system(22)is the globalasymptotic stability and has fi nite time convergence property.When Q1iis positive de fi nite,the every sequentialprincipalminorof Q1ihas a positive determinant.It can be represented by the formula as follows:

Thus we can getthe following solutions

Itis noted thatthe fi rstand second orderprincipalminor determinants of Q3iare the same as Q1i’s.Thus we only need to guarantee the determinant of Q3iis positive.It is noticeable that|Q3i|＞0 can be consequentially satis fi ed as long as|Q1i|＞0.

Next,we calculate the conditions to make sure that Q2iis positive de fi nite.By using the same method mentioned above,we getthe following conditions

Therefore,the solutions can be expressed as

Finally,we note thatthe fi rstand second order principal minor determinants of Q4iare the same as Q2i.Then we need only to guarantee|Q4i|＞0 is satis fi ed.By the same method above,we get

As mentioned above,the matrices Qji(j= 1,...,4;i=1,...,6)are positive de fi nite when(26),(28) and(29)are satis fi ed.Thus the origin x=012×1is an equilibrium point that is strongly globally asymptotically stable.

According to(23),we can obtain

whereλmin(Qji)is represented as the minimum eigenvalue of Qji.Consequently,derivative of Viwith respect to time can be rewritten as

Since the solution of differentialequation

is given as follows:

Thus we can obtain that Viconverges to zero in fi nite time and reaches zero atmostafter max(Tfi)units oftime.

4.Simulation results

4.1 Parameters initialization

The classical orbit elements of the target spacecraft and the service spacecraft are listed in Table 1.ωtb(ωsb)denotes the angular velocity of Fbt(Fbs)with respect to Ft(Fo).The initialization values of them are separately ωsb0=03×1(?)/s andωtb0=[-3.0 2.0 3.0](?)/s.The mass characteristics are in Table 2.The numerical simulations are performed with linearly increasing disturbance accelerations of atd=(1+kdt)[2.5 4.0 3.8]T×10?5m/s2and asd=(1+kdt)[2.0 4.2 3.8]T×10?5m/s2.

Table 1 Classicalorbit elements

Table 2 Mass characteristics ofthe service spacecraftand the target spacecraft

The docking port vectors on the service spacecraft and the targetspacecraftare expressed in respective body coordinate systems as follows:

The control parameters of standard ST and MST algorithms are designed separatelyBipropellantorbitand attitude controlengines are chosen to operate relative position and attitude maneuverssynchronously.Outputlimitations oforbitand attitude control engines are amax=0.2 m/s2and Tmax=0.8 N·m.The control object is xd=˙xd=06×1.In simulations,measurements ofrelative position,relative velocity and relative attitude angular velocity are assumed to be given by state estimator and so measure errors are ignored.

4.2 Results

When kd=0,the relative translation of docking port P0t

with respectto P0sis shown in Fig.3.Through comparing the two sliding mode control methods,illustrations show that the convergence time is approximate 30 s in MST while the standard ST needs more than 35 s to converge. The relative translationalcontrolaccuracy of the two controllers are both calculated with data from 65 s to 100 s. The controlaccuracies are expressed as errors in 3σ(σis the standard deviation).The relative position errors and relative velocity errors are listed in Table 3.As a result,the relative position errors are less than 2.4×10?6m(3σ)and relative velocity errors are less than 5.4×10?4m/s(3σ) in MST technique.By contrast,we can easily conclude that the MST algorithm has almost the same control effects withoutdisturbances butMST has fasterconvergence time than ST method.

Fig.3 Relative translation of docking ports in Fbs

Table 3 Relative translation errors in 3σ

Meanwhile,the relative rotation of the targetspacecraft with respect to the service spacecraftare shown in Fig.4. The control errors of relative attitude angles and relative angular velocity are listed in Table 4 by the same method as previously.Thusthe relative attitude angle errorsare less than 4.6×10?6(?)(3σ)and relative angularvelocity errors are less than 5.1×10?4(?)/s(3σ)in MST algorithm.It is shown clearly again thatthe controlaccuracies of MST are higher than ST.

The relative distance between CMs ofthe service spacecraft and the target spacecraft and that of the two docking ports are illustrated in Fig.5 by designed MST and ST algorithms respectively.The symbol|ρij|indicates relative distance of docking port

and distance between permanentCMs of the targetspacecraftand the service spacecraftis denoted by|ρ00|.The controlerrors of relative distance between the two docking ports are 2.038×10?6m(3σ)by MST method and 2.098×10?6m (3σ)in ST controller.It shows visibly that the MST controller has distinctadvantages in terms ofless convergence time and higher control precision than the basic ST algorithm.

Table 4 Relative rotation errors in 3σ

Fig.4 Relative rotation of T with respect to the service spacecraft

Fig.5 Relative distance of docking ports

Furthermore,the control forces and torques outputted by the orbitand attitude controlengines with saturation limits in MST and ST are illustrated in Fig.6.Meanwhile, the sliding surfaces are described in Fig.7,where Sx,Syand Szare relative position sliding surface components; Sq1,Sq2and Sq3are vector components of the relative quaternion.They presentobviously again thatthe convergence time with MST algorithm is less than thatused in ST method.

In order to verify the robustness of MST for linearly increasing disturbances,we increase kdseveral times and the control accuracies of relative translation are listed in Table 5 by comparison with ST controllerin the same other simulation conditions and the same precision calculation method.Simulation results indicate obviously thatthe proposed MST algorithm has stronger robustness for linearly increasing perturbations than ST method and the previous analytic analysis can be proofed accordingly.

Then we increase the inertia matrix by±15%to verify the robustness of MST for modeling uncertainty and inertial matrix parameters uncertainties.Emulation programs are performed again with kd=1.0 and the same other parameters as before and the control accuracies are shown in Table 6.The results show that the proximity operation can be performed with almost the same magnitude of precision as the previous simulation.The analysis of control accuracy can be obtained as follows.The precision of relative position is 3.983×10?6m(3σ),relative velocity accuracy is 5.354×10?4m/s(3σ)and control error of relative distance between two docking ports is 4.088×10?6m (3σ).Moreover,the accuracy of relative attitude angle is 9.790×10?6(?)(3σ)and the relative angular velocity erroris less than 8.6×10?4(?)/s(3σ).As a consequence,the strong robustness and high reliability of MST are demonstrated,which can guaranty the ARD process is collisionsfree.

Fig.6 Actuators outputs in Fbs

Fig.7 Sliding surfaces

Table 5 Controlprecisions in 3σof relative translation under linearly increasing disturbances

Table 6 Controlaccuracy with modeluncertainties in 3σ

5.Conclusions

In this paper,a coupled relative motion model was established for the docking port located in target spacecraft and another in service spacecraft with the coupled effects of relative rotation on relative translation.The considered coupling effects belong to both kinematical and dynamic coupled effects.On the basis of this dynamic model,a modi fi ed super twisting sliding mode controller with linear correction terms was proposed to operate the relative position and attitude synchronously for on-orbit servicing to a tumbling non-cooperative target spacecraft subjected to some disturbances.Furthermore,by using the second method of Lyapunov,the fi nite time convergence property of the closed-loop system was proved.Numerical simulations were presented to validate the previous analysis by contrast with the standard super twisting algorithm.Simulation results illustrated that the revised super twisting controllerhas highercontrolprecision,strongerrobustness and faster convergence velocity for linearly increasing perturbations and mode uncertainties than the basic one.

[1]A.Flores-Abad,O.Ma,K.Pham,et al.A review of space robotics technologies for on-orbit servicing.Progress in Aerospace Sciences,2014,68:1-26.

[2]K.Miller,J.Masciarelli,R.Rohrschneider.Advances in multimission autonomous rendezvous and docking and relative navigation capabilities.Proc.of the IEEE Aerospace Conference,2012:1-9.

[3]S.D’Amico,J.S.Ardaens,G.Gaias,et al.Noncooperative rendezvous using angles-only optical navigation:system design and fl ightresults.Journal of Guidance,Control,and Dynamics,2013,36(6):1576-1595.

[4]V.I.Utkin,A.S.Poznyak.Adaptive sliding mode control with application to super-twist algorithm:equivalent control method.Automatica,2013,49(1):39-47.

[5]X.G.Dong,X.B.Cao,J.X.Zhang,et al.A robust adaptive control law for satellite formation fl ying.Acta Automatica Sinica,2013,39(2):132-141.

[6]D.W.Gao,J.J.Luo,W.H.Ma,etal.Nonlinearoptimalcontrolof spacecraft approaching and tracking a non-cooperative maneuvering object.Journal of Astronautics,2013,34(6): 773-781.

[7]X.Y.Gao,K.L.Teo,G.R.Duan.An optimal control approach to robust control of nonlinear spacecraft rendezvous system withθ-D technique.International Journal Innovative Computing,Information and Control,2013,9(5):2099-2110.

[8]C.Pukdeboon.Finite-time second-order sliding mode controllers for spacecraft attitude tracking.Mathematical Problems in Engineering,2013,Article ID 930269.

[9]S.Janardhanan,M.Un Nabi,P.M.Tiwari.Attitude control ofmagnetic actuated spacecraftusing super-twisting algorithm with nonlinear sliding surface.Proc.of the 12th International Workshop on Variable Structure Systems,2012:46-51.

[10]H.Z.Pan,V.Kapila.Adaptive nonlinear control for spacecraft formation fl ying with coupled translational and attitude dynamics.Proc.ofthe IEEE Conference on Decision and Control,2001:2057-2062.

[11]S.Shay,G.Pini.Effect of kinematic rotation-translation coupling on relative spacecraft translationaldynamics.Journal of Guidance,Control,and Dynamics,2009,32(3):1045-1050.

[12]A.Capua,A.Shapiro,D.Choukroun.Robust nonlinear H∞output-feedback for spacecraft attitude control.Proc.of the AIAA Guidance,Navigation,and Control Conference-SciTech Forum and Exposition,2014:AIAA-0455.

[13]A.M.Zou,K.D.Kumar.Robust attitude coordination control for spacecraft formation fl ying under actuator failures.Journalof Guidance,Control,and Dynamics,2012,35(4):1247-1255.

[14]A.Levant.Sliding order and sliding accuracy in sliding mode control.International Journal of Control,1993,58(6):1247-1263.

[15]W.Perruquetti,J.P.Barbot.Sliding mode control in engineering.New York:Marcel Dekker,2002.

[16]A.Levant.Principles of 2-sliding mode design.Automatica, 2007,43(4):576-586.

[17]V.I.Utkin.About second order sliding mode control,relative degree,fi nite-time convergence and disturbance rejection. Proc.ofthe 11th InternationalWorkshop on Variable Structure Systems,2010:528-533.

[18]C.Pukdeboon.Second-order sliding mode controllers for spacecraftrelative translation.Applied MathematicalSciences, 2012,6(100):4965-4979.

[19]J.A.Moreno,M.Osorio.A Lyapunov approach to secondorder sliding mode controllers and observers.Proc.of the 47th IEEE Conference on Decision and Control,2008:2856-2861.

[20]B.L.Chen,Y.H.Geng.Relative motion coupled dynamic modeling between two docking ports.Systems Engineering and Electronics,2014,36(4):714-720.(in Chinese)

Biographies

Binglong Chen was born in 1984.He received his B.E.and M.E.degrees in the Schoolof Astronautics,Harbin Institute of Technology,in 2008 and 2010,respectively.He is now a Ph.D.student in the Research Center of Satellite Technology,Harbin Institute of Technology.His research interests are spacecraft navigation and spacecraft attitude and orbitcontrolmethods.

E-mail:chenbinglonghit@163.com

Yunhai Geng was born in 1970.He received his B.E.degree in engineering mechanics from Tongji University,M.E.and Ph.D.degrees in spacecraftdesign from Harbin Institute of Technology,in 1992, 1995 and 2003 respectively.Currently,he is a professor and doctoral advisor of spacecraft design, Harbin Institute of Technology.He has published about 60 journal and conference papers.His research interests include spacecraft attitude and orbit control,and spacecraftdynamics navigation guidance and controltechnology.

E-mail:gengyh@hit.edu.cn

10.1109/JSEE.2015.00039

Manuscriptreceived April02,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61104026).