1.Introduction

A synchronous control of relative attitude and position among spacecraft is increasingly demanded,such as disturbance-free,distributed,and drag-free spacecraft.The disturbance-free spacecraft was proposed by the Lockheed Corporation.The spacecraft is divided into a Support Module(SM)and a Payload Module(PM).1,2The control,energy,and propulsion systems which produce disturbances and micro vibrations are installed in the SM,while ultra-quiet equipment is installed in the PM.The two modules are separated within a gap of several centimeters,and their relative attitude and position are required to be accurately controlled in the designed gap to avoid any possible collisions.Currently,this problem is still unsolved yet.Drag-free spacecraft are frequently investigated in ultra-quiet space applications,such as gravity field measurement3,4and gravitational-wave detection.5The relative attitude and position control between the inner mass and the spacecraft body is strict in orders of micro radii and micro meters,respectively.6–8The performance of an independent control of relative attitude and position is limited due to the coupling between the attitude and the position.To provide an accurate relative attitude and position,a synchronous control method is increasingly demanded.

A distributed satellite is another novel spacecraft.9–12The Defense Advanced Research Projects Agency(DARPA)proposed the F6 project,13in which different small distributed modules are designed to replace large satellites by using near-formation flying.Moreover,one module can be conveniently replaced if failed.To guarantee the performance of the F6 system,an accurate control of relative attitude and position among different modules is significant.As in various reports,the coupling effect is severe in a highly accurate control of both the attitude and the position.14,15To enhance the control accuracy of both relative attitude and position among different spacecraft and modules,this paper will investigate a novel synchronous sliding mode control approach in which the coupling effectissolved using thetwistor technology.

Traditionally,a spacecraft’s position and attitude are manipulated by using vector algebra and Euler angles(or quaternions and Rodrigues parameters),respectively.16–18When the attitude and position need to be modeled simultaneously,the equations describing the attitude and position are written as synchronous equations.19–21Various algebra systems will increase the complexity of the modeling problem.Zhang and Duan22designed a six-degree-of-freedom control system based on the adaptive back-stepping control law to satisfy the relative attitude and position requirements simultaneously.Pukdeboon23developed an inverse optimal sliding mode control for spacecraft under external disturbances.In traditional modeling methods,synchronous equations are simply set as a group.The attitude and position parts are manipulated separately while designing control algorithms.Conversely,this paper will propose a twistor-based sliding mode control,in which the attitude and position are manipulated synchronously to deal with their coupling effect.

Recently,dual-quaternion-based methods have received more and more attentions among spacecraft position and attitude dynamics modeling.A dual quaternion is efficient,nonsingular,and compact in computations.24,25Dual quaternions are thus used to simultaneously model the attitude and position dynamics with the same form as the quaternion based attitude dynamics.Moreover,a composite attitude and position control can be obtained by extending attitude-only controllers based on quaternions.26For the relative attitude and position control problem,various controllers based on dual-quaternion representation have been investigated.Most of the literature employs the coupled dynamic model using dual-quaternion representation proposed by Brodsky and Shoham.27The control algorithms for the composite attitude and position motion are obtained by the coupled dynamics.To solve the spacecraft attitude control problem,Wang et al.28designed a terminal sliding mode control law using a dual quaternion and gave an enhancement method.Filipe and Tsiotras26proposed a nonlinear adaptive attitude and position controller of satellite proximity operations and discussed the analogies between quaternions and dual quaternions.A related control development was also presented.29Similar to quaternions,there also exists the normalization constraint problem in dual quaternions,because a dual-quaternion-based model represents a six-degree-of-freedom motion using eight parameters.The constraint needs to be guaranteed by additional operations,thus possibly causing inconvenience of computing.Additionally,most of the current investigations of the dual quaternion-based controller of a relative attitude and position motion need to redesign the control law,so a synchronous controller directly using the attitude and position remains unsolved.

Besides the widely used modeling methods,new methods are also investigated in modeling and control of the relative attitude and position motion.30–32The main objective is to derive general space motion equations from an attitude motion model and represent the coupled position and attitude model under the same mathematical framework.Dorst et al.33introduced a six-degree-of-freedom twistor representation and proposed a basic definition and kinematics equation using the twistor representation which was regarded as a complementary representation with respect to a dual quaternion.Twistors not only provide a uniform representation for a combined spacecraft rotational and translational motion,but also avoid the normalization constraint because of the dual quaternion representation.To solve the estimation problem of a combined translational and rotational motion,Deng et al.34presented a twistor-based unscented Kalman filter for a highly nonlinear observation model,and simulation results demonstrated that the proposed estimation method could achieve a higher estimation accuracy than that using the dual-quaternion-based extended Kalman filter.By using twistors,attitude and position controllers can also be designed as a single control law with desirable properties similar to those of the MRP-based attitude control law.Deng and Wang35established a coupled relative motion model and presented a PD control strategy based on twistors.It was demonstrated that the proposed PD controller was not robust to disturbances.The twistorbased PD controller design was implemented based on the Modified Rodrigues Parameter(MRP)-based attitude controller,but the stability could not be proven directly by extending the stability proof of the attitude controller.

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To solve the control problem of a combined spacecraft attitude and position relative motion as well as to guarantee the performance robustness of a system while considering mass uncertainty and external disturbances,a twistor-based sliding mode control method is investigated in this paper.The proposed twistor-based sliding mode controller is robust to mass uncertainty and external disturbances.The twistor-based sliding mode controller design and the proving of stability are implemented by modifying the MRP-based attitude control approach.More details about the MRP-based attitude control approach have been investigated by Crassidis and Markley.36

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This paper is organized as follows.Section 2 presents the problem statement.Section 3 proposes the twistor-based synchronous sliding mode control strategy as well as the stability analysis of the closed-loop system.Section 4 discusses simulation studies.Section 5 concludes this paper.

2.Problem statement

The purpose of this paper is to develop a twistor-based synchronous sliding mode control method and apply to the attitude and position control design as a unified control algorithm.The proposed control method is different from traditional methods in which attitude and position controllers are separately designed.

and the closed-loop system equation in the sliding phase is described as

A relative motion equation using twistors is given by our previous work as

The closed-loop system equation in the sliding phase is described as

whereis the twistor of frame X with respect to frame Y,andis the dual vector ofis the dual velocity of frame X with respect to frame Y expressed in frame Z,anddenotes the dual quaternion form ofis the force motor.is the dual mass operator.Subscript ‘d”denotes the disturbance force/torque,‘g” denotes the gravitational force/torque,and ‘u” denotes the control force/torque.

The definition ofandare given as

where f,τ are the external force and the external moment of the spacecraft,respectively.m is the mass and J is the inertia matrix of the spacecraft.I is identity matrix and ε is dual unit.

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OIXIYIZIis the inertial reference frame,OBXBYBZBis the body- fixed reference frame,and ODXDYDZDis the desired reference frame.Subscript BD is omitted in the following sections.

The twistor-based synchronous sliding mode controller design and the stability proof are implemented based on an attitude controller using MRPs representation.The details of the MRP-based attitude controller are presented in the appendix section.

3.Twistor-based synchronous sliding mode control

3.1.Controller design

Using the relative motion equations based on twistors in Eq.(1),this section designs an attitude controller based on twistors using the sliding mode method.Basic twistor operational rules rely on the operations of dual quaternions,which are related to quaternions.Therefore,we rewrite MRP-based equations using quaternion representation before expanding MRP-based attitude kinematics equations into twistor-based kinematics equations.An overbar is used to convert a threedimensional vector into a vector quaternion with a scalar quantity equal to zero.

Similar to Eq.(A2)in the Appendix A,considering p and ωBas quaternionsandwith a scalar quantity equal to zero,the sliding surface can be obtained as

In the same way,the sliding reaching law of the MRP-based attitude controller can be rewritten as

where K1andK2are 4×4 matrices related to K1and K2in Eqs.(A2)and(A3),respectively,as follows:

The symbolic function sgn of vector r=[x，y，z]can be defined as

Replacing the MRP and angular velocity in Eqs.(2)and(3)with a twistor and a dual velocity,respectively,the sliding surface and sliding reaching law based on twistors can be given by

whereandare constant dual operators.The operatoris defined as the vector part of the quaternion.is unit dual operators.The sgn function of the dual numbercan be defined as

where arand adare the real and dual parts of,respectively.Considering Eq.(1)and the time derivative of Eq.(6)and ignoring the dual disturbance forcethe control law can be derived as

The closed-loop equation of the sliding phase is also given by Eq.(15).We just need to prove the stability in the reaching phase while considering the mass uncertainty and external disturbances.The Lyapunov function in the reaching phase is still chosen as

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Then,the control law can be rewritten as wherecan be calculated by the twistor-based kinematics equations in Eq.(1).The coefficient dual operatorsandcan be designed by

Consider the gravitation J2perturbation,the gravity gradient torque,and the solar radiation pressure perturbation,which are regarded as external force and torque disturbances.

According to the control law in Eq.(11)and the relative motion equation in Eq.(1),the closed-loop system equation is derived as

The closed-loop system equation ofin the reaching phase is described as

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where

The relevant equations about the twistor-based controller are expressed in a dual form.To make the analysis easier,the closed-loop system equation is divided into a real part and a dual part.Then,the sliding surface can be rewritten as

where

is the dual part of the twistor and v is the dual part of the velocity motor.

Then,the sliding reaching law is rewritten as

Then,the closed-loop system equation in the reaching phase can be rewritten as

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3.2.Stability proof

Firstly,the stability proof of twistor-based sliding control is presented without considering the mass uncertainty and external disturbances.In the section,the stability of twistor-based synchronous sliding control is proven directly by expanding the stability proof of MRP-based sliding control.

To construct the Lyapunov function with a dual number,the following operations are defined:

According to Eq.(A5),the Lyapunov function in the reaching phase is chosen as

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Furthermore,

Then,the closed-loop equation of the reaching phase is given by

V1=0 holds only ifwhich is the equilibrium point of the closed-loop system in the reaching phase,and note thatdholds only ifUsing the LaSalle invariance principle,the closed-loop system is globally asymptotically stable at the equilibrium point

According to Eq.(A8),the Lyapunov function in the sliding phase is chosen as

Eq.(24)is rewritten as

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Considering Eqs.(13)and(15),the time derivative of V2can be calculated as V2=0 holds only ifwhich is the equilibrium point of the closed-loop system in the reaching phase,and note thatholds only ifand in that case,onlyare the solution of the system.Using the LaSalle invariance principle,the closed-loop system is globally asymptotically stable at the equilibrium pointin the sliding phase.Then,it is concluded that the motion of the closed-loop system in Eq.(13)is globally asymptotically stable while using the twistor-based synchronous sliding controller.

In the following section,we will prove the stability of the proposed twistor-based sliding controller while considering the mass uncertainty and external disturbances.

The mass parameterincludes a nominal partand an uncertain partas follows:

Using the relative motion equations in Eq.(1),the twistorbased dynamic equation can be rewritten as

whereis the nominal part of the dynamic equation,whileis the uncertain part and the augmented items caused by external disturbances.

Only the nominal part of the mass parameter is assumed to be known in the control law.Then,is given as

is given as

The closed-loop equations of the system can be derived as

Considering Eq.(14),the time derivative of V1can be calculated as

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According to the derivative operational rule of a dual quaternion,the time derivative ofcan be expressed as

Considering Eq.(32),the time derivative of V1can be calculated as

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is guaranteed only if the value of k2is large enough whenis bounded.V1=0 holds only if,which is the equilibrium point of the closed-loop system in the reaching phase,and note thatholds only if.Using the LaSalle invariance principle,the closed-loop system is globally asymptotically stable at the equilibrium pointthe reaching phase.Based on the analysis above,it is concluded that the closed-loop system in Eq.(31)is globally asymptotically stable while using the twistor-based synchronous sliding controller.

4.Simulation and discussion

In this section,a case study of separated satellite control is presented to illustrate the effectiveness of the proposed twistorbased synchronous sliding controller.

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4.1.Simulation condition

The separated satellite system includes a payload module and a support module which are mechanically de-coupled.The two modules fly in close formation and interact through six or more non-contact actuators.The payload module is controlled by using non-contact actuators which generate control force and torque by reacting on the support module.The support module is controlled to follow the payload module using external actuators.In addition,simulation results using PD control are also presented for comparison.

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The control task of the case study is to keep the states of the relative attitude and position between the body- fixed frame and the desired frame,converging to zero,in which the support and payload modules remain in the designed gap to avoid possible collision.

The origin of the desired frame is set in the Geostationary Earth Orbit(GEO).The desired attitude is coincident with the local orbit frame,the XDaxis protrudes geocentrically outward to the desired position,the ZDaxis is coincident with the direction of the angular momentum of the desired orbit,and the YDaxis is established by the right-hand rule.The coordinate definition and the separate satellite system con figuration are illustrated in Fig.1.

The Euler angle representation is used in the control of the support module.Considering the influence of flexible solar panels,the attitude equation of the support module and the vibration equation of solar panels can be obtained as

where Jmis the total moment of inertia of the support module,ω is the angular velocity vector,η is the flexible modal coordinates of the solar panels,and δ is the coupling matrix.C and K are the damping and stiffness matrices of the solar panels,respectively.Tdand Tuare the disturbance and control torques,respectively.Tpis the reaction moment of the payload module to the support module.

The inertia matrix of the support module is set to37

The coupling matrix δ between the support module and the flexible solar panels is set to37

The payload module is controlled using the proposed twistor-based sliding mode controller.The relative motion equation using twistors is given by Eq.(1).The nominal value of the uncertain mass and the inertia matrix of the payload module are set to

Fig.1 Coordinate definition and system con figuration.

Fig.2 Simulation results of the PD controller without disturbances.

where m0=85 kg,Jxx=85 kg·m2,Jyy=90 kg·m2,Jzz=95 kg·m2,Jxy=Jyx=0.85 kg·m2,Jxz=Jzx=0.9 kg·m2,and Jyz=Jzy=0.95 kg·m2.

The mass of the payload module and the inertia matrix with mass uncertainty are set to mp=m0(1+wmΔm)

where wm，wx，wy，wzare set to 0.01,and in simulation,

where k1and k2are positive real numbers.

The gravitation J2perturbation can be calculated as38

where m is the mass of the payload module,μ is the Earth’s gravitational constant,Reis the radius of the Earth,R is the distance between the payload module and the center of the earth,and x,y,and z are the coordinate components of the payload module under the inertial frame,respectively.

The gravity gradient torque can be calculated as39

where JBis the inertia tensor of the payload module to the center of mass,R is the radius vector of the payload module to the center of earth,and R1is the adjacent matrix of R.

The solar radiation pressure perturbation force and torque can be calculated as39

where

where I is the solar constant,c is the velocity of light,ψ is the incidence angle of solar radiation,and ρ，κ are the reflection and dissipation coefficients,respectively.A is the surface area of the payload module.d is the vector of dA relative to the center of mass.

The initial relative states including the attitude quaternion,angular velocity,position,and velocity are set to

Fig.3 Simulation results of the synchronous sliding mode controller without disturbances.

Table 1 Regulation time.

Relative motion PD controller(s)Sliding mode controller(s)Relative attitude(≤0.02°) 35.0 54.5 Relative position(≤2×10-4m) 33.2 47.3 Relative angular velocity(≤0.06(°)/s)43.7 61.4 Relative velocity(≤1×10-4m/s) 30.4 31.0

The spacecraft attitude and position with respect to the desired position are all located in the ODXDYDplane.

The three-axis control force and torque are constrained by

Fig.4 Simulation results of the PD controller with disturbances.

Furthermore,in order to minimize possible chattering in the control torques caused by the sgn function,a saturation function is used to replace the sign function,which is defined by

where σ is a small positive quantity,which is set to 0.1.

4.2.Simulation results

The control coefficients of the PD controller by using nominal spacecraft parameters are regulated to kp=3/200，

Table 2 Steady-state errors.

Relative motion PD controller Sliding mode controller Relative attitude(°) 4.8 × 10-37.2 × 10-5Relative position(m) 4.7×10-37.0×10-5Relative angular velocity((°)/s) 5.6 × 10-48.6 × 10-6Relative velocity(m/s) 1.1×10-51.6×10-7

Fig.5 Simulation results of the synchronous sliding mode controller with disturbances.

Firstly,a simulation of the nominal system is carried out without considering the mass uncertainty and external disturbances.Simulation results of the PD controller are presented in Fig.2.The three line types in the figure represent the components of the state variables on three axes respectively.The support module is stabilized under the disturbances from the payload module and the environment.The response of the support module is not provided in this paper.

The coefficients of the twistor-based sliding mode controller by using nominal spacecraft parameters are set to kα=0.1，kβ=1.

Fig.3 presents the responses of the relative attitude and position in the ODXDYDplane while using the proposed twistor-based sliding mode controller.

The control performance is evaluated using the regulation time of relative motion.Comparisons between the simulation results are given in Table 1.

Remark.The simulation results of a nominal system show that the PD controller is regulated to exhibit faster convergent rates than the sliding mode controller.Since the disturbances are not considered,the steady-state values of the simulation results in Figs.2 and 3 are equal to zero.In the following,the disturbance rejection performance will be investigated.

Considering the mass uncertainty and external disturbances,a simulation analysis is carried out with the controller parameters remaining the same vales.Simulation results of the PD controller and the proposed twistor-based sliding mode controller are presented in Figs.4 and 5,respectively.

For the PD controller,the regulation time of the relative attitude is 35.4 s,and the regulation time of the relative position is 15.4 s.For the proposed sliding mode controller,the regulation time of the relative attitude is 54.6 s,and the regulation time of the relative position is 50.1 s.

The comparisons of the steady-state errors are given in Table 2.

4.3.Discussion

The simulation results in Figs.2 and 3 demonstrate that the control performance of the PD controller is better than that of the sliding mode controller without considering the mass uncertainty and external disturbances.The regulation times of the relative motion states are smaller than those of the sliding mode controller.However,while considering the mass uncertainty and external disturbances,the numerical simulation results in Figs.4 and 5 demonstrate that the proposed twistor-based sliding mode control method can ensure smaller steady state errors of relative attitude and position compared to those of the PD controller.The relative attitude error of the proposed twistor-based sliding mode controller is reduced to 1.5%of that of the PD controller,and the relative position error is reduced to 1.49%of that of the PD controller.

The attitude and position stability of the proposed twistorbased sliding mode controller is enhanced by more than 98%compared to that of the PD controller.The attitude stability is improved by more than one order compared to that of the PD controller,and the position stability is also improved by more than one order compared to that of the PD controller.

The proposed twistor-based sliding mode controller gives both ultra-precision and ultra-stability in relative attitude and position.The relative attitude and position of the spacecraft are manipulated synchronously.The simulation results indicate that the proposed twistor-based controller is bene ficial to enhance both the precision and stability of the attitude and position simultaneously.

5.Conclusions

To provide ultra-precision and ultra-stability of both a spacecraft’s attitude and position,a twistor-based approach is presented toward synchronous control of the spacecraft’s relative attitude and position.Using a synchronous design method,the attitude and position are manipulated synchronously.The simulation results indicate the effectiveness of the proposed twistor-based controller.The proposed synchronous control design of relative attitude and position shows significant enhancement of disturbance rejection.The steady error of the proposed twistor-based sliding mode controller is reduced to 1.5%of that of the PD controller.

The proposed synchronous twistor-based controller gives ultra-precision and ultra-stability of both the relative attitude and position,which is bene ficial to drag-free spacecraft,disturbance-free spacecraft,and other separated spacecraft.

Acknowledgements

This work was supported by the National Natural Science Foundation ofChina (Nos.51675430,11402044,and U1537213).

Appendix A.MRP-based sliding mode control

In this section,a brief introduction of the classic MRP-based attitude controller using the sliding mode variable structure method is presented.The nonlinear model of spacecraft motion is given by

where p is modified Rodrigues parameters,ω is angular velocity,∏is matrix function of p,J is inertia matrix of the spacecraft.

The sliding surface is designed as

and the sliding reaching law is designed as

where K1and K2are the positive de finite coefficient matrices.The control law can be written as

The closed-loop motion equation in the reaching phase is given by Eq.(A3).The Lyapunov function in the reaching phase is chosen as

The time derivative of V1can be calculated as

V1=0 holds only if s=0,which is the equilibrium point of the closed-loop motion system in the reaching phase,and note thatholds only ifAccording to the LaSalle invariance principle,the closed-loop system is globally asymptotically stable at the equilibriumin the reaching phase.

The equation of the closed-loop motion system in the sliding phase is s=0,and then ωBis given by

The Lyapunov function in the sliding phase is chosen as

The time derivative of V2can be calculated as

V2=0 holds only if p=0,which is the equilibrium point of the closed-loop motion system in the reaching phase,and note thatholds only if p=0,and in that case,only p=0，ωB=0 are the solution of the system.According to the LaSalle invariance principle,the closedloop system is globally asymptotically stable at the equilibrium point p=0，ωB=0 in the sliding phase.Then we can conclude that the closed-loop system is globally asymptotically stable while using the MPR-based sliding controller.

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