Xin-hong Li ,Zhi-in Zhng ,*,Ji-ping An ,Xin Zhou ,Gng-xun Hu ,Guo-hui Zhng ,Wn-xin Mn

a Department of Aerospace Science and Technology,Space Engineering University,No.1 BaYi Road,HuaiRou District,Beijing,101416,China

b Beijing Institute of Remote Sensing Information,Beijing,100192,China

Keywords:Adaptive sliding mode control(ASMC)Time delay control Time delay estimation Modular self-reconfigurable spacecraft Uncertainty Coordinated control

ABSTRACT The reconstruction control of modular self-reconfigurable spacecraft(MSRS)is addressed using an adaptive sliding mode control(ASMC)scheme based on time-delay estimation(TDE)technology.In contrast to the ground,the base of the MSRS is floating when assembled in orbit,resulting in a strong dynamic coupling effect.A TED-based ASMC technique with exponential reaching law is designed to achieve high-precision coordinated control between the spacecraft base and the robotic arm.TDE technology is used by the controller to compensate for coupling terms and uncertainties,while ASMC can augment and improve TDE’s robustness.To suppress TDE errors and eliminate chattering,a new adaptive law is created to modify gain parameters online,ensuring quick dynamic response and high tracking accuracy.The Lyapunov approach shows that the tracking errors are uniformly ultimately bounded(UUB).Finally,the on-orbit assembly process of MSRS is simulated to validate the efficacy of the proposed control scheme.The simulation results show that the proposed control method can accurately complete the target module’s on-orbit assembly,with minimal perturbations to the spacecraft’s attitude.Meanwhile,it has a high level of robustness and can effectively eliminate chattering.

1.Introduction

Modular self-reconfigurable spacecraft(MSRS)is a new type of spacecraft system that can be maintained,repaired,and upgraded in orbit.It provides many advantages,including a low development cost,a short deployment time,and high system flexibility.The world’s space powers have put forward their own MSRS projects,such as Japan’s CellSat[1],HISat[2]of the United States,Germany’s IBOSS[3],and so on.This research group put forward the“Magic’s Cube”intelligent variable-structure spacecraft concept.On the one hand,the modules are linked by robotic arms,allowing the spacecraft to alter the configuration and enhance mission capability.On the other hand,the robotic arms can also be utilized to complete the on-orbit assembly and expansion of MSRS.

During the self-assembly phase of the spacecraft in orbit,the robotic arms must be precisely controlled to capture the target and complete the assembly task.However,the dynamic coupling between the arm and the spacecraft base,as well as external disturbances and load uncertainties,make coordinated control of the spacecraft base and the robotic arm more complex.To solve the above problems,researchers have proposed control methods such as sliding mode control(SMC)[4],adaptive control[5],fuzzy logic control[6],and neural network control[7,8].Robust control methods such as SMC suppress disturbances and uncertainties by outputting high control gains,but their high-frequency switching actions will lead to chattering.Some adaptive control methods can enhance SMC,but usually need to know the bounds of disturbances and uncertainties[9].Neural networks and fuzzy algorithms require a large amount of computation when estimating external disturbances and uncertainties.Although these schemes are theoretically feasible,they are challenging to implement in practice due to the requirements of system dynamics or sophisticated estimate procedures.

TDE technology provides a simple and efficient solution to the difficulties mentioned above.TDE is model-free,using previously observed information and control inputs to estimate the current unknown dynamics of continuous variable dynamic systems online,and is often used to eliminate complex dynamic uncertainties and external disturbances.Hisa and Gao[10,11]proposed an effective control method TDC for industrial manipulators based on TDE technology.TDC is simple and easy to implement,and has been successfully applied to a variety of scenarios,such as robots[12],underwater robots[13],and chaotic systems[14].However,due to the inherent measurement noise and limited sampling period in practical applications,the use of time-delay signals will inevitably result in estimation errors,known as TDE errors.When there are hard nonlinearities,such as saturation and coulomb friction,the TDE errors will increase significantly,leading to severe degradation of control performance.To compensate the TDE errors and achieve improved accuracy and robustness,TDE is usually combined with robust control strategies,such as adaptive SMC[15,16],terminal sliding mode control[17,18],etc.TDE is robust to uncertainties in system dynamics and does not require any prior knowledge of uncertainty bounds.Therefore,the control method based on TDE can guarantee high control performance.

Existing TDE-based adaptive sliding mode controllers are generally designed in three ways,the first being the adaptive design of robust terms[19-21],the second for the gain dynamics[22],and the third for the adaptive design of control gain[23,24].Baek proposed an ASMC scheme based on TDE technology,which adopted an adaptive law and achieved good tracking performance in the case of small jitter[19].However,the singularity will appear when the sliding variable crosses zero,and the tracking accuracy needs to be improved.Bae uses the fuzzy sliding mode control as an auxiliary control scheme,which has smaller chattering[20].The adaptive robust TDC(ARTDC)proposed by Roy does not require complete system knowledge or uncertainties and is robust to TDE errors.The evaluation of switching gains does not depend on any threshold,which alleviates the problem of over-estimation and under-estimation of switching gains[21].Lee proposed an adaptive integral sliding mode control(AISMC)with TDE,and the dynamic injection part uses adaptive gain dynamics to achieve applicable high tracking accuracy[22].

The above-mentioned TDE-based control schemes all use constant control gain,which will lead to a significant decline in control performance when time-varying complex disturbances occur.For space manipulators with a large load,the constant control gains may not meet the bounded condition of TDE errors.Baek proposed a practical adaptive ATDC scheme and designed adaptive rules for control gains to obtain good tracking performance[23].Wang proposed a new ATDC scheme,which uses the continuous chatter-less automatic setting algorithm to update the control gains.Through this adaptive mechanism,good control performance and effective noise suppression can be guaranteed at the same time[24].Nevertheless,it is difficult to find a fixed diagonal matrix for the control gainfor MSRS.To sum up,both the gain dynamics and the robustness term have a significant impact on the performance of the controller.The gain dynamics affect the steady-state performance of the system,and the robust term increases the robustness.However,only one of these has been examined in the available literature.

In this study,a new TDE-based ASMC is presented to improve tracking accuracy and reduce chattering.To increase the reaching speed,the sliding variable uses the exponential reaching law.The constant velocity reaching term is also employed as a robust term to compensate for TDE errors.The switching gains are updated by a new adaptive law,which can effectively reduce chattering and has no singularity problem.Adaptive gain dynamics is used to achieve high tracking accuracy and improve convergence performance.The gain dynamics is adjusted according to the thickness of the acceptance layer.The real-time estimation of the inertia matrix H is used as the control gain to ensure the boundedness of the TDE errors under heavy load.The main contributions of this paper are marked as follows:

(1)A TED-based ASMC scheme with exponential reaching law is proposed for the first time,which integrates the advantages of existing TDE control schemes and still has a fast convergence speed when sliding variables are small.

(2)The gain dynamics and switching gains can be adjusted adaptively at the same time to reduce chattering and improve tracking performance,and the new adaptive law does not have any singular problems.

(3)The Lyapunov method is used to prove that the tracking errors are UUB and have arbitrarily small bounds.

The remainder of the paper is laid out as follows.The dynamic equations of the MSRS are derived in section 2.Section 3 designs the sliding mode controller and adaptive sliding mode controller based on TDE technology and proves the stability of the system.Section 4 proves the effectiveness of the proposed method through comparative simulation.Conclusions are given in Section 5.

2.Dynamics modeling of MSRS

2.1.Modeling assumptions

Before introducing the dynamics model of the MSRS,the basic modeling assumptions are given as follows:(1)The entire system consists of rigid bodies.(2)The MSRS is weightless in space,meaning it is not affected by gravity.(3)The external disturbances and uncertainties acting on the system are bounded.

2.2.Notation definitions

As shown in Fig.1,the modular self-reconfigurable spacecraft BSANhas S modules and N arms.A module is selected as the virtual base of the MSRS,which is called the central module(denoted as B).The robotic arm k has nklinks and degrees of freedom,and the system has n1+n2+…+nN=nAlinks in total.The related symbols are defined in Table 1,and all vectors are represented in the inertial frame Σ

Table 1 Notation definitions.

Table 3 Inertial parameters of the MSRS.

I unless otherwise specified.

2.3.Dynamics modeling based on Lagrange method

The Lagrange method is used to create a dynamic model in this section[25].Select the generalized coordinates of the spacecraft system as,where,rBand θBare the position and attitude angle of the base,is the joint angle of the robotic arm.

Due to the weightless environment,the gravitational potential energy of the system is zero.The kinetic energy of Lkirelative to the inertial frame is

Fig.1.Schematic diagram of MSRS system.

Then the kinetic energy of the entire system is

The above formula can be expressed as

where H is the inertial matrix of the system;HBand HkMare the inertial matrices of base and Ak,respectively;HkBMis the coupling matrix between the base and Ak,the specific form are as follows:

where E is the identity matrix,is the antisymmetric matrix of r,and the remaining intermediate parameters are expressed as follows:

According to the Jacobian matrix of the end-effector of the robotic arm,the Jacobian matrix of link Lkican be obtained:

Substitute Eq.(5)into the Lagrange equation and obtain

where C is the generalized Coriolis and centrifugal force term of the system;CBand CkMare the Coriolis and centrifugal force terms that correspond to the base and Ak,respectively;τdstands for unknown external disturbances;τ is the generalized driving force and torque of the system,including the driving force and torque when the base is in attitude and orbit control,the driving force and torque when module v is in attitude and orbit control,and the joint driving torque of the Ak.

where JvBis the Jacobian matrix between the base and the module v,JvMkis the Jacobian matrix between Akand the module v.

When the spacecraft is self-reconfiguring on orbit,all joints except the service arm are locked,multiple modules form an equivalent unit,and the system degenerates into a single-arm space robot,as shown in Fig.2.The dynamics model can be simplified to the same classical form as[26].

To facilitate the design of the controller,the dynamic model of Eq.(19)can be rewritten as

Fig.2.Concept of MSRS.

3.New adaptive sliding mode controller design

3.1.SMC with exponential reaching law

The control objective in this study is to make the generalized coordinate q follow the reference coordinate qdprecisely,which means that the tracking error e=qd-q is suppressed as much as possible.To achieve such a control objective,we shall first define the following sliding variable

In order to improve the dynamic quality of reaching motion,the exponential reaching law(ERL)is used in sliding mode control[27,28].

where Ks=diag(Ks1，Ks2，…，Ksn)?Rn×nand G=diag(G1，G2，…，Gn)?Rn×nare positive diagonal gain matrices.Sign function sgns=[sgn(s1(t))，sgn(s2(t))，…，sgn(sn(t))]is defined as

The speed of the moving point approaching the switching plane is relatively slow in simple exponential approach,and it cannot be guaranteed to arrive in finite time.By adding a term of the equalvelocity reaching law,the velocity can be made non-zero when s approaches zero,which ensures the finite time arrival of the reaching motion[29].In exponential reaching law,to ensure fast approaching and restrain chattering at the same time,Ksshould be increased while G should be appropriately decreased.

According to Eq.(23),we can get

Substituting Eqs.(22)and(26)into(24),the sliding mode control with exponential reaching law can be obtained Since N contains unknown terms which cannot be obtained directly,an estimated valueis usually used instead of N.In other words,we have

Substituting Eq.(28)into Eq.(27)and omitting the time variable t,then the TDE-based SMC is obtained.

Substituting the control input Eq.(29)into(22),replacing the sliding variable s with Eq.(23),and rearranging terms yield

for all t≥0,the TDE errors are bounded by constant,i.e.||≤N*,then the stability of the system is guaranteed[30,31].Information such as the system's own nominal mass parameters and the maximum permitted payload is always available in practice,so it is always possible to obtain a matrixthat satisfies the condition Eq.(31).

In reality,due to the inherent measurement noise and limited sampling period,TDE errors cannot be completely avoided,which means that appropriate parameters should be selected to ensure the stability of the system.If the gain matrices KDand Ksare inappropriately small,the tracking performance will decrease.Conversely,if they become inappropriately high for fast response,it will not only cause system chattering but also may cause system instability.Nevertheless,it takes very time-consuming trial and error to select the appropriate gain matrices.From another point of view,if the fixed gain parameters can be self-tuned online,its performance will be greatly improved.

3.2.TDE-based ASMC

3.2.1.Controller design

To effectively suppress the TDE errors and ensure the system stability while reducing the chattering of TDE-based SMC,we propose a new ASMC based on Eq.(29),which improves the performance of the controller through the adaptive adjustment of the key parameters Ksand G.

The gain dynamics is updated by the following adaptive law:

where αiis the adjustable positive gain adapted to the speed,βiis the normalization factor related to the tracking accuracy,satisfying βi/βi,the gain dynamicsincreases,which reduces the TDE errors and tracking errors.If|si|

The switching gain is updated using the following adaptive law:

where φiis the adjustable positive gain that adapts to the speed,and ε is a small positive number,which determines the boundary of the increase or decrease of the switching gain.

The proposed adaptive law Eq.(34)does not require boundary information of uncertainties and is activated when the sliding variable s deviates from zero.To elaborate,for>0,the adaptive law has two different forms according to the output of the sign function:||s||∞≥ε and||s||∞<ε.When||s||∞≥ε,the switching gainincreases until||s||∞<ε.With the increase of switching gain,the sliding variable s approaches the sliding manifold more quickly.Once the sliding variable enters the vicinity of the sliding manifold,i.e.,||s||∞<ε,the switching gaindecreases while the sliding variable stays in the vicinity of the sliding manifold.The parameter ε plays a key role in the trade-off between tracking ability and chattering suppression.If ε is too small,the adaptive speed is slow and there is obvious chattering.On the contrary,if ε is too large,the tracking performance of the ASMC scheme is poor.

The above description shows that the proposed control scheme Eq.(32)does not require any dynamic parameters of the system,and the adaptive law has relatively high switching gain and relatively fast adaptive speed,which can provide better tracking performance and chattering suppression at the same time.The proposed TDE-based ASMC scheme can be described by block diagram,as shown in Fig.3.This controller can be viewed as an extended and enhanced version of the original TDC.If the gain matrixis constant,i.e.,αi=0,then the controller(32)degenerates into Baek’s formulation[19].If=0,the controller degenerates into Lee’s formulation[22].If bothandare constants,the controller degrades to Eq.(29).Ifis constant and=0,then the controller degrades to Hsia’s formulation[10,11],usually called TDC.

3.2.2.Stability analysis

Before showing the UUB property of the proposed control law(32),we introduce a Lemma that will be helpful in the Proof of the main results.

Lemma 1.For the system(22)controlled by(32),if the TDE errors are bounded,i.e.,

Proof.According to the definition,the range of parameter βiis βi/βi.

When|si|

The stability of the system(22)can be proved by the Lyapunov stability criterion.The range of the sliding variable siis assumed to be in?|si|>max(/βi，ε).Given the proposed method Eq.(32),Lyapunov function is defined as follows:

with time derivative as

Substituting the control input Eq.(32)into Eq.(13)yields

Substituting Eq.(38)into Eq.(37),and using adaptive laws Eqs.(33)and(34),we can get

Fig.3.Block diagram of the proposed TDE-based ASMC scheme.

The first time when the sliding variable s enters the region|si|≤max(/βi，ε),there is|si|≤max(N*/βi，ε).It can be proved that the Lyapunov function(36)is bounded

When the sliding variable s leaves the region|si|≤max(/βi，ε),becomes negative again and V decreases immediately.It follows then that we have quality information are detailed in Tables 2 and 3.It is assumed that the modules and links are homogeneous objects,so that their centroids are located at their geometric centers,which makes the data easy to be determined.

Table 2 DH parameters of 5-Dof robotic arm.

We suppose that the initial states of the spacecraft are all zero so that the initial position rB0=[0，0，0]Tm and the initial attitude θB0=[0，0，0]Trad.The initial joint angle of the arm is θM0=[10，20，60，35，-40]Tπ/180 rad.The target attitude of the spacecraft base is set as θBT=[0，0，0]Trad,while the target joint angle of the arm is set as θMT=[-25，65，50，65，-25]Tπ/180 rad.A fifthorder polynomial curve is used for joint trajectory planning and the terminal time is set as tf=20 s.The unknown external disturbances τd=[τdB，τdM]Tin Eq.(19)consist of the disturbances applied on the base of the MSRS τdB=[0.4sin(0.3t)，0.3cos(0.1t)，0.2sin(0.3t)]Tand applied on the robotic arm

.

Eq.(41)implies that the sliding variable s is UUB,and guarantees that the fluctuation of the sliding variable s in the vicinity of the sliding manifold is upper-bounded.

4.Simulation and results

To prove the performance of the proposed TDE-based ASMC,the on-orbit assembly process of the MSRS was simulated.The simulation object B9A1is composed of nine identical modules and a 5-DOF crawling robotic arm,as shown in Fig.2.The assembly process is that the robotic arm grabs the target module and moves to the specified position,while the spacecraft base is attitudecontrolled,namely rotation-floating[32].The DH parameters and

Fig.4.Trajectory planning and trajectory tracking curves.

Fig.5.Sliding variables generated by the proposed TDE-based ASMC.

Fig.6.Gain dynamics computed by the proposed TDE-based ASMC.

Fig.7.Switching gains computed by the proposed TDE-based ASMC.

To get more information,the scheme Eq.(32)proposed in this paper is compared with the fixed-gain TDE-based SMC in Eq.(29),the existing TDE-based ASMC[19],and the traditional TDC scheme[10].The parameters of the proposed controller are carefully tuned and detailed as follows: KD= diag(1，1，…，1),Ks=diag(10，10，10，30，30，30，30，30)G=5×10-5diag(1，1，1，20，20，20，20，20)α=105diag(20，20，20，1，1，1，1，1),β=5×106diag(20，20，20，1，1，1，1，1)(0)=diag(10，10，10，30，30，30，30，30), G(0)=5×10-5diag(1，1，1，1，1，1，1，1),φ=diag(1，1，1，1，1，1，1，1),ε=5×10-5.To ensure a fair comparison,apply the above parameters to the other controllers entirely.

To estimate the uncertainties and disturbances more accurately and reduce the TDE errors as much as possible,the sampling time is set as the simulation step L=dt=0.01 s.Assuming that the parameters of the MSRS are known,the estimated mass of the target module is=80 kg,and the estimated moment of inertia is=diag(12，12，12)kg?m2.The inertia matrixcan be obtained by Eqs.6-15.

The trajectory planning results and the tracking trajectories of the proposed controller are shown in Fig.4.It can be seen that TDE can accurately estimate the uncertainties and disturbances in the system,and the control accuracy is very high.

Fig.5 shows the change of the sliding variable s.The subscripts 1-3 correspond to the 3?of freedom of the base attitude,and 4-8 correspond to the 5?of freedom of the robot arm.Fig.6 shows the gain dynamics updated by the adaptive law Eq.(33).The gain dynamics increases gradually to speed up convergence,which is consistent with the conclusion in Eq.(39),that is,the greateris in the range of

Fig.7 shows the switching gains updated by the adaptive law Eq.(34).It can be seen from Figs.5 and 7 that when t<1.5 s,the errors are small,||s||∞<ε,and the switching gaindecreases,then||s||∞>ε,andincreases to compensate for the tracking errors.When t>9 s,the errors enter and maintain in the range of||s||∞<ε again,and the switching gain decreases to avoid unnecessary chattering.Compared with the adaptive law in Ref.[19],as shown in Fig.8,the parameters decline speed of the adaptive scheme proposed in this paper is more stable,especially when the sliding variable is small,it does not produce singularity.

Fig.8.Switching gains Ks computed by the existing ASMC scheme[19].

Figs.9 and 10 compare the tracking errors and control inputs of the four control schemes.To begin with,the proposed TDE-based ASMC scheme,TDE-based SMC scheme,and existing TDE-based ASMC scheme all have the potential to suppress TDE errors and outperform traditional TDC methods in error tracking,as shown in Fig.9.The proposed TDE-based ASMC method offers higher tracking accuracy than other controllers because it can achieve larger gain dynamics and switching gains while keeping low chattering.As illustrated in Fig.10,the traditional TDC scheme does not exhibit chattering due to the absence of a robust term;as a result,control accuracy and convergence speed are compromised.The TDE-based SMC scheme with fixed gain has an obvious chattering phenomenon,and both adaptive schemes have an obvious chattering suppression effect.In comparison to the fixed-gain control method,the proposed TDE-based ASMC method can adjust to parameter changes induced by spacecraft configuration modifications,and has superior tracking performance and less chattering.

Fig.9.The tracking errors of the MSRS:(a),(b)the proposed TDE-based ASMC;(c),(d)the TDE-based SMC;(e),(f)the existing TDE-based ASMC;(g),(h)conventional TDC.

The integral of time multiplied by the absolute value of the error(ITAE)and the integral of the square value(ISV)of the control input are applied to evaluate the control performance quantificationally.The ITAE is used as a numerical measure of tracking performance for the entire error curve,and the ISV shows the energy consumption[33].These are defined as follows:

The statistical results of the error indicators and control inputs of the four different controllers are shown in Tables 4 and 5,respectively.The TDE-based ASMC and the fixed-gain TDE-based SMC proposed in this paper have higher accuracy,and the former is more energy-efficient than the latter,which is the result of chattering reduction.The traditional TDC has no robust term and no chattering,so it saves more energy but has lower control accuracy.Simulation results verify the effectiveness of the proposed TDEbased ASMC scheme.

Table 4 ITAE of the tracking errors.

Table 5 ISV of the control inputs.

5.Conclusions

Fig.10.Comparison of the control inputs:(a),(b)the proposed TDE-based ASMC;(c),(d)the TDE-based SMC;(e),(f)the existing TDE-based ASMC;(g),(h)conventional TDC.

Aiming at the motion control problem of MSRS under uncertainties and external disturbances,a TDE-based ASMC method is proposed.A new gain dynamics adaptive law and switching gain adaptive law are designed,which can reduce chattering while enhancing convergence performance and tracking accuracy.The proposed TDE-based ASMC scheme does not require any upper bound information of uncertainties,and the tracking errors are uniformly ultimately bounded.The simulation results reveal that the proposed TDE-based ASMC system aggregates the advantages of traditional TDC and ASMC,resulting in reduced steady-state tracking errors,good tracking performance,and less chattering.

In the future,the free-flying MSRS control problem will be studied to precisely control the end trajectory of the robotic arm in Cartesian space.In addition,TDE technology will be used to design robust controllers under non-smooth and nonlinear conditions such as saturation and dead zones.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are grateful to the editor and all the anonymous reviewers for their valuable suggestions that helped to improve the quality of the manuscript.This study was supported by the National Defense Science and Technology Innovation Zone of China(Grant No.00205501).