Rong-Hua Lei,Li Chen

School of Mechanical Engineering and Automation,Fuzhou University,Fuzhou,350116,China

Keywords:Finite-time Terminal sliding mode Flexible links Vibration suppression Virtual control force

ABSTRACT The dynamic modeling,finite-time trajectory tracking control and vibration suppression of a flexible two-link space robot are studied.Firstly,the dynamic model of the system is established by combining Lagrange method with assumed mode method.In order to ensure that the base attitude and the joints of space robot can reach the desired positions within a limited time,a non-singular fast terminal sliding mode(NFTSM)controller is designed,which realizes the finite-time convergence of the trajectory tracking errors.Subsequently,for the sake of suppressing the vibrations of flexible links,a hybrid trajectory based on the concept of the virtual control force is developed,which can reflect the flexible modes and the trajectory tracking errors simultaneously.By modifying the original control scheme,a NFTSM hybrid controller is proposed.The hybrid control scheme can not only realized attitude stabilization and trajectory tracking of joints in finite time,but also provide a new method of vibration suppression.The simulation results verify the effectiveness of the designed hybrid control strategy.

1.Introduction

The space robots can save space launch costs and improve astronauts’work efficiency[1,2],since it can assist astronauts to complete a series of high-intensity and high-risk space missions,such as satellite orbit determination,rendezvous and docking of spacecraft and space science experiment[3-5],etc.Different from the ground robot with fixed-base,the base of the space robot is free-floating,i.e.the system is not affected by the external force or moment.Since the system meets the law of conservation of linear momentum,the motions of joints will interfere with the attitude stabilization of the base.

In order to realize the attitude stabilization and joint tracking control of the space robot,many control algorithms have been proposed.For the space robot with flexible joints,Yu[6]proposed a robust control method based on state observer.For the space tethered robot used to capture space debris,Zhang[7]designed an adaptive super-twisting sliding mode control algorithm.Kumar[8]presented an adaptive neural network controller for the space robot with uncertain inertia parameters.For the near-earth space robot considering microgravity effect,Qin[9]formulated an adaptive robust controller based on fuzzy logic system.However,these control schemes can only guarantee the asymptotic convergence of the trajectory tracking errors,i.e.the system states will converge to the desired states when the time tends to infinity.It is worth mentioning that some key space missions need to meet the real-time requirements strictly.For example,the spacecraft needs to adjust the deployment angle of solar array through attitude stabilization,so as to replenish electric energy in time.In order to obtain better dynamic response quality,it is of great engineering value and practical significance to investigate the finite-time control of the space robots.Du[10]studied the finite-time attitude stabilization of spacecraft,while the finite-time attitude synchronization stabilization of satellite formation flying are analyzed by Zhou[11].For a class of nonlinear systems,Feng[12]proposed a finite-time control method based on nonsingular terminal sliding mode.Polyakov[13]presented a fixed-time control strategy for the nonlinear systems by using the implicit Lyapunov function method.However,research on the finite-time control for the space robots is still rare.

For the purpose of improving the working space and dexterity of the space robot system,the arm of system is usually designed as an elongated structure,which can easily induce flexible vibration at the end of the manipulator[14,15].Therefore,in order to improve the motion stability of the space robot,the flexible vibration of the arm must be suppressed actively.Although the modeling methods of flexible link are mature,most of them are applied to the ground robot with fixed-base.Pereira[16]studied the integral resonance control of a flexible link robot.Meng[17]established the discrete model of the flexible link robot by the use of the assumed mode method,and designed a fast vibration suppression controller through the system energy theory and fuzzy genetic algorithm.But the above researches are all aimed at the flexible single-link robot,and the similar research on the flexible two-link counterpart is rarely reported.It should be noted that the vibrations of flexible two-link space robot is stronger than that of flexible single-link space robot,since the vibrations of the adjacent flexible links are coupled with each other.In literature[18],a flexible-link space robot system is decomposed into a rigid part and a flexible part by using singular perturbation method,and then a trajectory tracking sub-controller and a vibration suppression sub-controller are designed for them respectively.And the same control scheme can also be applied to a redundant parallel flexible-link robot in Ref.[19].Because the sub-controller depends on the corresponding sub-model,the structure of the final composite controller is complex.

With an effort to address the aforementioned several drawbacks,this work investigates the dynamic modeling,finite-time trajectory tracking control and vibration suppression of a flexible two-link space robot through only one control input.In order to ensure that the output trajectories of space robot can follow the desired positions within a limited time,a NFTSM controller is proposed.For the sake of abating the vibrations of flexible links,the concept of virtual control force is introduced so as to generate a hybrid trajectory which can describe the flexible modes and the trajectory tracking errors simultaneously.By updating the original control scheme,a NFTSM hybrid controller is formulated.The hybrid control scheme can not only achieve trajectory tracking control of the base attitude and joints but also restrain the mode vibrations of the flexible links by utilizing the same control input.

2.System dynamics modeling

Considering the baseB0is free-floating,the system meets the law of conservation of linear momentum,and the rotations of the joints will disturb the attitude stabilization of the base.Besides,the vibration of flexible linkB1is coupled with that of flexible linkB2.Hence the flexible two-link space robot is a highly complex nonlinear system,in order to realize the control objectives of attitude stabilization,joint trajectory tracking and vibration suppression,the first step is to establish the system model.

The plane structure of the flexible two-link space robot system is shown in Fig.1.The system consists of the baseB0,simplysupported beamB1and cantilever beamB2.OXYis the inertial coordinate system,whileoixiyi(i=0，1，2)is the local coordinate system ofBi.θ0is the attitude angle of the base,whileθi(i=1，2)is the rotation angle of joint linkBi.Oi(i=0，1，2)is the rotation center ofBi.l0is the distance between the rotation centersO0andO1,whileli(i=1，2)is the axial length of flexible linkBi.The mass and the moment of inertia of the base arem0andJ0respectively,while the linear density and the flexural rigidity of the flexible linkBi(i=1，2)areρiandEIirespectively.r0is the position vector ofO0inOXYframe,and rPi(i=1，2)is the position vector of arbitrary pointPi(i=1，2)of flexible linkBiinOXYframe.

Since the axial deformation and shear deformation of the flexible linkBi(i=1，2)are ignored,and only its bending deformation is considered,it can be regarded as Bernoulli-Euler beam.According to the assumed mode method[20],the elastic deformation of flexible linkBican be expressed as

Since the elastic deformation of the flexible beam is mainly composed of the low-order vibration modes,hence the first two modes can be selected for vibration analysis,i.e.

The position vectors rP1and rP2are

where,

Taking the differentiation of Eqs.(2a)and(2b)with respected to time t lead to

Ignoring the external gravitational force exerted by other planets,the space robot system satisfies the law of conservation of momentum.Without losing generality,it is reasonable to assume that the initial momentum of the system is zero,i.e.˙rC(0)=0;then,one has

Substituting Eqs.(3a)and(3b)into Eq.(4)yields

Solving˙r0from Eq.(5),one obtains

where,

Substituting Eq.(6)into Eqs.(3a)and(3b)lead to

Ignoring the gravitational potential energy of the system,the bending strain energy of the flexible linksVis the total potential energy of the system,i.e.

From Eqs.(8)and(9),the Lagrange function of the system can be obtained as

Defining the generalized coordinate q=[qrqf]T,where qr=[θ0θ1θ2]Tis the rigid generalized coordinate and qf=[δ11δ12δ21δ22]Tis the flexible generalized coordinate.The Lagrange equation of the system can be established as

where Q∈R7×1is the generalized force of the system.

Reorganizing Eq.(12),the dynamic equation of flexible two-link space robot system can be deduced as follows

3.NFTSM controller design and stability analysis

3.1.NFTSM controller design

In order to obtain the fully-actuated rigid subsystem,Eq.(13)can be rewritten as

where Drr∈R3×3,Drf∈R3×4,Dfr∈R4×3and Dff∈R4×4are the submatrices of D;Hr∈R3×1and Hf∈R4×1are the sub-vectors of H.

By eliminating¨qffrom Eq.(14),the dynamic equation of the rigid subsystem can be obtained as

where,

It should be stressed that Eq.(15)can directly describe the trajectory tracking motions of the base attitude and the joints.

Defining the state variable x=[x1，x2]T=[qr，˙qr]T,Eq.(15)can be expressed as the following nonlinear system

Assumption 1.The desired trajectories xd,˙xdand¨xdare continuously bounded.

Defining the tacking error as e1=xd-x1,hence the error dynamic equation of the system can be obtained

The traditional terminal sliding mode(TSM)and fast terminal sliding mode(FTSM)can be described by the following nonlinear differential equations respectively

And the convergence time of TSM and FTSM are[21,22].

For the TSM,when the tracking error e(i)=[e1(i)，e2(i)]Tis far from the equilibrium points(i)=0,the convergence speed is slow;while when the tracking error is close to the equilibrium point,the convergence speed is fast.For the FTSM,when the tracking error is far from the equilibrium point,αe1(i)can ensure the fast convergence of the error;While when the tracking error is close to the equilibrium point,β｜e1(i)｜γsgn(e1(i))can ensure the fast convergence of the error.

Taking the differentiation of stand sftwith respected to time t yields

Since-1＜γ-1＜0,one can know that whene1(i)→0,Therefore,the control algorithm adopting TSM or FTSM will induce calculation singularity.

Since the traditional terminal sliding mode is easy to cause calculation singularity or long convergence time,in order to avoid these shortcomings,a new nonsingular fast terminal sliding surface is defined as

whereαandβare positive constants;γ＞λ,1＜λ＜2.

Considering NFTSM is continuously differentiable,its differentiation with respect to time can be expressed as

Sinceγ-1＞λ-1＞0,then the control algorithm adopting NFTSM will not induce calculation singularity.Selecting the above gain parameters,the sliding surfaces(i)=0 obtained from Eq.(24)can avoid the instability of sliding surface caused by complex solution of system state whene1(i)＜0 ore2(i)＜0.

3.2.Stability analysis

Theorem 1.Consider the error dynamics system(17)and the nonsingular fast terminal sliding surface(24),if the control law is developed as

whereηis a positive constant.

Then the error vector e=[e1，e2]Twill converge to the equilibrium point in finite time.

ProofSelecting Lyapunov function as

and substituting Eq.(26)into Eq.(17)lead to

Taking the differentiation ofVwith respected to time t,and combining Eqs.(24)and(28)yield

Whens(i)≠0 ande2(i)≠0 are satisfied,then˙V= -the system error vector e=[e1，e2]Tarrives the sliding surface s=0 in a finite time.

When˙V=0,there are one of the following three situations:

(1)when s=0,which implies system error vector e=[e1，e2]Thas reached the sliding surface and converge to the equilibrium point in finite time;

(2)whens(i)＞0 ande2(i)=0,according to Eqs.(24)and(29),one obtainse1(i)＞0 and˙e2(i)＜0 respectively;

(3)whens(i)＜0 ande2(i)=0,according to Eqs.(24)and(29),one obtainse1(i)＜0 and˙e2(i)＞0 respectively.

The phase plane convergence trajectory of the tracking error is shown in Fig.2.It can be seen from Fig.2 that case(2)and case(3)mean that there is a infinitesimal scalarε＞0 in the phase plane,so that whens(i)＞0 ande2(i)∈[-ε，ε]hold,the system error vector e(i)=[e1(i)，e2(i)]Twill move away frome1axis to the second quadrant of the phase plane;whens(i)＜0 ande2(i)∈[-ε，ε]are satisfied,the system error vector e(i)=[e1(i)，e2(i)]Twill move away frome1(i)axis to the fourth quadrant of the phase plane and reach the switching planes(i)=0 in a finite time.Therefore,the system error vector represented by any position in the phase plane will arrives(i)=0 in a finite time under the control law(26)and the arrival timeTr(i)satisfies[23].

Fig.2.The phase plane convergence trajectory of the tracking error.

When the sliding surfaces(i)=0 is reached,the error convergence timeTs(i)can be solved from Eq.(24)as

Therefore,the system error vector e=[e1，e2]Twill converge to the equilibrium point in finite time.The Proof has been completed.

4.The NFTSM hybrid control scheme based on the concept of virtual control force

Since the above-mentioned NFTSM control scheme can only ensure the finite-time convergence of tracking errors of the base attitude and joints,but can not suppress the vibrations of flexible links;therefore,the original desired trajectory is modified by using the concept of virtual control force in this section,and a hybrid trajectory qhwhich can simultaneously reflect the tracking error of rigid trajectory e=[e1，e2]Tand flexible mode qδis generated.By using the modified NFTSM control scheme to track the new hybrid trajectory,the flexible mode can also be suppressed accordingly.

Based on the concept of virtual control force,a hybrid trajectory qhcan be defined and its tracking error with respected to the desired trajectory qdis eh=qd-qh.Then,the dynamics equation of tracking error eh=qd-qhis obtained

where a∈R3×3and b∈R3×3are positive-definite diagonal matrices;F is the virtual control force which is determined later.

Defining hybrid tracking error er=qh-qr,then the NFTSM controller based on virtual control force can be written as

where sr=er+α｜er｜γsgn(er)+β｜˙er｜λsgn(˙er)is the non-singular fast terminal sliding mode for the hybrid control scheme.

Substituting Eq.(33)into Eq.(15)leads to the dynamics equation of hybrid tracking error er=qh-qras follows

Substituting Eq.(34)into Eq.(32),the dynamics equation of tracking error e1=qd-qrcan be obtained

where Hp=P+a˙er+ber

From Eq.(35),one has

According to Eq.(14),one can obtain the dynamic subsystem representing the flexible vibration as

Substituting Eq.(36)into Eq.(37)yields

Combining Eq.(35)with Eq.(38),the following state equation can be obtained

where

Obviously,the state Eq.(39)includes both the vibration modes of the flexible links qfand the actual rigid tracking error e1.

Considering the nonlinear time-varying matrix E as disturbance,when E=0 holds,then Eq.(39)is a controllable linear system.Taking vibration modes qf,rigid tracking error e1and virtual control force F as optimization objectives,a performance indicator function based on optimal control theory can be constructed as

where M∈R14×14and N∈R3×3are symmetric positive-definite matrices.

According to linear quadratic optimal control theory,in order to minimize performance indicator function(40),virtual control force F can be designed as the state feedback optimal controller with the form as

where G is the unique positive-definite solution of the Ricatti equation given by

Substituting Eq.(41)into Eq.(39)leads to

When E=0 holds,the state feedback optimal controller(41)can guarantee the close-loop system(39)to be asymptotically stable.Combining the optimal controller(41)into Eq.(32),a hybrid trajectory reflecting both flexible vibration modes and rigid tracking error can be generated.When E≠0 holds,selecting Lyapunov function asV(z)=zTGz,then one has[24].

From Eq.(44),one can see that the closed-loop system(39)is still stable.

According to the analysis above,one can conclude that by adapting the NFTSM controller(33)based on the hybrid control scheme,the finite-time convergence of rigid tracking error and the vibration suppression of flexible links can be realized simultaneously.

From Eqs.(32)and(41),one can know that the hybrid trajectory qhcan reflect the flexible modes and the trajectory tracking errors simultaneously.Hence,Replacing e1 with er in controller(26),then the NFTSM hybrid controller(33),i.e.the single control input,is obtained.

5.Simulation results

The physical parameters of the flexible two-link space robot system are:l0=l1=1.5m,l2=1m,m0=40kg,EI1=EI2=50N/m2,ρ1=3.5kg/m,ρ2=1.1kg/m,J0=34.17kg·m2.The desired trajectory is qd=[0.1，0.4，0.8]T(rad),and the initial position and initial mode coordinates are qr(0)=[0.2，0.5，0.7]T(rad)and qf(0)=[0 0 0 0]T(m)respectively.

In order to illustrate the finite-time convergence characteristics of the proposed NFTSM algorithm,the simulation results are compared with these of the CTC algorithm proposed in Ref.[25]and these of the SMO algorithm proposed in Ref.[26].

The mathematical expression of the CTC algorithm is

wherekpandkdare positive constants;

while the SMO algorithm is

Similarly to the NFTSM hybrid controller(33),the CTC hybrid algorithm based on the virtual control force is

while the SMO hybrid algorithm based on the virtual control force is

Table 1Controller parameters of different control schemes.

The controllers(26),(45)and(46)are named conventional control schemes,while the controllers(33),(47)and(48)are called hybrid control schemes.The controller parameters are shown in Table 1.

5.1.Dynamic response based on the conventional control schemes

In this case,three kinds of conventional control schemes are respectively applied to the flexible two-link space robot system,and the simulation results are shown in Figs.3-6.Figs.3 and 4 are the trajectory tracking curves of the base attitude and joints under different control schemes respectively,while Figs.5 and 6 are the first two mode coordinates of the flexible link 1 and flexible link 2 respectively.

From Figs.3 and 4 and Table 2,one can observe that the proposed NFTSM algorithm has the fastest convergence speed,and the attitude stabilization of the base and the tracking control of joints can both be achieved within 1.5s.However,since the conventional control schemes don’t have the mechanism of vibration suppression,none of the three control algorithms can attenuate the mode coordinates of the flexible links to a lower level,as depicted in Figs.5 and 6.

5.2.Dynamic response based on the hybrid control schemes

While in this situation,three kinds of hybrid control schemes are respectively exerted to the flexible two-link space robot system.The gain matrices are selected as a=b=diag([4，4，4]),M=diag([1，1，…，1])and N=diag([1，1，1]);the initial position of the hybrid trajectory is qh(0)=[0.05，0.35，0.85]T(rad).The simulation results are presented in Figs.7-10.Figs.7 and 8 are the trajectory tracking curves of the base attitude and joints under different control schemes respectively,while Figs.9 and 10 are the first two mode coordinates of the flexible link 1 and flexible link 2 respectively.

Fig.3.Trajectory tracking curve of the base attitude under the conventional control schemes.

Fig.4.Trajectory tracking curves of joints under the conventional control schemes.

Fig.5.The first two mode coordinates of the flexible link 1 under the conventional control schemes.

Fig.6.The first two mode coordinates of the flexible link 2 under the conventional control schemes.

Table 2Comparison of convergence time of different control schemes.

From Figs.7 and 8 and Table 3,it can be seen that the convergence time of the proposed NFTSM hybrid control algorithm is 2.5s,which is significantly shorter than the other two hybrid control algorithms.Due to the introduction of the virtual control force,a hybrid trajectory reflecting simultaneously the flexible vibration modes and the rigid tracking error is constructed,and the proposed three hybrid control schemes can all limit the vibration amplitude of the first-order modal coordinate of the flexible links within 1×10-5m,which illustrates the effectiveness of the control schemes in vibration suppression.

6.Conclusion

The dynamic modeling,finite-time trajectory tracking control and vibration suppression of a flexible two-link space robot have been studied.In order to ensure that the base attitude and the joints of the space robot can reach the desired positions within a limited time,a NFTSM controller is designed.Subsequently,for the purpose of attenuating the vibrations of flexible links,a hybrid trajectory based on the concept of virtual control force is constructed,which can both reflect the flexible vibration modes and the rigid tracking error.By modifying the original control scheme,a NFTSM hybrid controller based on the hybrid trajectory is developed.

Fig.7.Trajectory tracking curve of the base attitude under the hybrid control schemes.

Fig.8.Trajectory tracking curves of joints under the hybrid control schemes.

Fig.9.The first two mode coordinates of the flexible link 1 under the hybrid control schemes.

Fig.10.The first two mode coordinates of the flexible link 2 under the hybrid control schemes.

Table 3Comparison of convergence time of different control schemes.

Simulation results illustrated that the NFTSM algorithm has faster convergence speed than the other two algorithms,but it can not restrain the vibration of the flexible links,while the NFTSM hybrid algorithm can realize finite-time tracking control of rigid subsystem and vibration suppression of the flexible links simultaneously through only one control input.The simulation results are consistent with the theoretic analysis,hence the effectiveness of the proposed hybrid control strategy is authenticated.Considering the proposed NFTSM hybrid controller has the characteristics of finite-time convergence and vibration suppression.The future work is to extend the control method of this research to the capture operation of the flexible two-link space robot.

Declaration of competing interest

None declared.

Acknowledgment

This work was supported by the National Natural Science Foundation of China(No.11372073).The authors want to thank the reviewers and the editors for their constructive comments on this paper.