Rong-Hua Lei,Li Chen

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

ABSTRACT Since the joint actuator of the space robot executes the control instructions frequently in the harsh space environment,it is prone to the partial loss of control effectiveness(PLCE)fault.An adaptive fault-tolerant control algorithm is designed for a space robot system with the uncertain parameters and the PLCE actuator faults.The mathematical model of the system is established based on the Lagrange method,and the PLCE actuator fault is described as an effectiveness factor.The lower bound of the effectiveness factors and the upper bound of the uncertain parameters are estimated by an adaptive strategy,and the estimated value is fed back to the control algorithm.Compared with the traditional fault-tolerant algorithms,the proposed algorithm does not need to predetermine the lower bound of the effectiveness factor,hence it is more in line with the actual engineering application.It is proved that the algorithm can guarantee the stability of the closed-loop system based on the Lyapunov function method.The numerical simulation results show that the proposed algorithm can not only compensate for the uncertain parameters,but also can tolerate the PLCE actuator faults effectively,which verifies the effectiveness and superiority of the control scheme.

Keywords:Space robot Actuator faults Uncertain parameters Effectiveness factor Fault-tolerant control

1. Introduction

A space robot is a kind of non-linear multi-body dynamic system composed of a carrier(spacecraft)and a manipulator.It is widely used in the rendezvous and docking of a space capsule,hovering and capturing of a small satellite and fuel filling of other spacecraft[1-4].With the rapid development of space technology,space robots will play a greater role in the exploration and development of space resources.At present,the dynamics and control of space robots have become the focus of aerospace technicians,and some research results have emerged[5-8].For a space robot with uncertain parameters,Yu[6]designed an augmented robust control algorithm.For a space robot with communication delays,Liang[7]proposed an improved computed torque control method based on Taylor series prediction.For a floating-based space robot with input constraints,Xie[8]introduces an anti-saturation fuzzy sliding mode controller.However,none of the above algorithms take into account the PLCE actuator fault of space robot.Considering the joint actuator is the core component of the entire control system,its failure is bound to cause unpredictable consequences.Therefore,it is extremely important to improve its own fault tolerance to maintain the normal operation of the control system.

Currently,there are abundant research results on fault-tolerant control of various dynamic systems[9-15].For a linear timevarying system with actuator failures,Rosalba[10]proposes a fault-tolerant strategy based on integral sliding mode and control allocation.For a linear multibody system with actuator faults,Zhu[11]designs a distributed fault-tolerant control scheme based on adaptive fault observer.For a class of linear time-delay system with PLCE actuator faults,Ye[12]developed a cost-guaranteed faulttolerant control algorithm based on linear matrix inequality(LMI)technology.Although the above control algorithms all achieve good fault-tolerant control effects,these control strategies are designed for linear systems.Since a space robot is a kind of MIMO(multiple inputs and multiple outputs)nonlinear system,it is quite hard to apply the above fault-tolerant algorithms to the motion control of space robot.It is worth mentioning that the researches on faulttolerant control of nonlinear systems need to be improved.For spacecraft with parameter uncertainties and actuator failures,Cai[13]proposed an indirect robust adaptive fault-tolerant controller.For spacecraft with input constraints and actuator failures,Xiao[14]designed an anti-saturation fault-tolerant velocity-free algorithm.It should be pointed out that both algorithms in Refs.[13,14]assume that the lower bound of the actuator effectiveness factor is known.However,for the actual spacecraft,the specific fault information of the actuator is difficult to be predicted,which limits the practical application of the algorithms.Geng[15]introduced a variable gain PID(proportion integral differential)fault-tolerant control scheme based on LMI technology for a spacecraft with actuator failure,which effectively improved the tracking performance of the system with time-varying inertial parameters,but the variable gain strategy increased the complexity of the algorithm.

Based on the current research situation,an adaptive faulttolerant control algorithm based on boundary estimation is designed for a space robot system with the uncertain parameters and the PLCE actuator faults.The mathematical model of the system is established based on the Lagrange method.The lower bound of the effectiveness factors and the upper bound of the uncertain parameters are estimated by adaptive strategy,and the estimated values are fed back to the control algorithm in real time.Compared with the fault-tolerant algorithms in Refs.[13,14],the algorithm does not need to pre-determine the minimum value of the actuator effectiveness factors,which is more in line with practical engineering applications.Moreover,the algorithm is simple in structure and has less computational complexity than the algorithms proposed in Ref.[15].

2. Dynamics modeling and problem description

The planar structure of a free-floating space robot system withn+1 degrees of freedom is shown in Fig.1.The system consists of a base carrierB0and some rigid linksBi（i=1, 2, …,n）.Oiis the rotation center ofBi;Ciis the mass center ofBi;l0is the distance from rotation centerO0toO1;li（i=1, 2, …,n）is the length of linkBialong theyiaxis;miis the mass ofBi（i=0,1,…,n）;Jiis the inertia moment ofBi（i=0, 1, …,n）with respect to its mass centerCi;θ0is attitude angle displacement of the base relative to the Yaxis;θi（i=1, 2, …,n）is the angular displacement of the ith link,i.e.the relative rotation angle betweenyiaxis andyi-1axis.

Combining the momentum conservation theorem with the Lagrange equation,the dynamic equation of the system can be derived as

Fig.1.Free-floating space robot system.

whereq=[θ0, θ1,…, θn]Tis the generalized coordinates of the system;D（q）∈R（n+1）×（n+1）is the symmetric positive-definite inertia matrix of the system;is the Coriolis/centrifugal force vector of the system;τ=[u1,u2, …,un]Tis the control torques of the joint actuators;is the uncertain parameters due to the high-frequency modes,measurement noise and the consumption of the liquid fuel.

Property 1.is a skew symmetric matrix,i.e.,

The dynamic Eq.(1)of the system can be expressed in the form of block matrices as follows

whereD11,D12∈R1×n,D21∈Rn×1andD22∈Rn×nare the submatrices ofD,H11,H12∈R1×n,H21∈Rn×1andqr=[θ1, θ2, …, θn]T.

Eq.(2)can be decomposed into

SinceD（q）is symmetric and positive-definite,thenexists.From Eq.(3),we have

Substituting Eq.(5)into Eq.(4),the dynamic equation of the joints can be obtained

Eq.(6)can be quasi-linearized as[16].

When the joints actuator encounters the PLCE fault,the dynamics model(1)can be rewritten as

where ρ=diag｛p1,p2, …,pn｝ represents the actuator effectiveness factor matrix with 0 ≤ρi≤1（i=1, 2, …,n）means the health status of the ith actuator.The case ρi=1 indicates that the ith actuator is working normally.0<ρi<1 corresponds to the case in which the ith actuator loses part of its effectiveness.While ρi=0 indicates that the ith actuator has lost its all control effectiveness.

The control objective of this work is to design an adaptive faulttolerant control algorithm for the space robot system(8)subjected to the joint actuator faults and uncertain parameters,so as to ensure the stability of the closed-loop system,i.e.,joint output trajectories can track the desired trajectories.

3. Adaptive controller design

In order to facilitate the design of the subsequent control algorithm,Eq.(8)can be rewritten as

Where Δρ=I-ρ,I∈Rn×nis the identity matrix;

Assumption 1.The uncertain parametersis bounded and satisfies

whereKis an unknown positive constant and||·||representsL∞norm in this paper.

Assumption 2.Desired trajectoriesqrd,andare normbounded.

Define the trajectory tracking error ase=qr-qrd.Then,the extended error is selected

where λ is a positive constant.

Next,the dynamic extended error can be further designed as

where χ can be seemed as the error between J and S and its derivative with respected to time iswhere sgn（S）= [sgn（S1）, sgn（S2）, …, sgn（Sn）]Tsgn（J）=[sgn（J1）, sgn（J2）, …, sgn（Jn）]T,k1andk2are two positive constants satisfyingk1≠k2.

For the real space robot,the lower bound of the actuator effectiveness factors min{ρi}and the upper bound of the uncertain parametersKare usually unknown.Therefore,it is necessary to design an adaptive laws to estimate the boundary values.The structure of the control system in this paper is depicted in Fig.2.

In order to estimate the minimum value of the actuator effectiveness factor,define

Fig.2.Block diagram of control system.

whereb=1-min｛ρi｝.

An adaptive fault-tolerant controller(AFTC)is designed as

where μ and β are positive constants.

Theorem 1. For the dynamic system(8)with joint actuator faults and uncertain parameters,supposing that Assumptions 1-2 hold and adopting the adaptive laws(14)and(15),the adaptive fault-tolerant controller(13)can ensure that the trajectory tracking error e=qr-qrd converges to zero asymptotically.

ProofChoose a Lyapunov function as

whereV1,V2andV3are different Lyapunov functions.

The process of proof can be divided into three steps.

Step 1Adaptive law Analysis

Select a Lyapunov functionV1as

Taking the time derivative ofV1,and utilizing Property 1,one obtains

Substituting Eq.(9)into Eq.(18)yields

Applying controller(13)into Eq.(19),one has

Substituting adaptive law(14)into Eq.(20)yields

Combining adaptive law(15)with Eq.(21),one obtains

Step 2Reach time analysis

In order to obtain the convergence time,a lemma is proposed as follows

Lemma 1. The dynamic extended error J exists and can converge to zero in finite timetJ[17].

After time tJ,dynamic extended error J=0;utilizingEq.(11),one has

Select a Lyapunov functionV2as

Taking the time derivative ofV2,we have

From Eq.(24),one obtains||S||=|2V2|1/2.Substituting||S||into Eq.(25)yields

Further,

Since whentreachestS,extended errorSwill converge to zero;which implies whent=tS,S=0;furtherV2（t）=V2（tS）,hence

Next,one obtains

Consequently,extended errorScan converge to zero in finite timetS.

Note that extended errorSand dynamic extended errorJcan both converge to zero in finite time,and dynamic extended error converges faster than extended error;i.e.,Since whent=tS,S=0;applying Eq.(10),we have

Step 3Tracking error analysis

Select a Lyapunov functionV3as

Taking the time derivative ofV3yields

Hence,the tracking error e is convergent.Based on the analysis results of the above three steps,one can see that ˙V≤0,which implies that the whole closed-loop system is stable.The proof is completed.

4. Simulation examples

In order to verify the effectiveness of the designed AFTC algorithm(13),numerical simulations of a planar two-link(n=2)space robot system are conducted using the fourth-order Runge-Kutta iterative method. The simulation results of the controller are compared with those of the nonsingular terminal sliding mode controller(NTSMC)proposed by Ref.[18]and the computed torque controller(CTC)proposed by Ref.[19]respectively.The NTSMC algorithm can only deal with model uncertainties,while the CTC algorithm can neither solve parameter uncertainties nor the PLCE actuator faults.

Fig.3.Angle displacement of the base attitude under AFTC algorithm.

Fig.4.Tracking performance of the AFTC algorithm and the NTSMC algorithm.

The mathematical expression of the NTSMC algorithm is

where α,φ,σ1and σ2are positive constant,1b,c1>1,0

Fig.5.Tracking error of the AFTC algorithm and the NTSMC algorithm.

Fig.6.Tracking performance of the CTC algorithm.

Fig.7.Angle displacement of the base attitude under the AFTC algorithm.

Fig.8.Tracking performance of the AFTC algorithm and the NTSMC algorithm.

wherekvandkpare positive constant.

The dynamic parameters of the space robot system arel0=1m,l1=l2=3m,m0=40kg,m1=m2=3kg,J0=34kg·m2,J1=J2=1kg·m2.

The control gains of AFTC algorithm are chosen ask1= 0.15,k2=0.2,μ=0.5,β=0.001,ε=1,λ=3,χ=[0.1 0.1]T;NTSMC algorithm are set as σ1=2,σ2=3,α=1.8,φ=3,a=2,b=5/3,c1=1.1,c2=0.1;and CTC algorithm arekv=0.28,kp=0.4.

The desired trajectories of the link joints are:θ1d=sin（0.2πt）,θ2d=cos（0.2πt）.The uncertain parameters are:0.05

Fig.9.Tracking error of the AFTC algorithm and the NTSMC algorithm.

4.1. Control performance in healthy status

In this case, all the joint actuators are fault-free, i.e.,ρ=diag｛1, 1｝.The simulation results are shown in Fig.3 to Fig.6.Angle displacement of the base attitude under AFTC algorithm is illustrated in Fig.3.Fig.4 is the tracking performance comparison between the AFTC algorithm and the NTSMC algorithm,while Fig.5 is the tracking errors comparison under the two algorithms.Fig.6 is the tracking performance of CTC algorithm.

It can be seen that both the AFTC algorithm and NTSMC algorithm can achieve trajectory tracking control of the joints,as shown in Fig.4;From Fig.5,one can further observe that the two algorithms can also limit the joint tracking errors to a small range of 0.01 rad.Since the CTC algorithm does not have the mechanism of compensating for uncertain parameters,the tracking errors of the closed-loop system can not converge.

4.2. Control performance in failure status

4.2.1. Scenario 1

In this case,the PLCE actuator fault scenarios are considered and simulated.The actuator mounted in joint 1 loses 30%of its normal power at 5 s,while the actuator mounted in joint 2 lose 20%normal power at 8 s;i.e.,

The simulation results are shown in Fig.7 to Fig.9.Angle displacement of the base attitude under AFTC algorithm is depicted in Fig.7.The tracking performance comparison between the AFTC algorithm and the NTSMC algorithm is shown in Fig.8,while Fig.9 is the tracking errors comparison under the two algorithms.

Fig.10.Angle displacement of the base attitude under the AFTC algorithm.

Fig.11.Tracking performance of the AFTC algorithm and the NTSMC algorithm.

One can observe that although all the joint actuators are subjected to the PLCE faults,the link joints can still reach their desired positions with a tracking accuracy of 0.01 rad when the proposed AFTC algorithm is implemented to the space robot,as illustrated in Fig.8 and Fig.9(a).However,the closed-loop system is turn to unstable when NTSMC algorithm is applied to it,since the NTSMC algorithm can not resist the PLCE actuator faults.

4.2.2. Scenario 2

In this case,a more serious PLCE failure occurred to the joint actuator under these situations:1)The actuator mounted in joint 1 decreases 52%of its normal value after 5 s;2)The actuator mounted in joint 2 undergoes 68%loss of effectiveness in 8 s;i.e.,

Fig.12.Tracking error of the AFTC algorithm and the NTSMC algorithm.

The simulation results are shown in Fig.10 to Fig.12.Angle displacement of the base attitude under the AFTC algorithm is depicted in Fig.10.The tracking performance comparison between the AFTC algorithm and the NTSMC algorithm is shown in Fig.11,while Fig.12 is the tracking errors comparison under the two algorithms.

One can clearly see that although all the joint actuators encounter serious PLCE faults,the proposed AFTC algorithm can still manage to compensate for the PLCE faults and acquire the same tracking accuracy as Scenario 1,as presented in Fig.11 and Fig.12(a).However,with the deterioration of the joint actuator fault,the tracking performance of the NTSMC algorithm becomes worse than that in Scenario 1,as depicted in Fig.9(b)and Fig.12(b).Hence,it can be known that the proposed AFTC algorithm is robust to the PLCE actuator faults.

5. Conclusion

An adaptive fault-tolerant control algorithm is designed for freefloating space robot system subjected to uncertain parameters and the PLCE actuator faults.Since the lower bound of the effectiveness factors and the upper bound of the uncertain parameters are estimated adaptively,the ADFTC algorithm does not need to obtain the specific information of the worst actuator failure as the traditional fault-tolerant algorithms did,which means it possesses a huge potential for engineering applications.In addition,the algorithm has a simple structure and few adaptive parameters,so it can greatly reduce the computational load of the on-board computer.In the future,the author decides to extend the algorithm from planar system to three-dimensional counterpart and further validate the feasibility of the algorithm by semi-physical simulation experiments.

Acknowledgment

This work was supported by the National Natural Science Foundation of China(11372073,11072061).