Xiaoqin Huang and Li Chen

(1.College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China; 2.School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China)

Abstract: Under the conditions of joint torque output dead-zone and external disturbance, the trajectory tracking and vibration suppression for a free-floating space robot (FFSR) system with elastic base and flexible links were discussed. First, using the Lagrange equation of the second kind, the dynamic model of the system was derived. Second, utilizing singular perturbation theory, a slow subsystem describing the rigid motion and a fast subsystem corresponding to flexible vibration were obtained. For the slow subsystem, when the width of dead-zone is uncertain, a dead-zone pre-compensator was designed to eliminate the impact of joint torque output dead-zone, and an integral sliding mode neural network control was proposed. The integral sliding mode term can reduce the steady state error. For the fast subsystem, an optimal linear quadratic regulator(LQR) controller was adopted to damp out the vibration of the flexible links and elastic base simultaneously. Finally, computer simulations show the effectiveness of the compound control method.

Key words: space robot with flexible links; elastic base; dead-zone; proportional-integral sliding mode control; active vibration suppression

Free-floating space robots (FFSRs) are complex, nonlinear and strong coupling dynamic systems, and they plays an indispensable role in space exploration. They can replace astronauts performing a variety of space tasks on orbit. Therefore, they have received extensive attention[1-3]. Ref.[4] explored the application of nonlinear model predictive control (NMPC) strategies in the redundant space robot, and the simulations showed its effectiveness. Ref.[5] studied the free-floating space manipulator system with the initial angular momentum not assumed to be zero and its influence on system behaviour.

The FFSR with flexible links is getting increasing attention because of its characteristics of light weight, long arm, heavy load, and more[6-8]. In order to complete space tasks with high precision, in addition to the trajectory tracking, vibration suppression must also be considered. Ref.[9] studied active damping strategies and relevant devices that could be used to reduce the structural vibrations of a space manipulator with flexible links during its in-orbit operations. Ref.[10] used the singular perturbation method to divide the space manipulator system with flexible links into slow and fast components, and simulation results showed that the link vibrations had been stabilized effectively.

In order to expand the working range of a FFSR on space stations, it is installed on a mobile base which can move along the guide rails assembled by a truss. The system inevitably causes the elastic vibration of guide rail which would drive the vibration of the base in the operation process[11]. The tracking trajectory deviation is caused by the coupling relationship among rigid motion, base elasticity, and flexible vibration, which affects the control precision of the FFSR. Hence the base elasticity should be considered as a factor in the high-precision control of FFSR. Under the influence of base elasticity, Ref.[12] adopted a cascade control method to suppress the vibration of the elastic base and flexible joints of flexible-joint FFSR.

Futhermore, there is a dead-zone nonlinearity in the space manipulator actuator system, which is caused by the insensitivity of physical devices (such as sensors or drives, etc.) to some small signals. It gives rise to steady-state tracking error, oscillation of the limit cycle, and even control failure. Therefore, it is worthwhile to design a suitable dead-zone compensation method to improve the control accuracy. Ref.[13] took the space manipulator with dead-zone as the research object, and designed a recursive CMAC control method and a dead-zone pre-compensator to realize the trajectory tracking. Ref.[14] studied an adaptive backstepping control scheme and a dead-zone estimator for the robot manipulator with asymmetric dead-zone. However, neither of them considered the flexibility of the links and the base.

In light of the above research, under the joint torque output dead-zone and external disturbance, the trajectory tracking and vibration suppression for a FFSR with multiple flexible links and elastic base are discussed. Based on the singular perturbation theory, a slow subsystem describing the rigid motion and a fast subsystem corresponding to vibration are obtained. For the slow subsystem, a dead-zone pre-compensator is designed to eliminate the impact of joint torque output dead-zone, and an integral sliding mode neural network controller is proposed. For the fast subsystem, an optimal LQR controller is adopted to damp out the vibration of the flexible links and elastic base, which guarantees the stability and tracking accuracy. Numerical simulations of a FFSR system with an elastic base and two flexible links show the effectiveness of the proposed scheme.

1 Dynamics Modelling of a FFSR with Elastic Base and Multiple Flexible Links

Fig.1 FFSR system with elastic base and flexible links

The structure model of FFSR with an elastic base andnflexible links is shown in Fig.1. The system consists of the free-floating baseB0and the flexible linksBi(i=1,2,…,n). In order to simplify the operation, the elasticity of the guide-way is simplified as a light spring to indicate the elasticity of the base.x′ is the elastic displacement of the spring. The local coordinateOixiyiof each fragmentBi(i=0,1,…,n) is established.O0coincidesOc0, which is the mass center ofB0. The rotary jointO1ofB1is connected withB0by the light spring.Oi(i=2,3,…,n) is the rotational center of the revolute joint betweenBi-1andBi. Thexi-axis is collinear withOiOi+1fori=0,1,…,n-1, and thexn-axis is tangent toBnatOn. The distance betweenO1andO0isl0. The length ofBi(i=1,2,…,n) along thexi-axis isli. The mass and moment of inertia about the centroid of the base arem0andJ0, respectively. A translational inertial reference frame (O-XY) is established by taking the arbitracy pointOof the space as the origin.

Assumption

① The spring is a massless spring;

② The spring can only stretch;

③ Elastic coefficient of the springkx′is constant;

④ The initial displacementx′ of the spring is zero.

The flexible links are slender and homogeneous. The bending deformation is mainly considered. The axial and shear deformation can be ignored, while transverse vibration is performed in the plane.

(EI)iis the flexural rigidity of linkBi(i=1,2,…,n), andρiis the uniform density of the flexible linkBi(i=1,2,…,n). According to the assumed mode method[15], the flexible links can be regarded as Bernoulli-Euler beams. Then, their elastic deformations can be described as

(1)

wherewi(xi,t) is the transverse elastic deformation ofBiat sectionxi(0≤xi≤li).φij(xi) isjth order modal function of linki.δij(t) is the modal coordinate associated withφij(xi).γiis the truncation of the flexible linkBi(i=1,2,…,n).Bi(i=1,2,…,n-1) andBnare regarded as simply-supported beams and a cantilever beam, respectively. According to Ref.[16], their modal functions can be expressed.

According to a Lagrange equation of the second kind, the dynamic equation is obtained as

(2)

2 Fast and Slow Subsystem Decomposition Based on Singular Perturbation

The dynamic system is decomposed into a slow subsystem representing rigid motion and a fast subsystem which expresses the elasticity of base and the vibration of flexible links. Eq.(2) with block matrices can be rewritten as

(3)

Since M is a symmetric, positive definite matrix, its inverse exists and can be written as

(4)

where

(5a)

(5b)

Define the combination control law

(6)

In order to obtain the slow subsystem, the singular perturbation model is evaluated withε=0, then the following equations are obtained

(7a)

(7b)

where the matrix or variable with a crossed “-” means the corresponding slow variable component.

(8)

Substituting Eq.(8) into Eq.(5a), the slow subsystem is obtained as

(9)

(10a)

(10b)

Settingε=0, thus the fast subsystem is

(11a)

(11b)

3 Combined Control Law Design

3.1 Dead-zone pre-compensator design

The joint torque output dead-zone is used to describe the system’s insensitivity to small signals. When the signal enters the dead-zone, there would be considerable loss, which leads to the control deviation.

(12)

wheredi-anddi+are left and right breakpoints of dead-zone, respectively, which are positive and unknown but bounded.

Fig.2 Adaptive dead-zone compensation principle

(13)

(14)

3.2 Design the slow subsystem controller

While there exists the external disturbanceτd, the slow subsystem can be written as

(15)

We define the system tracking error

e=θd-θ

(16)

whereθd=[θ0d…θid…θnd]Tis the desired trajectory of rigid motion.

We define the filter trajectory error

(17)

whereΛ∈R(n+1)×(n+1)is a symmetric and positive definite constant matrix.

The slow subsystem Eq.(15) can be expressed as a form of error equation about s

(18)

(19)

(20)

wherecjis the center of thejth basis function, andbjis the width of the corresponding basis function.

(21)

where W*is the optimal value,ψis the optimal approximation error, and ‖ψ‖≤α.

The control law is designed for the slow subsystem

(22)

Eq.(22) is substituted into Eq.(18), thus it can be rewritten as

(23)

Theorem For a slow subsystem of the FFSR with an elastic base and flexible links given by Eq.(15), s is eventually consistent bounded, i.e., the tracking error e converges to zero in an arbitrary small neighborhood if the control law is given by Eq.(23) and the parameter adaptation laws are given by

(24)

(25)

Proof Constructing the following Lyapunov function

(26)

Taking the first derivative ofLwith respect to timet

(27)

Substituting from Eqs.(23)-(25) yields

(28)

(29)

(30)

wherekmin=λmin(K1) is the minimum singular value of matrix K1.

Therefore

(31)

3.3 Linear quadratic controller of fast subsystem

Based on the linear quadratic regulator, the control scheme for the fast subsystem is designed to suppress the vibration of the elastic base and the flexible links actively at the same time.

(32)

where

As shown in Eq.(32), the fast subsystem is a linear system, and the system state variableζcan be adjusted to zero by the optimal control method, so that the base elasticity and the multiple links flexible vibrations can be suppressed. For a linear system, if the performance index function is defined as an integral quadratic function of the state variables and control variables and the control quantityτfis obtained when the function is minimum, then the system can obtain the optimal performance.

The linear quadratic optimal performance index function is introduced as

(33)

By the linear quadratic optimal control theory, in order to minimize Z, the controller should be designed as

τf=-R-1BTPζ

(34)

where P satisfies the following Riccati algebraic equation

PA+ATP-PBR-1BTP+Q=0

(35)

4 Simulation Examples

Taking the model of a FFSR system with elastic base and two flexible links shown in Fig.3 as an example, numerical simulations are carried out.

Fig.3 FFSR system with elastic base and two flexible links

The system inertia and structural parameters are:l0=l1=1.5 m,l2=1.0 m,m0=40 kg,J0=34.17 kg·m2,ρ1=3.5 kg/m,ρ2=1.1 kg/m, (EI)1=50 N·m2, (EI)2=50 N·m2,kx′=500 N/m.

The large amplitude vibration is mainly composed of the first few modes, so the truncation of flexible links is chosen asγ1=γ2=2. The external disturbance of the system is taken as

The dead-zone widths are estimated as

+=diag(10,10),-=diag(5,5).

The desired anglesθdof the base and two flexible links respectively are:θ0d=-1 rad,θ1d=0 rad,θ2d=1 rad. The initial angles are taken as:θ0(0)=-1.2 rad,θ1(0)=-0.3 rad,θ2(0)=1.2 rad. The initial displacement of the base spring isx′(0)=0. Simulation time is given byt=50 s.

Fig.4 Trajectory tracking of the base’s attitude and the two links’ joints

Under the conditions of dead-zone and external disturbance, the integral sliding mode neural network control scheme is adopted for the FFSR with flexible links and an elastic base. Fig.4 shows the actual trajectory (solid line) and the desired trajectory (dashed line) of base and two links. Fig.5 shows the elastic vibration of the base, in which the elastic displacement is 0.018 m at 0.3 s and then decays to zero after 35 s. Figs.6-7 show the first and second order modal of the two flexible links. The first order mode of the linkB1attenuates from 0.366 m in the initial 1.0 s to zero after 45 s, while the second order mode of the linkB1attenuates from 0.032 m in the initial 0.1 s to zero after 30 s. The first order mode of the linkB2attenuates from 0.215 m in the initial 1.1 s to zero after 45 s, while the second order mode of the linkB2attenuates from 0.045 m in the initial 0.2 s to zero after 20 s. From these simulation results, it can be seen that the actual trajectory of the base and the two joints can track the desired trajectory well and the vibrations of the base and two flexible links are suppressed.

Fig.5 Elastic displacement of the base

Fig.6 Mode of flexible link B1

Fig.7 Mode of flexible link B2

In order to illustrate the effectiveness of the vibration suppression method, Figs.8-9 show the first and second order modal of the two flexible links whenτfis closed. It can be seen from these figures, the first-order mode of the linkB1vibrates is about ±0.08 m after 25 s, while the second-order mode of the linkB1vibrates is about ±0.004 m after 10 s. The first-order mode of the linkB2vibrates is about ±0.06 m after 10 s, while the second-order mode of the linkB2vibrates is about ±0.005 m after 25 s. Fig.10 simulates the base elastic displacement whenτfis closed. It can be seen that the base always vibrates between ±0.004 m. All the above can not suppress the vibration; thus, the effectiveness of the vibration suppression scheme is verified.

Fig.8 Mode of flexible link B1 (close τf)

Fig.9 Mode of flexible link B2 (close τf)

Fig.10 Elastic displacement of the base (close τf)

For the sake of showing the effectiveness of the dead-zone pre-compensator, the trajectory tracking error of closing it is shown in Fig.11. It can be seen clearly that the error of two joints is always unable to converge. Because of the multiple coupling relationship, there is a tracking error between the desired trajectory and the actual trajectory of the base angle, although there exists no dead-zone in it.

Fig.11 Tracking error (off dead-zone compensation)

5 Conclusion

Considering the multiple coupling among rigid motion, base elasticity and flexible vibration, the dynamic model of the FFSR system is derived. Based on the singular perturbation method, in the case of dead-zone and external disturbance, an integral sliding mode neural network control law of the slow subsystem and an optimal LQR controller of the fast subsystem are designed. A dead-zone pre-compensator is proposed to eliminate the impact of dead-zone, and the integral sliding mode term can reduce the steady-state error. The neural network can approximate the uncertainty of the dynamic equation and external disturbance. The simulation results show that the scheme can meet the control requirements.