The Fuzzy Neural Network Control Scheme With H∞Tracking Characteristic of Space Robot System With Dual-arm After Capturing a Spin Spacecraft

Jing Cheng and Li Chen

Abstract—In this paper, the dynamic evolution for a dualarm space robot capturing a spacecraft is studied, the impact effect and the coordinated stabilization control problem for postimpact closed chain system are discussed.At first,the pre-impact dynamic equations of open chain dual-arm space robot are established by Lagrangian approach, and the dynamic equations of a spacecraft are obtained by Newton-Euler method. Based on the results, with the process of integral and simplify, the response of the dual-arm space robot impacted by the spacecraft is analyzed by momentum conservation law and force transfer law. The closed chain system is formed in the post-impact phase.Closed chain constraint equations are obtained by the constraints of closed-loop geometry and kinematics. With the closed chain constraint equations,the composite system dynamic equations are derived. Secondly, the recurrent fuzzy neural network control scheme is designed for calm motion of unstable closed chain system with uncertain system parameter. In order to overcome the effects of uncertain system inertial parameters, the recurrent fuzzy neural network is used to approximate the unknown part,the control method with HHH∞tracking characteristic. According to the Lyapunov theory, the global stability is demonstrated.Meanwhile,the weighted minimum-norm theory is introduced to distribute torques guarantee that cooperative operation between manipulators. At last, numerical examples simulate the response of the collision,and the efficiency of the control scheme is verified by the simulation results.

I. INTRODUCTION

AS the exploration of space continuously advancing, the space robot has been employing to accomplish more onorbit service missions [1]. There is harsh operating environment in outer space, space robot system assisted astronauts complete space missions, can protect the astronauts’ life from dangers. Space robotic systems have received a large number of significant attentions[2]?[5]in the past decades.The space robot was employed to accomplish complicated tasks, such as fueling, maintaining of spacecraft in earth orbit, clearing of orbital debris, etc. [6]?[8]. Therefor it is becoming increasingly important for space robot system to have the capability of capturing a satellite. As space robot system with dualarm possesses bigger carrying capacity and better structure rigidity,it is more expected to fulfil capturing spacecraft tasks.Since the weightlessness condition in outer space,it makes the dynamics and control problems related to capturing satellite operation by space robot system with dual-arm to be extremely complicated compared with the counterpart of fixed-base robot system and single arm space robot system, and there are some unique characteristics, such as, nonholonomic dynamics restriction, change of system configuration, transfer of linear momentum, angular momentum and energy, topology transfer from open to closed loop system, and the constraints of closed-loop geometry and kinematics during capturing satellite operation.It develops many challenging problems that need to be solved.

Space robot system is essentially a coupling, time-varying and nonlinear system[9]?[11],if there is uncertain parameter exists in the system will make control scheme design more difficult [12], [13]. Vafa et al. [14] proposed a concept of virtual manipulator approach for the kinematics and dynamics analysis of space robot. Nakamura et al. [15] proposed a concept of generalized Jacobi matrix, and studied the relation between the velocity of joint space and the velocity of working space for single arm space robot. Huang et al. [16]?[18]studied the application of tethered space robots capturing a target in future on-orbit missions, they analyzed the impact dynamic modeling and proposed coordinated stabilization scheme, adaptive postcapture backstepping control, et al. for tumbling tethered space robot-target combination. Abad et al.[19] predicted the best capturing time and configuration of the target, and found an optimal control solution to guide the robot to reach the predicted location with a minimal attitude disturbance. Rekleitis et al. [20] proposed a planning and control methodology for manipulating passive objects by cooperating orbital free-flying servicers in zero gravity.

During the process of capturing operation,the end-effectors of dual-arm space robot will inevitably collide with the captured target. The collision lead to instability and rolling of space robot in the weightlessness environment which is harm for the precise instrument on the servicing spacecraft,even lead to the failure of space missions. Because of the aforementioned challenging problems, the current studies are mostly focus on single arm space robot [21]?[23]. In fact,dual-arm space robot is similar to human arms; it is more suitable for capturing operation. Patolia et al. [24] discussed the coordinated motion planning problems for the dual-arm space robot system. Chen et al. [25] proposed an hybrid position and force control method based on radial basis function neural network for a closed chain space robot. Shah et al.[26] found the optimal motion planning strategy for dual-arm space robot to reduce the influence of capturing operation.Aforementioned studies give primary attention to trajectory planning and control of dual-arm space robot in pre-impact phase, none of them have considered that impact effort and calm control method for the composite system with closed chain after the collision.

This paper is devoted to dynamic evolution analysis of dual-arm space robot capturing a spin spacecraft, and the impact effort and control scheme of the closed chain with uncertain inertial parameters are discussed. First, based on the Lagrangian approach and Newton-Euler method, the preimpact dynamic models of dual-arm space robot and a spin spacecraft are established respectively. The dual-arm space robot and the spacecraft form a closed chain system in the post-impact phase. According to the closed chain constraint equations, the dynamic equations of the closed chain system are obtained.The impact effect of closed chain system after the collision analyzed by momentum conservation law and force transfer law. Then, for the unstable closed chain system with uncertain system parameter,the recurrent fuzzy neural network control scheme is designed for calm motion. The recurrent fuzzy neural network is used to approximate the unknown part of the system[27],[28].For the approximate error,the control scheme which is satisfied the H∞performance indicators requirement [29], [30]. The stability of the system with H∞tracking characteristic is demonstrated according to Lyapunov theory. For the existence of the controller redundancy, The introduction of the weighted minimum-norm theory is applied to distribute the torques harmoniously of each joint. Finally,the numerical examples simulate the response of collision for closed chain system and confirm that the proposed control scheme is reliable for the calm motion of unstable system.

II. DYNAMIC EVALUATION ANALYSIS FOR DUAL-ARM SPACE ROBOT CAPTURING A SPACECRAFT

Fig.1. Dual-arm space robot system and target system in pre-impact phase.

According to the geometric relations, the relationship between the velocity of end-effectors and coordinates can be expressed as

Vector ql= [xlylθl]Tis represent the location of the mass center and base attitude which is defined as generalized coordinates for target spacecraft.The dynamic equations of the target can be derived base on the Newton-Euler formulation

According to Newton’s third law, we have

Impact forces can be decomposed as follow:

Invoking (3), (4), (5), and (6), we can obtain that

Integrating (7) over the momentary period of collision

The period of impact is transient: ?t →0. The generalized coordinates is approximated as constant value, and the generalized velocity is changed in a limited range.There is no input torques during the momentary period of collision.Besides,the impact force is huge so that FI[31]can be ignored.According to the theorem of impulse, (8) can be further reformulated as

Target and end-effectors are fixedly connected after the capturing operation, the whole system can be regard as a composite system with closed chain.

In order to obtain the constraint equations of the closed chain system, the composite system is cut at point bLwhich is shown in Fig.2. Since cut points have the same movement velocity and angular velocity, relationship of left arm angles and right arm angles reference to base-fix coordinate x0O0y0can be derived as

Noting that

where In×nis a n×n unit matrix, On×nis a n×n zero matrix.

With the (11), (12), and (13) we have

Solving (9), the response of composite system is obtained

Differentiating (13), we can obtain that

Fig.2. Closed chain system in pre-impact phase with virtual cut-point.

Substituting (14), and (16) into (7), we have

The dynamic model of unstable closed chain system with closed-loop constraints has been obtained so far. The unstable motion is harm for the space equipment, it is necessary to propose a proper control scheme for the calm motion of whole system.

III. RECURRENT FUZZY NEURAL NETWORK CONTROL FOR CLOSED CHAIN SYSTEM WITH H∞TRACKING CHARACTERISTIC

A. Recurrent Fuzzy Neural Network

The neural network [32] and fuzzy logic [33] have their advantages, the learning ability of neural network is used to adjust the fuzzy control by combine neural network with fuzzy control [34], [35]. The neural network has characteristics, and the relationship between input and output of the controller is expressed quantificationally by adjusting the parameters continually to complete control purpose; the fuzzy logical system is dependent on expert experience qualitatively. The fuzzy neural network has combined the advantages of fuzzy neural network and fuzzy logical system and but also overcomes some disadvantages of them. This kind of combination characterized the fuzzy control as self-learning and the neural network as reasoning. In this paper, the neural network is used to realize fuzzy reasoning to constitute the fuzzy neural network control scheme and realize trajectory control. The four-layer recurrent fuzzy neural network which includes input layer, member function layer, rules layer, and output layer is shown in Fig.3.

Input layer:

superscript is corresponding to the layer input and layer output.

Member function layer: Gauss function is chosen as membership function, aijand bijare center value and base width of jth language set.

Fig.3. Structure of recurrent fuzzy neural network.

Fuzzy reasoning layer:

Output layer: the layer of defuzzification:

Output can be written as follow:

where Wr= [w1rw2r... wlr]Tis a vector of weights,Φ = [x1x2... xu]Tis the corresponding vector of basis functions.

The recurrent fuzzy neural network feedback unit memorize the previous rules, therefore, the network has characteristics of dynamic nature and simple configuration.

B. Control Design for Closed Chain System

Noting that

Substituting (12) into (23), we have

The internal force have no influence on closed chain system,the elements in the vector ˉFIare zeros which can be obtained by means of matrix calculating. The reduced-order form of dynamic equation for composite system are reformulated as follow

In order to control the position and attitude of both the base and the load, the output matrix of system is defined as

Differentiating (28) yields

where J0(qf)∈R6×6is the Jacobian matrix.

Defining the output error as follow

where the desired trajectory Xdis bounded and continuous.

Substituting (29) into (27) the dynamic equations of closed chain system can be rewritten as:

Since fuel consumption and the other technology factors of space robot, it is very difficult to obtain the exact dynamic model of space robot. Considering the uncertain parameters of system, (31) can be further reformulated as

Using recurrent fuzzy neural network to approximate the unknown part, χ can be expressed as

where W?is the optimal weight matrix. Φ is a column correlation with basis function, ?δ is approximate error.

For the unstable composite system with uncertain inertial parameters, the control law is given as

where Kvand Kpare positive definite coefficient matrixes.μ is H∞r(nóng)obust control item which is used to eliminate the influence of approximation error.

Substituting (34) into (32) leads to

The system state space equations are developed as

Fig.4. Time history of base position after impact.

Fig.5. Time history of base attitude after impact.

Fig.6. Time history of load position in the x-direction after impact.

Fig.8. Time history of load attitude after impact.

Fig.9. Motion of the closed chain system after impact.

Fig.10. Time history of base position without compensation control.

Fig.11. Time history of base attitude without compensation control.

Fig.12. Time history of load position in the x-direction without compensation control.

Fig.13. Time history of load position in the y-direction without compensation control.

Fig.14. Time history of load attitude without compensation control.

To show the performance of the proposed control scheme,a comparison experiment is carried out which close the compensation control.The simulation result are shown in Figs.10–14.Figs.10 and 11 show the tracking curves of the base position and attitude. Figs.12–14 show the tracking curves of the load position and attitude. From the observation of simulation results, in the condition of uncertain inertial parameters, the control performance is poor. Control system cannot achieve control objective by closing the compensation control.

V. CONCLUSION

The dynamic models of dual-arm space robot and target spacecraft are established by multi-body theory. With the theorem of impulse and force transfer law, the response of dual-arm space robot system impacted by a spin spacecraft is analyzed. The closed chain system model is derived by the constraints of closed-loop geometry and kinematics. In order to accomplish calm motion control for composite system, the recurrent fuzzy neural network control scheme is designed,the control method with H∞tracking characteristic. The composite system is unstable in the post-impact phase. The proposed control method inhibits the rolling trend of whole system and remarkable tracking performance is shown in simulation results. The control scheme dispense with accurate system model or linear parameterization of the system dynamic equations.