Adaptive Neural Network Robust Control for Space Robot with Uncertainty

The trajectory tracking problems of a class of space robot manipulators with parameters and nonparameters uncertainty are considered. An adaptive robust control algorithm based on neural network is proposed by the paper. Neutral network is used to adaptive learn and compensate the unknown system for parameters uncertainties, the weight adaptive laws are designed by the paper, System stability base on Lyapunov theory is analysised to ensure the convergence of the algorithm. Non-parameters uncertainties are estimated and compensated by robust controller. It is proven that the designed controller can guarantee the asymptotic convergence of tracking error. The controller could guarantee good robust and the stability of closed-loop system. The simulation results show that the presented method is effective.


Introduction
The main task of the free-floating space robots is used to instead of the astronauts to complete satellite capture, to repair space station facilities.Because positions and attitude of the base of space robot that are entirely free, as will result in economization of fuel, increase lifespan of space robot, and decrease in launch expenditure, free-floating space robot would have played an more and more important role in the future.In the free floating condition, space robots are different from ground robots in dynamic characteristics and constraints: dynamics coupling of machine arms and the base, dynamic singularity,a limited supply of fuel and restrictions of the attitude control system.
Therefore not like robot with fixed-base on ground, space robots can't adopt a general control method.Meanwhile, there are many uncertainties existing in the space robot dynamic model, for example, the dynamic model of manipulator mass, inertia matrix and load quality can not be accurately acquired, and external disturbance signals have a certain impact on the controller.To eliminate impact of these non-linear factors, all sorts of advanced control strategies [1][2][3][4][5].suchas adaptive control [6][7], fuzzy control [8][9][10][11][12][13] and neural network control [14][15][16][17] have been used in robot tracking control.
Ref [6] and [7] proposed adaptive control methods, but required complex pre-calculation of the regression matrix, and adaptive control depends on accurate estimate of the unknown parameters.However, it is often difficult to be achieved in practical application, because of the  [18], the method does not require an accurate model for the control object, but the method requires too much adjustment parameters, the reason increases the computer burden and affects the real time.A Radial Basis Function (RBF) neural network control method proposed by [19][20], neural network is used to identificate to uncertainty model, but the control scheme can only guarantee the uniform ultimate bounded (UUB).A sliding mode variable structure base on neural network control scheme is proposed by [21], neural network is used to learn unknown parameters system model.However, the scheme needs to know the upper bound of the unknown uncertainty.
An adaptive robust control scheme base on neural network is proposed to overcome these shortcomings.Robust controller is used to estimate real-time the non-parameters uncertainty upper bound, neural network controller is used to approximate the parameters uncertainty, sliding mode controller is designed to eliminate the approximation error.This kind of control scheme does not need the accurate mathematical model, and overcome the defects that only robust controller usually needs to predict the non-parameters uncertainties, such as errors and disturbances upper bound.The global asymptotic stability (GAS) of the closed-loop system based on Lyapunov theory is proved by the paper.PID feedback strategy is introduced to makes the control scheme more easy to be implemented in Engineering.The simulation results show that the scheme is effective.

Dynamic Equation of Space Manipulators
Free-floating space robot dynamic equation can be written as Equation 1 [20][21]: Where, n R    q,q,q are joint position, velocity and acceleration vectors； is the sum of the external disturbance and the friction, Space robot dynamics equation (1) has the following properties [21][22][23].
P1: Inertia matrix M(q) is a symmetric positive definite matrix.
P2: By suitably choosing  C(q,q) , the matrix 2    M(q) C(q,q) can be the skew symmetric matrix.The following research will use the following assumptions:  and 3  make the friction and disturbance d norm satisfy If space robot system (1) does not exist the unmodeled dynamics and parameters uncertainty, where,the reference trajectory is definited as r q , the position tracking error is definited as e ,the following error tolerance is definited ass , positive definite matrix is Then Where, , and are defined separately.The controller designed as Equation 4.
s e e Where, v K is positive definite matrix, M(q) and ˆ C(q,q) are its separate estimation.
If the system all parameters are known accurately and there isn't non-parameters uncertainty conditions.Namely, the estimated value M(q) and ˆ C(q,q) are ( ) M q and ( , ) and non-parameters uncertainty(the external disturbance and the friction) are 0. The controller (2) and Lyapunov theory are used ,then However, the system all parameters can not be measured accurately, only estimated parameters can be obtained.and non-parameters uncertainties(the external disturbance and the friction) are not equal to 0. A new control law needs to be designed.

Adaptive Robust Controller Designed base on Neural Network
If the system all parameters are known accurately and there isn't non-parameters uncertainty conditions.Namely, the estimated value M(q) and ˆ C(q,q) are ( ) M q and ( , ) and non-parameters uncertainty(the external disturbance and the 10 friction) are 0. The controller (2) and Lyapunov theory are used, then, considering the uncertainty space robot system (1), then the error equation of closed loop system is written as Equation.
Where, the system unknown parameters and non-parameters uncertainty are considered.f is designed to compensate unknown parameters uncertainty.
Where, p K and d K are positive definite matrix, rc  is adaptive robust controller, it designed to compensated unknown non-parameters uncertainty as Equation 9.The optimal output of neural network is definited as f , the optimal network weights matrix is definited as *  , the neural network approximation errors is defined as  .
Then * ( ) || || exp( ) Where, Gauss type function is definited as ( ) x  .The j basis function center is definited as j c , Basis function width is definited as j According to the approximation ability of fuzzy basis function network,the paper makes following assumptions: A3: For any given small positive constant  M , there exist the optimal weight vector *  ,so that ( ) But actual output of neural network should be f , actual weights matrix of network should be θ.
Then ˆ( ) For approximation errors of neural network, Sliding mode controller is designed to compensate approximation errors of neural network.

( )
In order to eliminate the chattering of sliding mode controller and convenient application in engineering, the saturated function is used to instead of sign function.Then tanh( ) Thus,the adaptive robust controller base on neural network is designed as Where, f is Neural network controller to compensate parameters uncertainty, rc  is adaptive controller to compensate non-parameters uncertainty .
Adaptive learning algorithm of neural network weights matrix is designed as Where,

System Stability Analysis
Theorem: If the system (1) meets the assumption 1 and assumption 2, controller (17) and weights adaptive learning law (18) are adopted.Then global asymptotic stability (GAS) of the closed-loop system can be guaranteed.
Lyapunov function is defined to prove the stability of the closed loop system.
The two sides of equation ( 19) is derivatived.
s Ms e K e e K e θ θ (20) According to the error of closed-loop system equation ( 6), then , the adaptive law (18),control law ( 9) and ( 10) are substituted,assumption 2 is considered, following equation can be got is satisfied in a very small neighborhood near the origin.As can be known form the Lyapunov stability theory system, signal s 、 e and  θare uniformly bounded.Because of     s e e ,  e are uniformly bounded..According to assumption 1 and assumption 2, r  q 、 r  q 、 q and  q are bounded.
Because the 1 ( ) V t is bounded and the greater than zero, 1 ( ) 0 V t   . Base on Barbalat theory ,

Simulation and Analysis
In this section, simulation for a planar two-link space robot is studied.
 : the inertia of the link-rod relative to its centroid; i m : the mass of i B .Simulation parameters of the two-link space robot shows: 0 300 The estimate values of parameters are assumed: . disturbances and friction: The desired trajectory: The initial value of joint position and velocity: Network initial weight value, basis function width and the basis function center were randomly selected in the group (0 ~ 0.01).The simulation results are shown in Figure 1 ~ Figure 5. Figure 1 shows parameters uncertainty model and its neural network estimated value; Figure 2 shows position tracking curves of space robot joint 1, Figure 3 shows position tracking curves of space robot joint 2; Figure 4 shows control torque curves of space robot joint 1; Figure 5 shows control torque curves of space robot joint 2. As can be seen from the figure 1,after the initial learning, neural network can get to complete learning and good approximation for parameters uncertainty model in less than 2s, it shows not only the design of adaptive law is effective, also the radial basis function neural network has good generalization ability and fast learning speed Figure 2 and Figure 3 show position tracking chart that this control method can reach better control effect only in about 3s, Figure 3 and Figure 4 show position tracking chart that this control method can reach better control effect only in about 3s.Control torque of space robot joints aren't big, As can be seen from the figure 4 and Figure 5.

Conclusion
The trajectory tracking problems of a class of space robot manipulators with parameters and non-parameters uncertainty are studied by the paper.An adaptive robust control scheme based on neural network is put forward by the paper.1) An adaptive robust controller is designed to compensate to non-parameters uncertainty, neural network controller is designed to approach and estimate the upper bound of parameters uncertainty real-time; 2) The sliding mode variable structure controller is designed to compensate the approximation error, saturated function instead of sign function is used to eliminate chattering; 3) The adaptive weight learning algorithm of neural network is designed to ensure online realtime adjustment, offline learning phase is not need; 4) Globally asymptotically stable(GAS) of the closed-loop system is proved based on the Lyapunov theory.

TELKOMNIKA
Vol. 11, No. 3, September 2013: 513 -520 520 Simulation results show that the designed adaptive robust controller based on neural network is effective and has good engineering value.
is the estimated value of  ; 1  and 2  are normal constant.RBF (Radial Basis Function) neural network has good local generalization ability.It can accelerate greatly the learning speed and avoid local minimum problem.So it is used to approach the unknown parameters uncertainty in the system. ISSN: 1693-6930 TELKOMNIKA Vol.11, No. 3, September 2013: 513 -520 516