RBF Neural Network Adaptive Control for Space Robots without Speed Feedback Signal

Transcription

1 Trans. Japan Soc. Aero. Space Sci. Vol. 56, No. 6, pp. 37 3, 3 RBF Neural Network Adaptive Control for Space Robots without Speed Feedback Signal By Wenhui ZHANG, Xiaoping YE and Xiaoming JI Institute of Technology, Lishui University, Lishui, China Received July 3rd, ) Tracking control problems for space robots are studied under conditions without speed feedback signals. An adaptive RBF neural network control method with a speed observer is proposed. Specially, we conduct the following. ) A dynamic model of space robots is established. ) A speed observer based on a neural network is designed to reconstruct speed information. 3) A controller based on a neural network is designed to compensate the nonlinear model of system. 4) A weight adaptive learning laws of the neural network is designed to ensure on-line tuning without an off-line learning phase. 5) The uniformly ultimately bounded state of the closed-loop system is proved based on Lyapunov theory. Simulation results show that the adaptive neural network control method with the speed observer can achieve good precision. This has important engineering value. Key Words: RBF Neural Network, Speed Observer, Adaptive Learning Laws, Adaptive Control, Space Robots. Introduction Space robots will have an increasingly important role in the future. In the free-floating condition, space robots are different from ground robots in terms of dynamic characteristics and constraints; 4) specially in terms of dynamic coupling of machine arms and the base, dynamic singularity, a limited supply of fuel and restrictions of the attitude control system. Therefore, unlike robots with a fixed base on the ground, a general control method cannot be adopted for space robots. 5 7) 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 cannot be accurately acquired, and external disturbance signals have a certain impact on the controller. To eliminate the impact of these nonlinear factors, various advanced control strategies. 8 5) The above control strategies were proposed under the condition that the speed information of joints can be accurately obtained. However, in practical engineering control, although joint speed measurements can be obtained by means of tachometers, the speed information is often contaminated by noise disturbance. In particular, joint motors of space robots are usually run under low speed conditions. Moreover, moment fluctuations and other high frequency effects are engendered because of discontinuity of the stator magnetic field in tachometers. These reasons further reduce the quality of the speed signal. 6) Moreover, tachometers increase the weight of moving parts and reduce the efficiency of space robotic manipulators. Therefore, it is difficult to achieve the high control precision in practical engineering. The method becomes especially important in without speed feedback signals. Ó 3 The Japan Society for Aeronautical and Space Sciences There has been some success in robot research based on speed observers. A position feedback control strategy was put forward, 7,8) but the design of the observer requires exact knowledge of robot dynamics, which is difficult to acquired. An output feedback controller was designed, 9,) where the design of the observer does not need an exact dynamics model owing to use of a neural network. All signals in the closed-loop dynamic can be uniformly ultimately bounded UUB); however, the exact inverse of the inertia matrix is still required by the observer, and too much calculating is required. Control problems for space robots are studied under conditions without speed feedback signals and an adaptive neural network control method with a speed observer is proposed in this paper. A radial basis function RBF) neural network control strategy without speed signals is presented for space robots. A dynamic model of space robots is established. Then, the neural network is used to compensate the nonlinear model and a speed observer based on the neural network is designed to reconstruct the speed information. A controller based on the neural network is designed to approach the unknown model. A weights adaptive learning law of the neural network is designed to ensure on-line tuning. Stability of the closed-loop system based on Lyapunov theory is analyzed.. Dynamic Model of Space Robots A dynamic equation of space robots can be written as 3) MðqÞ q þ Cq; _q _q þ F ¼ : ðþ where joint position, velocity and acceleration vectors are defined respectively as q, _q and q R n, the inertia matrix is defined as MðqÞ R nn, Coriolis forces are defined as

2 38 Trans. Japan Soc. Aero. Space Sci. Vol. 56, No. 6 Cq; _q _q R n, the external disturbances are defined as F R n and control torques are defined as R n. The following properties are defined for space robots system. 4) P: choosing Cq; _q, the matrix _MðqÞC q; _q is a skew symmetric matrix. 3. Design of Adaptive RBF Neural Network Control System x and x are defined as x ¼ q x ¼ _q: Equation ) can be described as: _x ¼ x _x ¼ M ð3þ ðx Þ Cðx ; x Þ _q F : A neural network is used to reconstruct speed signals, and approach the unknown system. The control structure is shown in Fig.. ^q and _^q^q are defined respectively as joint position and speed estimation value, and ~q and _~q~q are defined respectively as the position estimation error and speed estimation error. ~q ¼ q ^q ð4þ _~q~q ¼ _q _^q^q: Further, Eq. ) can be described as _x ¼ x _x ¼ h o ðx ; x ; _x Þþ: h o ¼ðI mm Mðx ÞÞ_x Cðx ; x Þx F ð6þ The RBF neural network belongs to a local generalization network, so it can greatly accelerate the learning speed and avoid local minimum. h o ðx ; x ; _x Þ ¼ f ðx ; x ; W o Þ¼Wo T ðxþ: ð7þ where x ¼ðx ; x Þ, W o is defined as the weight matrix of observer and ðxþ is a Gaussian function. According to the approximation ability of the RBF network, assume A: The optimal weight Wo is bounded and meets k Wo kw o;m; W o;m is the positive constant. ðþ ð5þ A: For any given small positive constant " o;m, there exists a approximation error " o ðxþ, where k " o ðxþ k" o;m. The RBF neural network is used to approach h o ðx ; x ; _x Þ. The optimal approximation can be written as h o ðx ; x ; _x Þ ¼ f o x ; x ; Wo ¼ W T o ðxþþ" oðxþ: ð8þ where x ¼ðx ; x Þ. The functional estimation of Eq. 8) with ^x, ^x is described as ^f o ^x; ^W o ¼ ^W o T ðþ: ^x ð9þ where ^W o is the estimation weight. ^x, ^x are defined as ^x ¼ ^z ^x ¼ ^z þ k ~x : ðþ Defining, ~x ¼ x ^x and ~x ¼ x ^x, Eq. ) has the following form. _^z^z ¼ ^x þ k ~x _^z^z ¼ ^W o T ðþþ ^x ðþ þðk þ k k I mm Þ~x : where K ¼ K T are positive definite matrixes, k > and k >. Further, the following expression can be obtained. _^x^x ¼ ^x þ k ~x _^x^x ¼ ^W o T ðþþ ^x ðþ þðk þ k k I mm Þ~x þ k _~x~x : The observer error dynamics can be obtained by subtracting Eq. 4) from Eq. ). _~x~x ¼ ~x k ~x _~x~x ¼ Wo T ^W o T ðþþ" ^x ð3þ o ðxþk ~x k ~x : Since Wo TðxÞ ^W o T ðþ¼w ^x o T ~ þ ~W o T ^, the following equation can be obtained. _~x~x ¼ ~x k ~x _~x~x ¼ Wo T ~ þ ~W o T ^ ð4þ þ " o ðxþk ~x k ~x : where ~ ¼ ^, ¼ ðx; w ; c Þ, ^ ¼ ðþ. ^x The weight estimation error is defined as ~W o ¼ Wo ^W o. The adaptive learning law is designed as _~W~W o ¼K Wo ~x T ^ þ o K Wo k ~x k ^W o : ð5þ where K Wo is defined as the diagonal matrix and o is defined as a positive real constant. 4. Speed Observer Based on RBF Neural Network e and _e are defined respectively as the position tracking error and the speed tracking error. e ¼ q d q ¼ q d x ð6þ _e ¼ _q d _q ¼ _q d x : Modified filtered tracking error ^s is defined as Fig.. Control system structure. ^s ¼ _^e^e þ e ð7þ where is defined as the positive definite matrix. According to Eqs. 6) and 7) and ~x ¼ x ^x. The

3 Nov. 3 W. ZHANG et al.: RBF Neural Network Adaptive Control for Space Robots 39 speed tracking error can be written as _e ¼ ^s ~x e ð8þ According to Eqs. 7) and 8) and Eq. ), the space robots dynamic system error equation can be written as M_^s^s ¼C^s þ h c ð9þ h c ¼ M q r þ _~x~x þ C _qr þ ~x F: The neural network optimal approach of h c ðx ; x Þ can be written as h c ðx ; x Þ¼ f c x ; x ; Wc ¼ W T c ðxþþ" cðxþ: ðþ A3: Network approximation error " c ðxþ and the optimal weight Wc are bounded by k " cðxþ k" c;m, k Wc kw c;m. ^x ¼ ð^x ; ^x Þ is defined, so the estimated output of neural network can be written as f c ^x ; ^x ; ^W c ¼ ^W c T ðþ ^x ðþ The control law is designed as ¼ ^W c T ðþþk ^x p e þ K d _^e^e þ ^v; ^v ¼ k v ^s: ðþ ð3þ where K p and K d are defined as positive definite matrixes and k v >. Further, Eq. 9) can be written as Mðx Þ_^s^s ¼Cðx ; x Þ_^s^s þ Wc T ð ^x; ^w; ^c Þþ" c ðxþ ^W c T ð ^x; ^w; ^c ÞK p e K d _^e^e ^v ð4þ Since Wc T written as ^ ^W T c ^ ¼ W T c ~ þ ~W T c ^, Eq. 4) can be M_^s^s ¼C_^s^s þ Wc T ~ þ ~W c T ^ þ " c K p e K d _^e^e ^v ð5þ The weight adaptive learning law is designed as _~W~W c ¼K Wc ^^s T þ c K Wc k ^s k ^W c : ð6þ where K Wc is defined as diagonal matrixes and c is defined as the positive constant. 5. UUB Analysis Based on Lyapunov Theory UUB can be proved using the following system based on Lyapunov theory. Proof: The Lyapunov function is defined as V ¼ V o þ V c : where V o ¼ ~xt K ~x þ ~xt ~x þ tr V c ¼ et k p e þ ^s T M ^s þ tr ~W T o K W o ~W T C K W c The derivative of V o can be written as _V o ¼ ~x T K _~x~x þ ~x T_ ~x~x þ tr ~W o T _ ~W~W o ~W o ; ð7þ ~W C : ð8þ ð9þ where k ~W o T ^ ¼ d and Wo T ~ þ ~W o T ^ þ " o ðxþ ¼d. Then _V o ¼k ~x T K ~x þ ~x T d ~x T k ~x þ ~x T d þ tr ~W o o k ~x k ^W o The derivative of V c can be written as ð3þ _V c ¼ e T k p _e þ _^s^s _Mðx Þ^s þ ^s T Mðx Þ_^s^s þ tr ~W C T K W c _~W~W C ð3þ According to Eqs. 5) and 6) and P, Eq. 3) can be written as _V c ¼ e T k p _e þ ^s T Wc T ~ þ ~W c T ^ þ " c ðxþ ^s T K p e ^s T K d _^e^e ^s T ^v tr ~W C T ^^s T c k ^s k ^W c ð3þ where d c ¼ Wc T ~ þ " c ðxþ, since _^e^e ¼ ^s e and _e ¼ ^s ~x e. Eq. 3) can be written as _V c ¼e T K p x e T K p e c ^s T K d ^s þ ^s T K d e ^s T ^v þ ^s T d c þ tr ~W C T c k ^s k ^W c : ð33þ Based on ab a þ b Thus _V k ~x T K ~x þ ~x T d þ ~x T d ~x T k ~x xt K px ^s T K d ^s ^s T ^v þ ^s T K d ^s þ et K d e et K p e e T K p e þ ^s T d c þ tr ~W o o k ~x k ^W o þ tr ~W C T c k ^s k ^W c : ð34þ Based on tr A T B k A kk B k and ~W ¼ W ^W Eq. 34) satisfies _V ~x T d ~x T k K ~x ~xt ðk I mm þ K p Þ~x þ ~x T d þ ^s T d c ^s T K d K d þ I mm ^s k v e T K p K p K d e þ o k ~x k k ~W o k k Wo kk ~W o k þ c k ^s k k ~W c k k Wc kk ~W c k : ð35þ where K p K p K d> must be defined, because p k ^ k ffiffiffi n and dc, d and d are bounded. Using A A3, the upper bounds of d c, d and d are defined respectively p as d c;m, d ;M and d ;M. where d c;m ¼ ffiffi n Wc;M þ " c;m, d ;M ¼ pffiffi p k n Wo;M and d ;M ¼ 3 ffiffiffi n Wo;M þ " o;m. Then Eq. 35) can be written as

4 3 Trans. Japan Soc. Aero. Space Sci. Vol. 56, No. 6 n o s _V ks k K s k s kþ c k ~W c k s n o k~x k K x k ~x kþ o k ~W o k x k~x kfk x k x k g: where s ¼ c s þ d c;m, ¼ o x þ d ;M, ¼ d ;M, s ¼ W c;m, x ¼ W o;m, K s ¼ K d K d þ I mm, k v K x ¼ k K and K x ¼ k I mm þ K p. K s >, K x >, K x > is defined. If k s k > rffiffiffiffiffi s s or k ~W c k > þ s K s c k ~x k > rffiffiffiffiffiffi or k ~W o k > þ x K x then k x k > K x ; _V : This shows that _V is negative. Therefore, according to Lyapunov theory all the signals ^s, ~x, ~x, ~W o and ~W c are UUB. 6. Numerical Analysis A simulation study of a planar two-link space robot is presented. The simulation parameters of the two-link space robot are given in Table. Unmodelled dynamics and external disturbances can be expressed as T: F ¼ q _q : sin t; q _q : sin t The desired trajectory can be expressed as q d ¼ : þ :5ðsin :t þ sin :4tÞ q d ¼ : þ :5ðcos :t þ cos :4tÞ: The number of hidden neurons: n ¼ 5. The corrected filter tracking error parameters are as follows. ¼ diag ð5; 5Þ; K ¼ diag ð; Þ; k ¼ ; k ¼ ; K p ¼ diag ½4; 4Š; k v ¼ :5; K d ¼ diag ð6; 6Þ; K Wo ¼ diag ð; Þ; o ¼ :5; K Wc ¼ diag ð; Þ and c ¼ :: The initial conditions of all the states were zero, i.e.: Table. o Parameters of the -DOF space manipulator. m 4 kg a.75 m m kg a.5 m m 7kg b.5 m I 66 kg m b.75 m I.5 kg m b.5 m I.5 kg m q ðþ ¼; q ðþ ¼ ^q ðþ ¼; _q ðþ ¼_^q^q ðþ ¼; q ðþ ¼ ^q ðþ ¼ and _q ðþ ¼_^q^q ðþ ¼: The network initial weight value, basis function width and the basis function center are randomly selected in the group.). Fig. shows the comparison between the desired trajectory q d, q d of two joints and the actual trajectory q, q. Fig. 3 shows the situation of two-joint position tracking errors e ¼ q d q. Fig. 4 shows the situation of the two-joint speed estimation errors _~q~q ¼ _q _^q^q. Fig. 5 shows the output control torque of the two-joint. From the simulation results, if can be found that the designed observer is able to preferably reconstruct the actual Joint trajectory trackingrad) Joint trajectory trackingrad) Joints position errorsrad).5.5 Desired trajectory Actual trajectory Times).5.5 a) Joint trajectory tracking Desired trajectory Actual trajectory 5 5 Times) Fig b) Joint trajectory tracking Trajectory tracking curves of space robots. Joint Joint Times) Fig. 3. Position tracking errors curves of the joints.

5 Nov. 3 W. ZHANG et al.: RBF Neural Network Adaptive Control for Space Robots 3 Observer velocity estimated errorsrad/s) Joint torquen.m) Joint torquen.m) Times) speed information of joints at 4 s, and that the output feedback controller can effectively track the desired trajectory at 5 s. In addition the control torqueses of joints are not large. This indicates that the neural network controller can overcome and compensate for the uncertain effects, and has good robustness. From further simulations, if can be also found that within the proximity of zero, the choice of weight values and initial values for structure parameters of the neural network has little effect on system stability. The learning rate has greater impact on the system. A large learning rate will affect system stability. For control parameters, the observer gain and controller gain become greater, and the approximation error bounds " M of the neural network and k v become smaller. The convergence radius of ^s and ~x become smaller and the tracking effect improves. 7. Conclusion Fig. 4. Speed estimated errors curves. Joint Joint Times) 5 Fig. 5. Control torque of joints. Tracking control problems for space robots were studied under conditions without speed feedback signals. An adaptive neural network control method with a speed observer was proposed in this paper. ) A dynamic model of space robot was established ) A speed observer based on a neural network was designed to reconstruct speed information 3) A controller based on the neural network was designed to approach the unknown model 4) A weights adaptive learning law of the neural network was designed to ensure on-line tuning 5) UUB of the closed-loop system based on Lyapunov theory was proved. The simulation results showed that the proposed neural network control strategy without speed signal was able to achieve higher control precision. This has important value engineering for applications. Acknowledgments This paper was supported by Zhejiang Provincial Education Department Science Research Project No. Y33), Zhejiang Provincial Natural Science Foundation No. LZF) and Zhejiang Provincial Science and Technology Project No. 3C3). References ) Chen, L.: Adaptive and Robust Composite Control of Coordinated Motion of Space Robot System with Prismatic Joint, Proc. of the 4th World Congress on Intelligent Control and Automation, Shanghai, China,, pp ) Zhang, W., Ye, X., Jiang, L., Zhu, Y., Ji, X. and Hu, X.: Output Feedback Control for Free-floating Space Robotic Manipulators Base on Adaptive Fuzzy Neural Network, Aerospace Sci. Technol., 9 3), pp ) Cheah, C. S., Kawamura, S. A. and Lee, K.: H Tuning for Task- Space Feedback Control of Robot with Uncertain Jacobian Matrix, IEEE Trans. Autom. Control, 46 ), pp ) Zhang, W., Liu, W., Ye, X., Zhu, Y. F. and Hu, X. P.: Robust Adaptive Control for Free-floating Space Manipulators Based on Neural Network, J. Mech. Eng., 48 ), pp ) Zhang, W. H., Qi, N. M., Ma, J. and Xiao, A. Y.: Neural Integrated Control for a Free-floating Space Robot with Suddenly Changing Parameters, Sci. China Inf. Sci., 54 ), pp ) Wang, S. D. and Lin, C. K.: Adaptive Control of Robot Manipulator Using Fuzzy Compensator, Fuzzy Sets Syst., ), pp ) Wang, C. H., Tsai, C. H. and Lin, W. S.: Robust Fuzzy Model-following Control of Robot Manipulators, IEEE Trans. Fuzzy Syst., 8 ), pp ) Lin, C. K.: Non-singular Terminal Sliding Model Control of Robot Manipulators Using Fuzzy Wavelet Networks, IEEE Trans. Fuzzy Syst., 6 9), pp ) Yoo, B. K. and Ham, W. C.: Adaptive Control of Robot Manipulator Using Fuzzy Compensator, IEEE Trans. Fuzzy Syst., 8 ), pp ) Zhang, X., Jia, Q., Sun, H. and Chu, M.: The Research of Space Robot Flexible Joint Trajectory Control, J. Astronaut. Chin., 9 8), pp ) Lin, C. K.: H Reinforcement Learning Control of Robot Manipulators Using Fuzzy Wavelet Networks, Fuzzy Sets Syst., 9), pp ) Hong, Z., Yun, C. and Chen, L.: Modeling and Trajectory Tracking Control of a Free-Floating Space Robot with Flexible Manipulators, Robot Chin., 9 7), pp ) Lewis, F. L. and Kim, Y. H.: Intelligent Optimal Control of Robotic Manipulators Using Neural Networks, Automatica, 36 ), pp ) Psillakis, H. E. and Alexandridis, A. T.: Adaptive Neural Motion Control of n-link Robot Manipulators Subject to Unknown Disturbances and Stochastic Perturbations, Proc. Inst. Electr. Eng. Control Theory Appl., 53 6), pp ) Hong, Z. and Chen, L.: Self-learning Control of Space Flexible Manipulator Base on Gauss Function Fuzzy Neural Network, Eng. Mech. Chin., 9 8), pp ) Feng, W. and Postlethwaite, A.: Simple Robust Control Scheme for

6 3 Trans. Japan Soc. Aero. Space Sci. Vol. 56, No. 6 Robot Manipulators with Only Joint Position Measurements, Int. J. Robotics Res., 993), pp ) Li, S., Wan, J., Feng, Z. and Xiao, Y.: A Study on the Uncertain Robust Control of Robot Based on Dynamic Observe, Acta Armamentar II Chin., 6 5), pp ) Berghuis, H. and Lewis, F. L.: Robust Control of Robots Vialinear Estimated State Feedback, IEEE Trans. Autom. Control, ), pp ) Kim, Y. H. and Lewis, F. L.: Neural Network Output Feedback Control of Robot Manipulators, IEEE Trans. Robotics Autom., 5 999), pp ) Niu, Y. G., Wang, X. Y. and Hu, C.: Neural Network Output Feedback Control for Uncertain Robot, Proceedings of 4th World Congress on Intelligent Control and Automation, Shanghai, China,, pp ) Manoj, M., Chuang, C. H. and Juan, J. N.: Nonlinear Control of Space Manipulators with Model Uncertainty, AIAA, the American Institute of Aeronautics and Astronautics, 994, pp ) Xu, Y. S. and Kanade, T.: Space Robotics: Dynamics and Control, Kluwer Academic Publishers, 99. 3) Xie, J., Liu, G. I., Yan, S. Z., Xu, W. F. and Qiang, W. Y.: Study on Neural Network Adaptive Control Method for Uncertain Space Manipulator, J. Astronaut. Chin., 3 ), pp ) Ortega, R. and Spong, M. W.: Adaptive Motion Control of Rigid Robots: Atutorial, Automatica, 5 989), pp