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[Elearnica.ir]-Adaptive ANN Control of Robot Arm Using Structure of Lagrange Equation

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  Proceedings of the American Control Conference San Diego, California June 1999 Adaptive ANN Control of Robot Arm Using Structure of Lagrange Equation Masaki YAMAKITA and Takashi SATOH Department of Control and Systems Engineering Tokyo Institute of Technology Abstract Several adaptive control algorithms for robot manipu- lators using artificial neural network ANN) have been pro- posed and an algorithm using physical structure of robot manipulators also has been proposed. But the structure of ANN should be modified according to the change of the de- gree of freedom of the system. In this paper, we propose the algorithm in which structure of ANN is independent of the degree of freedom and compare it to some other algorithms by simulation and experiment. 1 Introduction Adaptive control algorithms for robot manipulators have been studied for a decade, e.g., [2] and they usually use a property that the dynamic equation of the system can be represented by a multiplication of a known regressor matrix and an unknown dynamic parameter vector. It is not easy, however, to obtain the regressor matrix if the kinematic pa- rameters are unknown. Several adaptive control algorithms using Artificial Neural Network ANN) have been also pro- posed to overcome the problem. The explicit calculation of the regressor matrix is not required in these algorithms [l] [3]. The learning capability of the network was very low be- cause the structure of the dynamic equations was not taken into account. In [4] an adaptive control algorithm which considers the structure of the property has been proposed, however, since the allocation of the neural elements was fixed, the structure of the neural network should be modified according to the degree of the system and neural elements may not be used effectively. In this paper a new adaptive ANN controller which over- comes the problem is proposed. It uses a structure of dy- namic energy of the system. The efficiency of the proposed method is compared to other constructions of the neural net- works by the numerical simulations and experiments. 2 Structure of ANN For computational efficiency we use RBF Radia1 Basis Function ) neural network which is known that it can ap- proximate any continuous non-linear function with any ac- curacy for a bounded set of the d0main.A neural network 4~ which has n input, one output and p functions can be represented by where B E Rp,< E Rp,p; E R . Of course, multi output functions can be represented by a combination of the net- works. We introduce the following assumption as in [3]. [Assumption] Let 4 ~) e a continuous function to be estimated and as- sume that there exists a known matrix function Y, satisfying in the domain where 11 11 stands for a matrix induced norm or the Frobenious norm and & is an unknown constant matrix. Please notice that this condition is satisfied by the RBF neu- ral network and also by three layers neural networks if we assume bounds of weighting matrices as in [3]. 3 Design of Controller We consider the system whose dynamic equation is repre- sented by M(q)i + (C(q7 4 + D)i + G(q) = 5) where q is a generalized coordinate vector, is a general- ized force vector and M(q), C q, , D, G(q) are inertia, Col- ioris/centrifugal and damping, gravity terms, respectively. Let qd, qd, i[d E L, be desired trajectories and e := q- qd, s := i+Ae (A > 0), r := qd he. If we use a control input defined by where B = -r eYT(q, 41 r)s, re > 0 8) = raYallSll, r > 0, 9) and 8 is estimated function of 8, then q, converges to the desired signal qd,qd. The convergence of the error can be shown as in [l] 3] under the assumption. That is proved in appendix. 0-7803-4990-6199 10.00 0 999 AACC 2834 Downloaded from http://www.elearnica.ir  as .......... GONTlL0LLE-R; :r-- Figure 1: Control System The remained problem of the control design is how to con- struct 4~. n the following 4 alternative methods are com- pared and Method 3 and Method 4 are methods proposed in this paper. Method1 [l] 4~ is constructed by a single NN with the knowledge of passivity structure of 4, and it is constructed by and d, G can be derived as in Method 2. Please notice that the allocation of the neural elements is not fixed in this Method. k s constructed by a multiplication of estimated ma- trices and neural elements and the potential function U s constructed by the another NN as Met ho d4 proposed) n M = xkits i l O = eG<C. where A? is a matrix. The block diagram of .the, net- work is equal to that of Method2.) Though C, can be derived as in Method 2, usually the number of the estimated parameters increases compared to Method3. 4 Simulation and Experiments noted shadow area in Figure 1 has no structure in this Method. In the following Methods the knowledge of the structure, JN = k d + D)qr + G is used for the construction of ANN, Method2 [4] Each element of M(rn,,) is approximated by each NN and potential function U s done by another NN. C q, 4 and G q) are computed by partial derivative operations of M q) and V(q) as Method3 proposed) To use full structure of the whole energy function of the system, first kinetic energy h and potential energy U are constructed by two NNs, and the required ele- ments are calculated by partial derivative operations 4.1 Computer Simulation Each algorithm in the previous section is applied to a two- link mainpulator system in a vertical plain. In the following figures, the responses to the desired angle and anglar ve- locity are shown. The center of the each RBF is specified randomly in the operation range of the desired signal and the total number of RBFs in each Method is 100 n the sim- ulations. From the figures it can be seen that Method2 - give bet- ter performance. The Method3 gives the best performance for the velocity tracking from the Figure 6 - 9. 4.2 Experiment We applied each algorithm to an industrial robot arm under the same condition. It was observed that Method2, 3, 4 give better perfomance, however, the differences between Method2, 3 and 4 are not significant. The reason may be in- fered as that dumping effect in the real robot arm was very big and dominant. 5 Concluding Remarks In this paper an effective way to construt regression vec- tor for an adaptive controller using a neural network has been proposed. The proposed Methods were compared with other Methods by numrical simulation and experiments, and it was convinced that the proposed Method gives good track- ing performance. In the practical applications robustness of the adaptive scheme, e.g., sigma modification so on, should be combined. In the propoed Method RBF neural network was used for comutational simplicity, however, the perfomance of the 2835  network depends on the allcation of the center of each basis function, pi. Further study shoud consider adaptive alloca- tion of the center, and also another type of neural network, e.g., three layered neural network shold be compared to the RBF neural network. APPENDIX A.1 The Proof of Stability where Choose a candidate of Lyapunov function V = Vi + %, where = 8 , = & . The time derivative of VI an be derived as 2 = sTMi+-sTMs 2 = sTM(i &) + -sTMs 1 2 = sT(r C D)q G Mir) + -sTMs ST(-&& 6 + b)qr sT(-Y DS + v KdS) -sTDs TKds + STV STY TI = = = DS + v KdS) [3] Wei-Der Chang, et. al: “ADAPTIVE ROBUST NEURAL-NETWORK BASED CONTROL FOR SISO SYSTEM” IFAC,13th Triennial World Congress, San Fran- cisco,USA, 1996) [4] Shuzhi S Ge: “ROBUST ADAPTIVE CONTROL WARKS” IFAC,13th Triennial World Congress, San Fran- cisco,USA,1996. OF ROBOTS BASED ON STATIC NEURAL NET- Simulation results The respances for desired angle Figure 2 - 5) 5 5 -sTDs TKds + sTv Ty + ~ -sTDs TKds + sTv TY + BTYalls/l. Figure 4: Method3 Figure 5: Method4 The respances for desired anglar velocity Figure 6 - 9) he time derivative of V2 can be also calculated as r2 = iTrg-fB+ Tr -l a -8Tre-1j Tra-li. ~~~~~ , ?ad sec Q ad SCE am m tm >m &m Im am om Im ,m >m uo xm m method 4 10 Therefore the time derivative of V is am v = Vi+ ,a am am -sTDs STKds + sTv Tra-iii + ‘YfiIIsll 1.4 I 4 = -sTDs TKds + sTv GTre-’ii + &‘Y&lls11 Figure 6: Method1 Figure 7: Method2 +tiT Yn 1311 = -sTDs TKds :R;R d rad SCC In a am to the transfer function from s to e. Furthemore it can be om nm zm >m <m x ~0 om ~m tm rm ~m ~m method 0. 4 O Vis negative semi-difinite. Hense s, , are bounded and s E L2. s - as 1 - CO because e E L2 n L,, i E L2 accoding 4 1 11 [ 4 seen that d 0 because L E L, from the boundedness of i. 0 Figure 8: Method3 Figure 9: Method4 References [I] F.L.Lewis, et. al: “Neural Net Robot Controller: Structure and Stability proofs.” Proc. of IEEE CDC’93, 2785/2791 1993) [2] Jean-Jacques E.Slotine Weiping Li: “On the Adap- tive Control of Robot Manipulators” Int. Jontal in Robotics Reserch, vo1.6 49/59 (1987) 2836
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