Calculation of Model of the Robot by Neural Network with Robot Joint Distinction

Transcription

1 Calculation of Model of the Robot by Neural Network with Robot Joint Distinction J. Możaryn and J. E. Kurek Warsaw University of Technology, Institute of Automatic Control and Robotics, Warszawa, ul. Sw.Andrzeja Boboli 8, , Abstract. There is presented the design of the feedforward neural network for calculation of coefficients of the robot model. Proposed method distinguishes the degrees of freedom and improves the performance of the network using information about the control signals. A numerical example for calculation of the neural network model of Puma 560 robot is presented. 1 Introduction Mathematical model of industrial robots can be calculated using the Lagrange- Euler or d Alambert equations [1]. However, it is rather difficult to obtain the data of physical properties of the robot: inertia momentums, masses, etc. without disassembling of the robot. The mathematical model of robot is highly nonlinear and it is very difficult to identify its coefficients. For the reason the neural networks can be used for calculation of the model coefficients [3], [5], [6]. This technique has advantages such as approximation and generalization. One of the problems in designing the neural network structures for robot model calculation is the distinguishing between the degrees of freedom. This information is significant for proper identification of the coefficients of equations describing the dynamics of each degree of freedom. In the paper we present the method for the neural networks design using data from the positions of robot links and control signals. The proposed method can be used for coefficients identification for each degree of freedom of robot. The organization of the paper is as follows. The discrete time robot model based on the Lagrange-Euler equations is given in the Sect. 2. Then, in Sect. 3 and Sect. 4 the neural network structures for identification of the robot model coefficients are presented. In the Sect. 5 the computer simulations are described. The conclusions are given in Sect Discrete Time Robot Model The discrete time model of robot with n degrees of freedom, based on Lagrange- Euler equations can be presented as follows [6]

2 q(k + 1) = T 2 p M 1 (k)(τ(k) V (k) G(k)) + 2q(k) q(k 1), (1) where τ(k) R n is a vector of input signals, q R n is a vector of generalized joint coordinates, M(k) = M[q(k)] R n n is a robot inertia matrix, V (k) = V [q(k),q(k 1)] R n is a vector of Coriolis and centrifugal effects, G(k) = G[q(k)] R n is a vector of gravity loading, k is a discrete time, T p is sampling time, t = kt p. 3 Neural Network Robot Model Using (1) and the following notation E(k) = [e i (k)] n 1 = T 2 p M 1 (k)(v (k) + G(k)) + 2q(k) q(k 1) (2) X(k) = [x ij (k)] n n = T 2 p M 1 (k) (3) the equation for m-th degree of freedom can be written as follows q m (k + 1) = e m (k) + n x mi (k)τ i (k) (4) where e m (k) = e m [q(k),q(k 1)] and x mi (k) = x mi [q(k)]. The neural network structure which will be used for calculation of the model (4) coefficients is shown in the Fig.1. i=1 4 Neural Network Robot Model with Robot Joint Distinction Usually it is impossible to distinguish which degree of freedom is modelled using the neural networks structure based on (4). The trajectories in each degree of freedom used in training process can be the same, and therefore each structure of neural networks does not model the exact coefficients of (2). In order to distinguish the degrees of freedom there is proposed to consider the additional information about the control signal of the link which is modelled. This information is unique for this degree of freedom. Therefore the both sides of (4) we multiply by the control signal τ m (k) for the modelled joint q m (k + 1)τ m (k) = e m (k)τ m (k) + n x mi (k)τ i (k)τ m (k) (5) This equation will next be used for calculation of robot model coefficients. The neural networks structure for calculation of coefficients of the (5) is shown in the Fig.2. i=1

3 5 Computer Simulations In order to train and test the proposed neural network there were generated the data of inputs and outputs of the robot Puma 560 with 6 degrees of freedom [4]. The robot was simulated with given trajectory in time interval T = 10[sec], with sampling time T p = 0.01[sec]. Thus, there were 1000 data samples for training and 1000 data samples for testing of the neural models. For calculation of the training and testing data, the trajectory for every joint was set according to the following formula: q i (k) = acos(sk) + a + q i2 if q i1 > q i2 q i (k) = acos(sk + π) + a + q i1 if q i1 < q i2 (6) q i (k) = q i1 if q i1 = q i2 where a = qi1 qi2 2, q i1 is the start position in the link i, s = 2π Tp T. The values of the q i1, q i2 are different for the training and testing trajectories and are given in the Table 1. The training and testing data are presented in the Fig.3 and Fig.4. Table 1. Values of the lowest and highest positions in every degree of freedom links link 1[ ] link 2[ ] link 3[ ] link 4[ ] link 5[ ] link 6[ ] q 1, training trajectories q 2, training trajectories q 1, testing trajectories q 2, testing trajectories Both models, that are presented in the Fig.1 and Fig.2, were trained with the same number of neurons. In all nonlinear layers (NL) the neurons described by the sigmoid function (7) were used: y = f nl (x) = 1 1 (7) 1 + e x In linear layers (L) there are neurons described with the linear function: y = f l (x) = x (8) There were 2 neurons in each nonlinear layer and 1 neuron in each linear layer. Neural networks were trained using the backpropagation method and the Levenberg-Marquardt method to update weights in all layers [2]. The performance of each neural network model for each degree of freedom of the robot was checked. Good performance was obtained, if the input-output data for both neural network structures was the same. Thus, in the neural network from Fig.2 the control signal τ m (k) = 1. The quality index for the models was the

4 maximum absolute difference between the trajectory in i-th link q i and output of the network y i : Q i = max y i (k) q i (k + 1), (9) where k is the number of the data sample. The values of Q i for neural network models of (4), (5) and each degree of freedom of robot Puma 560 are given in the Table 2. Table 2. Values of the neural networks quality indexes Q i Q i link 1[ ] link 2[ ] link 3[ ] link 4[ ] link 5[ ] link 6[ ] NN Model of (4), training trajectories NN Model of (5), training trajectories NN Model of (4), testing trajectories NN Model of (5), testing trajectories Concluding Remarks The results obtained during the simulations show, that presented neural network models which use information from control signals have good approximation properties for the training data. Unfortunately if the testing data are used the results are not very accurate. We plan further research in order to improve the model. References 1. Fu K. S., Gonzalez R. C., Lee C. S. G. Robotics: control, sensing, vision, and inteligence, McGraw-Hill Book Company, (1987) 2. Osowski S. Neural Networks, OWPW, Warszawa (In Polish) 3. (1994) Kurek J. E.(1998) Neural Net Model of Robot Manipulator, Proceedings Neural Computing NC 98, Vienna, Austria, (1998) 4. Corke P. I. Matlab Robotics Toolbox (release 5), CSIRO, Australia, (1999) 5. Kurek J. E. Calculation of Robot Manipulator Model Using Neural Net, European Control Conference ECC 99, Karlsruhe, Germany, (1999) 6. Możaryn J., Wildner C., Kurek J.E. Calculation of the Model of the Industrial Robot Using Neural Networks, XIV National Control Conference KKA 2002, Zielona Gora, Poland, (2002)(In Polish)

5 Fig. 1. Feedforward neural network structure proposed for calculation of coefficients of (4) Fig. 2. Feedforward neural network structure proposed for calculation of coefficients of (5)

6 Fig. 3. Training trajectories in every degree of freedom Fig. 4. Testing trajectories in every degree of freedom