Natural Frequency Estimation of a Free Vibrated Robotic Arm Using Artificial Intelligence

Transcription

1 Natural Frequency Estimation of a Free Vibrated Robotic Arm Using Artificial Intelligence 1 Israa Rafie Shareef, 2 Iman Ahmed Zayer, 3 Mohammed Yahya Edrees 1, 2 &3 Department of Mechatronics Engineering, Al Khwarizmi College of Engineering, University of Baghdad, Baghdad, Iraq. Abstract This paper studies the natural frequency of a robotic arm in free vibration case ; firstly analyzing it using the classical analytic method with the well- known Euler Bernoulli Theory. Then, simulation method using one of the artificial intelligence ways presented by the neural network and applying it through MATLAB programming. For the purpose of the analysis; the fabrication materials will be selected according to the recently used materials, in the field of robots manufacturing like PLA and ABS. These materials' properties will represent the inputs of neural network, to be treated, and find its effect on the natural frequency of the robotic arm. Finally, Comparison between the two methods to calculate the error percentage between them. The results show that the neural network performances is optimum, in term of minimizing the analysis time cost and simplify the works. Keywords: Natural Frequency, Robotic Arm, Neural Network, PLA & ABS materials. INTRODUCTION Every system in nature has an infinity number of vibration frequencies and mode shapes, the estimation of these frequencies and their mode shapes is essential to solve the vibration phenomenon which is induced in engineering domain problems [1]. In fact, analyzing the system's vibration is a critical step required in designing miscellaneous systems [2]. Increasing interest is noticed in artificial intelligence controlled systems like using neural networks, to reduce the complicated computations of robotic arms and manipulators besides their behaviors like natural frequencies to deal with the nonlinear systems that would be difficult to model mathematically [7,8]. Also, the ability to process control computers and data makes it easy to generate neural network solutions for modeling and controlling. Neural Network (NN) solutions are accepted in controlling systems since they are cost-effective, well-understood, nonlinear (especially the multi layered neural networks), datadriven, and parallel distributed structure with learning ability [9,10]. Many of neural networks for systems control are feed forward networks such as the multilayer. The Recurrent neural networks with feedback connections, such as the Hopfield networks, have also been applied for kinematic control[7].a supervised teach NN is used in this paper as a learning rule. In this study, f n which represents the natural frequency will be computed theoretically depending on the principle of Bernoulli-Euler beam theory, where the robotic arm will be viewed as a single supported cantilever beam carrying a concentrated mass M at its free end [3]. The study of the robotic arm as a cantilever beam is useful in knowing the oscillation amplitude and the frequency of flexible tool. Predicting natural frequency of system's parts is available using different software programs like neural networks, requiring knowledge of the properties for the materials used besides geometry of the parts [4]. ROBOTIC LINK MODELING As mentioned previously, the robotic link will be viewed as a single supported cantilever beam carrying a concentrated mass M at its free end. The analysis of beams has been performed over the years mostly using Bernoulli Euler beam theory [5]. Where in Bernoulli Euler beam theory, the shear deformation is negligible because the ratio between length and thickness is large enough to reduce the error and the results will be acceptable. EI d4 W(x) d x 4 - ρaω 2 W(x) = o...(1) Where ω is the circular natural frequency of vibrated system, W(x) is the normal mode or modal displacement, β = ρaω2 EI.....(2) W(x) is the original load, W(x) is the slope, (W(x) ) is the shear moment, while the fourth differentiation of the load((w(x) ) ) will be back to its origin. ω = β 2 EI ρa....(3) The robotic link will be considered as a single supported cantilever beam with a concentrated mass M at its free end for its tool or end effector. 5393

2 THE CHOSEN MATERIALS In this work, multi materials will be selected for robotic link fabrication like Stainless Steel and Aluminum. Besides, the PLA (Poly lactic Acid) and ABS (Acrylonitrile Butadiene Styrene) are added because of their importance in robots manufacturing domain. PLA & ABS, both known as thermoplastics, they become soft and moldable when heated, and return to solid when cooled, this process can be repeated. ABS is distinguished due to its strength,machinability, flexibility and higher temperature resistivity. While PLA is available in different colors, it has higher printing speeds, sharper printed corners, lower layer heights and low warping [6]. With PLA and ABS, the density can be controlled according to the desire and requirement off the work using the three dimension printing. The density of PLA & ABS could be changeable in the three dimension printing, thus strength will be changeable alternatively and it can be loaded according to this property. The Training Process Flow Chart The following figure represent the process of training using the N.N to find the first four circular natural frequency of robot link in case of free vibration where, MSE: (mean square error), W: (weight), b:( bias). Figure 1: The Training Process Flowchart 5394

3 ANALYTIC METHOD RESULTS The robotic link which is treated as a beam with concentrated mass (tool or gripper), analyzed according to Euler-Bernoulli Equation to get its circular natural frequency for its four circular natural frequencies.as shown in table (1). For the materials stainless steel, aluminum, PLA & ABS, the rectangular and circular sections were taken, I(kg.m^2) is the moment of inertia, L(m)is the proposed length of the link, where two cases of lengths were studied, m(kg)is the concentrated mass which could be changed also for further study and data A (m^2) is the area of the section, ρ (Mg/m^3) is the density, ω (Hz) is the circular natural frequency. of ABS was considered as 1.21Mg/m^3, and low density ρ=1.01mg/m^3, while PLA was considered with 1.25Mg/m^3. No. Material Type Table 1: Robotic link Parameters and Analytic Results Data Section Shape I L M A ω 1 w 2 ω 3 ω 4 1 Stainless steel rectangle 4.8 e e Aluminum rectangle 4.8 e e PLA rectangle 4.8 e e ABS 5 ABS rectangle 4.8 e e rectangle 4.8 e e Stainless steel Circle 5.4e e Aluminum Circle 5.4e e PLA Circle 5.4e e ABS 10 ABS Circle 5.4e e Circle 5.4e e Stainless steel rectangle 4.8 e e Aluminum rectangle 4.8 e e PLA rectangle 4.8 e e ABS 15 ABS rectangle 4.8 e e rectangle 4.8 e e Stainless steel Circle 5.4e e Aluminum Circle 5.4e e PLA Circle 5.4e e ABS 20 ABS Circle 5.4e e Circle 5.4e e Figure (2) show the first four mode shapes for the robot link after normalizing the circular natural frequency to one, noticing that the natural frequency will be approximately doubled for each mode. 5395

4 Figure 2: First four mode shapes. Artificial Neural Network (ANN) A learning algorithm is an adaptive method by which the network will self-organizes to reach the desired behavior. This is happened in some learning algorithms by introducing examples of the desired input/output of the network. The correction step will be executed iteratively until the network learns to produce the desired response alone. The learning algorithm can be represented as shown below Figure 3: Learning process in a parametric system [ 11] While, the following figure represents the general model of ANN followed by its processing Figure 4: The general model of MIMO ANN[12] 5396

5 For the previous general model of ANN, the net input could be calculated as shown in the following equation: y in = x1.w1 +x2.w2 +x3.w3 + + xm. wm i.e., Net input yin = Σ xi.wimi The output would be calculated by applying the activation function over the net input. Y= F(yin) Output = function (net input calculated) b is the bias [12] Each of the outputs (circular natural frequencies ω1, ω2, ω3 and ω4) would be the results of the neural network, according to the input parameters (material type, robotic link cross section shape and its area,moment of inertia, concentrated mass value,and the length of the proposed robotic link). Thus a MIMO (multi input multi output) neural network is to be processed using the nntool (neural network toolbox) GUI (graphical user interface) application in MATLAB software. Each of the inputs and outputs were inserted as matrices of 4 20 (4 variables, 20 data samples) as in figure (5): Figure 5: The designed GUI of the proposed neural network. Where a good training performance is obtained as in figure (6),(7), knowing that this performance could be enhanced using various ways, just like changing the neural network type, updating the weights, or changing the training, adaption learning, transfer, and performance functions. Even the number of layers and neurons could be changeable till reaching to the acceptable training performance. In fact, increasing the data itself will increase the performance effectively. Figure 6: The neural network training performance 5397

6 Figure 7: The training state for the proposed N.N. The best regression is shown in figure (8), where good convergence between outputs and targets were gained through neural network training. Figure 8: Best Regression for the N.N. Neural Network natural frequencies of the robotic link As the circular natural frequencies of the proposed robotic link were calculated using the Euler Bernoulli theory, those results were considered as targets in THE designing of the neural network, and after training, the outputs which act as the neural network results, these last results would be compared with the theoretical ones that had been found before. Table 2 : Error Percentage between theoretical and simulated neural network for the first circular natural frequency No. Theoretical frequency Simulation NN frequency Error Percentage *100%

7 Best Training Series1 Theoreticalfre Series2 Neural network freq. Figure 9: Comparison between theoretical and N.N frequencies. 0 A very acceptable error percentage was noticed between theoretical and neural network results of the first circular natural frequency as shown in figure (9), and this is right also for each of the second, third, and fourth circular natural frequencies. CONCLUSION In this study, a model of a robotic link is presented to be analyzed for its free vibration response and get its natural frequencies, firstly adapting the traditional Euler Bernoulli method, and secondly simulating one of the artificial intelligence method which is neural network method. The proposed ANN was noticed to be trained in a good manner and acceptable solutions were achieved, the differences between actual and predicted target values using this neural network show that reasonable values of error were gained, So, the free vibration natural frequencies' results showed the usefulness and effectiveness of the proposed neural network, where less computing time and cost were reached besides taking the recent trend of using artificial intelligence methods to replace the classical ones. For the purpose of the another work part will be important which is the experimental part, by building the robotic arm from different materials and different physical properties to get natural frequencies' results. REFERENCES [1] C. Y. Wang and C. M. Wang, 2014, Structural Vibration, Exact Solutions for Strings, Membranes, Beams and Plates, CRC Press, Taylor and Francis Group, Boca Raton,. [2] J. A. HOFFMANN AND T. WERTHEIMER, 2000, CANTILEVER BEAM VIBRATION, Journal of Sound and 229(5), [3] B. Karlik, E. Özkaya, S. Aydin and M Pakdemirli, 1998, Vibrations of a beam-mass systems using artificial neural networks, Comput. Struct., vol. 69, pp [4] J. A. HOFFMANN, 1999, CANTILEVER BEAM VIBRATION, California. [5] Yusuf Yesilce, 2011, Free vibrations of a Reddy-Bickford multi-span beam carrying multiplespring-mass systems, IOS Press, Izmir, Turkey, Shock and Vibration [6] Makerbot PLA and ABS Strength Data, July 2015, Available at: Inhalte/Support/Datenblatt/MakerBot. [7] Youshen Xia and Jun Wang, February 2001,"A dual neural network for kinematic control of redundant robot manipulators", IEEE Transactions on systems, Man and cybernetics - PART B, Vol. 31, no.1, pp [8] Leila FallahAraghi, M. H. koorayme, March (2009)"Neural network controller for two links- robotic manipulator control with different load", Proceedings of the International Multi Conference of Engineers and Computer Scientists 2009, Vol. II, IMECS, Hong Kong, pp.978. [9] Omid M. Omidvar and David L. Elliott, February, 1997, Neural Systems for Control, Academic Press, Elsevier, pp [10] Zhao-Hui Jiang, March 2010, Hiroshima Institute of Technology, "Trajectory control of robot manipulators using a neural network controller", Robot Manipulators Trends and Development, pp [11] Raul Rojas, 1996, Neural Networks A systematic Introduction Springer Berlin. [12] Artificial neural network tutorial points, 2017, available at www. Tutorialspoints.com. 5399