Vibration of a robot arm

Transcription

1 Vibration of a robot arm Livija Cveticanin Faculty of Technical Sciences, Trg D. Obradovica 6, Novi Sad, Serbia cveticanin@uns.ac.rs Abstract Robot arm represents the fundamental working element of the most robotsand manipulators. To enlarge the efficiency of the robot system the velocity of the arm has to be increased. Unfortunately, it causes vibrations whose intensity increaseswith working velocity of the arm. Vibrations have negative effect on the basic motion. Namely, a useful part of energy, which has to be used for the working process, is dissipated due to oscillations. Besides, vibrations decrease the accuracy of the realization of the task. It is the reason that vibrations of the arm have to be analyzed and eliminated. If during the working process axial vibrations occur, the arm is modeled as a clamped-free beam. Mathematical model of the motion is a second order partial differential equation. Due to nonlinear properties of the robot arm, the model is strong nonlinear. An analytical solving procedure for the problem is developed. The method is based on the generating solution of thestrong nonlinear ordinary second order differential equation. Applying the suggested procedure, frequency of the robot arm with pure cubic nonlinearity is determined. Dependence of the frequency of vibration on initial and boundary conditions is proved. output mechanism are taken into consideration. Vibrations which appear in robot arm are usually of axial type. They have a negative effect on the accuracy of the working process of arm. Because of that, vibrations have to be eliminated or put into limits. In the papers [21]-[23] axial vibrations of a beam are investigated. In this paper the obtained results will be included for solving the aforementioned problem. II. MODEL OF THE ROBOT ARM In Fig.1, a surgery robot arm is shown. It can be modeled as a clamped-free beam as one end is free, while the other is fixed (Fig.2). Cross-section properties of the beam are smaller than its length. The robot-arm moves right-line and the most important accompanying vibrations are in axial (longitudinal) direction. Keywords Robot Arm; Vibrations; Nonlinearity;Mathematical model; Clamped-free Beam I. INTRODUCTION Robotic arm is fundamental working part of robot which fulfills variety types of tasks. Robot arm has a wide application in industry [1] for cutting, painting, welding [2], in underwater operations [3], for snapping (used as camera-robot arm) [4], etc. However, the most responsible task for the robot arm is in surgery. Nowadays, for a significant number of medical treatments the robot arms are used. Thus, in vascular surgery the mechanical robot arm is applied [5]. Passive robot arm is suitable for orthognathic surgery planning [6] and for orthognathic model surgery [7]. Development of robot arm improves the mandible reconstruction surgery [8]. The robot for trans nasal surgery has needle-sized tentacle-like arms [9]. Robot arm may assist in renal surgery [10], [11] and liver surgery [12]. Robot arm is used in laparoscopic surgery, too [12]. Unfortunately, recent investigation report about perioperative complications of robot-assisted laparoscopic surgery which use robot arms [13]. It is stated that the problem is with the accuracy of the mechanical system. To exceed the problem the control of the robot arm has to be improved. Results of investigation in robot control (see for example, [14]- [17]) have to be incorporated into control of robot arm. Analyzing the state of the art in dynamics of mechanical arm, it can be concluded that some additional investigation are necessary which will include nonlinear properties of the system and also vibrations which accompany the motion. In the recently published paper [18]-[20] the nonlinearities in the Fig. 1. Surgery robot arm. Axial deflection u depends on time t and position x. analysing the elementary part with length dx and mass ρadx, where ρ is the density of material of arm, A is the cross-section the inertial force is ρadx. Usually the material of the robot arm is with strong nonlinear properties and the stress-strain relation is, where E is the elasticity coefficient, ε is the deformation, and α is the order of nonlinearity obtained experimentally. Elastic force is. Equating the elementary elastic force df and the elementary inertial force the equation for longitudinal vibrations are obtained The boundary conditions are 31

2 wherel is the length of the beam. For p(x)=dx/dx=x and d 2 X/dx 2 =p p=p(dp/dx), equation (7) transforms into a first order equation which solution is [ ] Finally, [ ] wherek 1 and K 2 are arbitrary constants. For kx/ =s have, we Fig.2. Model of the clamped-free beam. Mathematical model is a second order partial differential equation with strong nonlinearity. To give a valid analysis of the beam motion, it is necessary to solve this equation. Unfortunately, it is not an easy task. III. SOLVING PROCEDURE Let us introduce a solution which separates the variables Substituting (3) into (1) and (2) we have ( ) and wherek 2 is an unknown constant value, =(E/ρ)k 2, and X =dx/dx. After some transformation of (4), two ordinary strong nonlinear differential equations follow Our task is to solve equation (6) 2 and to calculate the constant k according to the boundary conditions (5). It has to be outline, that k is a necessary value for calculating of ω but we can denote the frequency of vibration only if the solution of the nonlinear equation (6) 1 is determined. It represents a significant difference in comparison to solving of the linear differential equations. A. Solving of the equaiton with displacement function We rewrite the expression (6) 2 as [ ] and, in general, the solution is X=Φ(x,K 1,K 2,k). Applying the boundary conditions (5), we obtain k as a function of constants K 1 and K 2. This property of the constant k is due to nonlinearity of the system. If the system is linear, constant k is independent on K 1 and K 2, as it will be proved. Remark 1: For the linear case, when α=1, we obtain the known solution (11) For the boundary condition (5) 1, we have sin(kk 2 )=0, i.e., K 2 =0. Substituting (12) into (5) 2, the frequency equation follows which is satisfied for wheren=1,2, In general, the X function is wherek n satisfies the relation (14). Remark 2: The rewritten Eq.(7) is ( ) and the corresponding first integral 32

3 Using the boundary conditions (5), we obtain ( ) Using the expression (16), values of K 1 and K 2 and their relation is obtained. B. Solving of the equation with time function One of the most important properties which have to be determined for the oscillating system is its frequency. As equation (6) 1 is strong nonlinear, the frequency of vibration depends on the initial values To obtain the frequency we need not to know the period of vibration. Using the procedure presented in [24] and [25], we transform (6) 1 into Relation (18) corresponds to an oscillator with exact period of vibration i.e., ( ) Frequency of vibration strongly depends on the initial and boundary conditions of the beam and also on the order and coefficient of nonlinearity. Eq. (6) 1 has an exact solution in the form of the Ateb function [25]. Nevertheless, in this paper the approximate solution in the form of a harmonic function is assumed (see [24]) with amplitude T 0 and frequency ω. C. Approximate solution Using the solutions (15) and (26) with (14) and (25) the approximate solution of (1) is the sum ( ) ( ( ), Substituting, relation (19) gives with the first time derivative ( ) Integrating (20) and applying the definition of the complete beta function ( ) ) For initial values and using the condition of orthogonal functions we obtain the period is obtained ( ) For, where Γ is the gamma function, and Γ(1/2)=, we have while (29) 2 is identically satisfied. IV. BEAM WITH PURE CUBIC NONLINEARITY Let us assume that the nonlinearity is pure cubic and the mathematical model of the axial vibration of the beam is ( ) Based on (23) the frequency of vibration in axial direction is obtained ( ) According to the suggested procedure (31) is rewritten into two ordinary differential equations with separated variables. Specifying the nonlinearity, the equation (9) is [ ] 33

4 Modifying (10) into the form nonlinearity and coefficient of nonlinearity but also on the initial and boundary conditions. It is a quite new result in solving the nonlinear partial differential equation. Future investigations have to prove the result numerically. we have the approximate solution and its derivative [ ] ) * + Introducing the boundary conditions into (34) and (35), we obtain [ i.e., n solutions for k are obtained ( ) wheren=1,2, According to (37), n frequencies of vibrations follow ( ) Frequencies depend on initial constants. Approximate solution for time variable cubic function is [ ( ) + Due to (34), (37) and (39) the axial vibrations are [ ( ) + ( ) [ ( ) ]. (40) As it is previously presented, constants in (40) are calculated according to the initial conditions (29). Then, the obtained values for constants are substituted into the frequency relation (38). V. CONCLUSIONS Axial vibration of the robot arm is considered. System is assumed to be nonlinear. Mathematical model of the motion is given with a strong nonlinear partial differential equation. In spite of the nonlinearity the solution of the equation is a product of two functions which depend on two independent variables: a displacement and a time function. Due to nonlinear properties of the system the constant of separation depends on the order of nonlinearity and on boundary conditions. Besides, the frequency of vibration is also dependent on the order of REFERENCES [1] Haikal A.Y., Elhosseini M.A., A smart robot arm design for industrial application, Studies in Infromatics and Control, 23(1), , [2] Zhai J., Ying C., Zhang T., Xu X., Optimization on two cooperative robot arm s trajectory for welding, Transactions of China Welding Institution, 36(1), 91-95, [3] Ambar R.B., Sagara S., Imaike K., Experiment on a dual-arm underwater robot using resolved acceleration control method, Artificial Life and Robotics, 20(1), 34-41, [4] Ognibene D., Baldassare G. Ecological active vision: Four bioinspired principles to integrate bottom-up and adaptive top-down attention tested with a simple camera-arm robot, IEEE Transactions on Autonomous Mental Development, 7(1), , 3-25, [5] Wengerter K.R., Veith F.J., Self-retaining retraction techniques for vascular surgery. Use of a mechanical robot arm, Journal of Vascular Surgery, 8(1), 14-20, [6] Theodossy T., Bamber M.A., Model Surgery with apassive robot arm for orthognathic surgery planning, Journal of Oral and Naxilofacial Surgery, 61(11), , [7] Omar E.A.Z., Bamber M.A., Orthognathic model surgery by using a passive robot arm, Saudi Dental Journal, 22(2), 47-55, [8] Zhao H., Duan X., Development and experiments of a multi-arm robot for mandible reconstruction surgery, Information (Japan), 16(6B), , [9] Gilbert H., Hendrick R., Remirez A., Webster R., A robot for transnasal surgery featuring needle-sized tentacle-like arms, Expert Review of Medical Devices, 11(1), 5-7, [10] Rogers C.G., Laungani R., Bhandari A, Shrivastava A., Menon M., Maximizing console surgeon independence during robot-assisted renal surgery by using the fourth arm and TilePro, Journal of Endourology, 23(1), , [11] Tontochi S., Elgin R., Monahan M., Johnston W.K., Journal of Endourology, 28(8), , [12] Baca I., Robot arm in laparoscopic surgery, Chirurg, 68(8), , [13] Yim G.W., Kom S.W., Nam E.J., Kim S., Kim Y.T., Perioperative complications of robot-assisted laparoscopic surgery using three robotic arms at a single institution, Yonsei Medical Journal, 56(2), , [14] Mester Gy., Introduction to control of mobile robots, Proc. YUINFO 2006, 1-4, ISBN , Kopaonik, Serbia & Montenegro, [15] Mester Gy., Obstacle avoidance and velocity control of mobile robots, Proc. 6th Int. Symp. Intelligent Systems and Informatics SISY 2008, , IEEE Catalog Number: CFP0884C-CDR, ISBN , Library of Congress: , DOI /SISY , Subotica, Serbia, Sept , [16] Rodic A., Mester Gy., Virtual WRSN Modeling and simulation of wireless robot-sensor networked system, Proc. 8th IEEE Int. Symp. Intelligent Systems and Informatics, SISY 2010, , DOI: /SISY , ISBN: , Subotica, Serbia, [17] Corradini M.L., Giantomassi A., Ippoliti G., Longhi S., Orlando G., Robust control of robot arms via quasisliding modes and neural networks, Studies in Computational Intelligence, 576, , [18] L. Cveticanin, M. Kalami-Yazdi, Z. Saadatnia, H. Askari, Application of Hamiltonian approach to the generalized nonlinear oscillator with fractional power, International Journal of Nonlinear Sciences and Numerical Simulation, 11(12), ,

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