Research on Inverse Dynamics and Trajectory Planning for the 3-PTT Parallel Machine Tool

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1 06 International Conference on aterials, Information, echanical, Electronic and Comuter Engineering (IECE 06 ISBN: Research on Inverse Dynamics and rajectory Planning for the 3-P Parallel achine ool ing Hu and Suoxian Yuan ABSRAC In this aer, dynamics modeling of the 3-P arallel machine tool is carried out by the Lagrange method. he cycloid curve is adoted in the lanning of the trajectory. Combined with a ractical examle, the comuter simulation is carried out. It rovides a theoretical basis for the control of the machine tool. he work will rovide hel for the control and structural otimization design of the arallel machine tool. INRODUCION Recently, lots of research has focused on the automatic grinding rocess [] []. he 3-P arallel machine tool was designed for grinding [3]. In arallel machine ool technology, inverse dynamics modeling is one of the most imortant asects for controlling a arallel machine tool with high osition recision [4]. he mechanism of this machine tool is shown in Fig.. his mechanism consists of a moving latform, three sliding rails, three slider joints, as well as a translational- mechanism that ing Hu, School of echanical Engineering & Automation, Northeastern University, Shenyang 0004, P.R. China Suoxian Yuan, School of echanical Engineering & Automation, Northeastern University, Shenyang 0004, P.R. China 34

2 Slider Rail ranslational-motion mechanism Link oving latform Figure. 3-P horizontal arallel machine tool. equivalent to one link, so this mechanism has three links equivalently, which are same the length, for the manufacture and control of convenience. the three sliders are located in three rail resectively, to drive three-bar moving along the horizontal rails by serve motors, three sliding rail evenly saced arallel. As the arallel the mechanism restricting, moving latform for equilateral triangle is always arallel to the rails. he link and the slider are connected by universal joints, the link and the moving latform are connected by universal joints, and the mechanism has three translational DOF. he 3-P arallel machine tool is simlified as shown in Fig. to obtain kinematics for convenience. he base coordinate system O(x, y, z on Rails is set u, three sliders are regarded as the three oints A (x,-a, 0, A (x, 0, 0, A3 (x3, a, 0 A(x,-A,0 A x A o A3(x 3,A,0 y l l l A(x,0,0 z (x,y,z Figure. he Schematic diagram of 3-P arallel mechanism tool. resectively under the O-xyz coordinates. he moving latform located under the base coordinate system is the oint (x, y, z. he kinematics can be determined as follows 34

3 ( x ( x ( x x x x 3 ( y y ( y A z A z l z l l ( Where, A=a-b, b is the side length of moving latform for equilateral triangle. l is the length of the link. Jacobian matrix is simle and can be determined by J Ay z l z ( Ay l z ( Ay y z l z y l z y Ay Ay l z ( Ay l z ( Ay ( DYNAICS According to the Lagrange method, Dynamics equations can be determined as follows F Dqq ( Hqq (, Gq ( Above, vector F F F F3 3, q x x x3 and q x x x q x x x (3 reresents the inertia force of slider, vector reresents of slider osition, 3 slider seed and slider acceleration. he moving latform vector is assumed as V [ V, V, V ], q is also exressed as q J V x y z D(q,H(q, q and G(q can be obtained resectively as follows D( q cj J sj J A( m 3 (4 D(q is Joint sace quality matrix. dj dj dj dj dj J Hqq (, c( JJ s( JJ qq c q dt dt dt dt dq dj dj dj3 s ( Jx Jx J3x3 ( x x x3 (5 dq dq dq 343

4 dz Gq ( g (6 dq Where, J J J J 3, J is first line of the Jacobian J matrix row vector, m is the mass of moving latform, m,m,m3 is the mass of three links resectively. 5 s ( m m m3 (7 6 c (3 mmm m3 (8 6 m 0 0 Am ( 0 m m 3 (9 ( mmm m3 (0 It can be seen that the dynamics equation is comosed of acceleration, seed and gravity. his shows as a driving force on slider for machine tool associated with slider acceleration, seed and quality of arallel mechanism and end-effector. RAJECORY PLANNING here are joint and Cartesian sace methods on trajectory lanning of arallel machine tool. ethod of Cartesian sace trajectory lanning has a clear concet and can be carried out directly from the task, equation of motion of the arallel machine tool is simle, less comutation, has non-singular oints in a sace[4]. rajectory lanning based on Cartesian is suitable for the machine tool. Because the grinding machine with constant ressure grinding, end-effector are required to maintain a certain seed in entire oerating area for the avoidance of grinding burn and grinding quality. he cycloid curve is used to connect corner of track, linear, circular curve and curvature is continuous at the oints change, first derivative and second derivative are also continuous. End-effector can smoothly through the track corner under the condition of constant seed, high recision and quality can be ensured. he ordinary exression of the cycloid curve is as follows t x x0 a cos( bt dt 0 t y y0 a sin( bt dt 0 ( 344

5 Above, (x0, y0 is coordinates of origin of cycloid curves, a is length coefficient φis angle between x axis and tangent in the initial oint of curves, b is roortional coefficient Figure 3. cycloid curves. he cycloid curves is shown in Fig.3, the whole curve is continuous, its first derivative, second derivative is continuous. he curve at the zero oint is zero. Curvature is roortional to the arc length of the curve. his is characteristic of cycloid curves. he curvature is not zero in other oints on the clothoid curve, so one clothoid curve can only ensure the oint on the curve and line connections to meet the requirement of continuous curvature. he trajectory is designed as shown in Fig.4, grinding oerations from A oint to F oint to the end,ab, DC and EF are linear segments, BC and DE are two segments in the dual form of the cycloid curve, N, is the contact oint resectively. When the straight line and the cycloid curve are connected, he use of two starting from the origin of the tail of the clothoid curve docking, so corner curve and two straight lines are connected to meet the requirement of continuous curvature. rajectory lanning using swing curves as the corner a billet grinding machine for the tyical job trajectory. 4 0 A B 6 D C N E F Figure 4. Straight linear and clothoid curve connection. DYNAICS SIULAION Set of arallel machine tool structure arameters following as table I 345

6 echanism arameter m=50kg able I. Simulation arameters. otion arameters =0.mm/s m= m= m3=45kg z=700mm l=800m A=850mm End-effector starting oint coordinates for A(0, 5 to end oint D(6, 5 Along the curve in Fig.4, motion arameters of the trajectory of the oerating sace can be transformed into joint sace coordinates by the kinematic equation (, structural arameters of arallel machine tool and the converted motion arameters are brought into the equation (4. he drive force of slider is simulated as shown in Fig.5. he corner is connected with cycloid curve, the Force of slider driving rod can be steady changed, the vibration and imact of the robot can be effectively reduce CONCLUSIONS In this aer, dynamics modeling of the 3-P arallel mechanism tool is carried out. he trajectory lanning of the 3-P arallel machine tool is suitable for lanning in the oeration sace. hrough the simulation research and theoretical analysis for the secific oeration of grinding, the corner is connected by a cycloid curve, the velocity -000 V F/N F F F 3 and acceleration of the end-effector are continuous. he vibration and imact of the machine tool can be reduced effectively and continuously. REFERENCES t/s Figure 5. Dynamic simulation of the arallel machine tools.. Berend Denkena, a Develoment trend of grinding technology J. ool Engineering,. 05, Weiyang Lin, Bing Li, Xiaojun Yang and Dan Zhang odeling and Control of Inverse Dynamics for a 5-DOF Parallel Kinematic Polishing achine J. Robot. syst., Vol. 0, Wang, X. and Hu,. he Kinematics Analysis of the 3-P Parallel achine ools J. achinery Design & anufacture, Adam Y. Le, James K. ills and Beno Benhabib. Dynamic modeling and control design for a arallel-mechanism-based meso-milling machine tool J. Robotica. V 3, 04, Zhang,.Z, Research on the smooth design of robot corner orbit- cycloid Curve continuation method, J. Robot,

UNIVERSITY OF OSLO. Faculty of Mathematics and Natural Sciences

UNIVERSITY OF OSLO. Faculty of Mathematics and Natural Sciences Page 1 UNIVERSITY OF OSLO Faculty of Mathematics and Natural Sciences Exam in INF3480 Introduction to Robotics Day of exam: May 31 st 2010 Exam hours: 3 hours This examination paper consists of 5 page(s).