Analysis of the Effect of Number of Knots in a Trajectory on Motion Characteristics of a 3R Planar Manipulator

Transcription

1 Analysis of the Effect of Number of Knots in a Trajectory on Motion Characteristics of a Planar Maniulator Suarno Bhattacharyya & Tarun Kanti Naskar Mechanical Engineering Deartment, Jadavur University, Kolkata, India suarno.bhattacharyya@gmail.com, tknaskar@mech.jdvu.ac.in Abstract - The aer resents a method of trajectory lanning and motion characteristics of a robotic maniulator. Main objective is to study the motion characteristics of a maniulator and to exlore the scoe of minimization of jerk. 8 th order olynomial is considered for the trajectory design and the effect of number of intermediate knots between start and final ositions of a maniulator within the worksace is studied. Dislacements, velocities, accelerations and jerk of end-effectors on a linear ath are resented. The simulation for motion of the maniulator is done with the hel of AutoLISP on AutoCAD latform. Key words - trajectory; sline; knots; jerk; maniulator. I. INTODUCTION obotic Maniulators are widely used in almost every leading manufacturing industry - welding shos, assembly sections, machining and many more. In each of these fields one common objective is to move the maniulator according to reuirement along a secified trajectory. The roblem of trajectory lanning is an active area of research in the field of robotics. N. A. Asragathos [] worked in the area of generation of Cartesian trajectory under bounded osition deviation. Gasaretto and Zanotto [] develoed a new method for smooth trajectory lanning. Saramago and Ceccarelli [] roosed the otimization of trajectory lanning taking into account robot actuating energy and grasing forces in maniulator grier. There are many other similar works in the area of robotic trajectory lanning. However, there is a little number of aers ublished on the effects of a articular trajectory-curve on acceleration and more articularly on jerk of maniulators in motion. This work is carried out to find the effect of number of knots in olynomial sline used as robotic trajectory on the motion characteristics of maniulator. Many works are there on effects of knots in olynomial sline on kinematics arameters like acceleration and jerk of cam followers []. Introduction of knots in olynomial sline and B-sline can lay very imortant role in minimizing acceleration, jerk and ing (time derivative of jerk) of cam followers [5-6]. Taking lead from these works higher order olynomial sline with multile knots is designed in this work for robotic trajectory. Considering the jerk at initial and final ositions as zero, 8 th order olynomial is used to join the knots in the joint sace and olynomial sline is constructed. Effects of number of knots on dislacement, velocity, acceleration and jerk for the linear ath of end-effectors motion are resented. A case study is done with a lanar maniulator tracing the newly designed trajectory-curve. The simulation of the motion of the maniulator is done with the hel of AutoLISP rogram on AutoCAD latform. The roblem is aroached in the following way: (i) Start oint and end oint are assumed on a secified maniulator trajectory; (ii) the trajectory is defined in the coordinate sace; (iii) some via-oints (called here knots) are assumed on the trajectory; (iv) inverse kinematics analysis at these oints is done and the oints are lotted in the joint coordinate system for each link of the robotic arm; (v) these oints are joined by 8 th order olynomials to construct the sline; (vi) at each oint forward kinematics analysis is done; (vii) simulation of motion is done sace using AutoLISP to see whether the maniulator follows the trajectory in the coordinate. II. THEOETICAL ANALSIS Fig. shows a lanar maniulator along with its work-sace B and target trajectory xy, a straight line with two via-oints and (called intermediate knots). B is a ring with inner and outer diameters. The number of intermediate knots can be as many as user s choice. Fig. (a) shows a sline with two end and intermediate knots. Inverse kinematics analysis at the oints a,,, b International Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 677, Volume-, Issue-, 5

2 Analysis of the Effect of Number of Knots in a Trajectory on Motion Characteristics of a Planar Maniulator is done and the oints are lotted in the joint coordinate systems for each link - a and b being the end knots. In a general way if there is a oint P(x, y) in the coordinate sace the inverse kinematics analysis reuires to find the joint angles (θ,θ,θ ) as a function of wrist osition and orientation (x,y,φ) as shown in Fig.. Solving for θ we rewrite the nonlinear using a change of variables as follows: x=l c L c y=l s L s x = k c k s y= k s k c where k = L Lc and k=ls. Finally we comute θ using the two argument arctangent function θ =atan(y,x)-atan(k,k ) xy-l-l θ =atan(s,c )=atan(± -c, ) LL θ =φ-θ-θ Fig. : lanar maniulator: work-sace and trajectory. Fig. (a) : Sline with knots Fig. These oints are lotted in the joint co-ordinate system against time. θ and θ for a,,, b oints are determined. Fig. Fig. Fig.5 International Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 677, Volume-, Issue-, 5

3 Analysis of the Effect of Number of Knots in a Trajectory on Motion Characteristics of a Planar Maniulator Velocity, acceleration and jerk are considered to be zero at end knots a and b. So we see velocity, acceleration, jerk and osition at end knots are known - a total of 8 boundary conditions. Hence these oints are joined by 8 th order olynomials for trajectory design. We lan to join the knots (a,,, b) by three 8 th order olynomials (for k number of knots we use k- olynomials). In general if there are k number of knots and m order olynomial then number of unknown uantities: m(k-), smoothness euations: (k-)(m-), interolation euations: k- and boundary conditions: m []. Here 8 th order olynomials are connected by knots. So the above numbers will be,, and 8 resectively. We consider here three olynomials to construct the olynomial sline. Es. (-) are the olynomials, Es. (-7) the smoothness euations, Es. (8-9) the interolation euations and Es. (-7) the boundary condition euations. Polynomials: = a at at at at at 5 at 6 at 7 for <t<t () ( ) = b b t t b t t b t t b t t b t t b t t b tt for t <t<t () ( ) = c c t t c t t c t t c t t c t t c t t c7 tt 5 6 Smoothness euations: for t <t<t () b =a a t a t a t a t a t a t a t () 5 6 = a a t at a t 5a5t 6a6t 7a7t (5) 5 = a 6ata t a5t a6t a7t (6) =6a at 6a5t a6t a7t (7) =a a 5t6a 6t 8a 7t (8) =a 7a t 5a t (9) =7a 65a7t () ( ) =b b t-t b t-t b t-t b t-t b t-t b t-t b t-t () ( ) =b b t-t b t-t b t-t 5b t-t 6b t-t 7b t-t =b 6b t-t b t-t b t-t b t-t b t-t 6 7 ( ) 5 6 =6b b t-t 6b t-t b t-t b t-t 7 ( t-t ) =b b t-t 6b t-t b () () () (5) =b 57b6 t-t 5b7 t-t (6) =7b 65b7( t-t ) (7) Interolation euations: b =θ at t= t (8) c=θ at t=t (9) Boundary conditions: a= (At t = ) () a= (Since at t =, =) () a = (since at t =, = ) () a = (since at t =, = ) () ( ) (at b c c t -t c t -t c t -t c t -t b b b b c t -t c t -t c7 t -t = 5 b 6 b b 5 6 b b b c c t -t c t -t c t -t 5c t -t 6c t -t 7c t -t = 5 b 6 b 7 b (since at t=t b =) 5 b b 5 b c 6c t -t c t -t c t -t c t -t c t -t = (since at 6 b 7 b t=t b =) t = t ) () (5) (6) International Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 677, Volume-, Issue-, 5

4 Analysis of the Effect of Number of Knots in a Trajectory on Motion Characteristics of a Planar Maniulator ( ) b 5 b 6 b 6c c t -t 6c t -t c t -t c t -t = 7 b (7) (since at t=t b =) We ut t=t and t=t in Es. (-9) the resulting euations along with Es. (-7) generates a set of simultaneous euations, which can be written in matrix form: M X x NX = UX (8) where N X =[ a.. a7 b.. b7 c.. c7] T. time erasing the revious entity on the grahics screen are done raidly at a seed selected by the user. This gives an effect of animation in the robotic maniulator on the grahics screen. AutoCAD Figs. (6-7) show simulation of maniulator motion in two ositions. In course of simulation number of knots in the trajectory curve is varied and effect of these variations on velocity, acceleration and jerk is comuted using MATLAB (9b). The results are shown in Figs. (8-). Figs. (-) show the variations of maximum values of velocity, acceleration and jerk with the variation of number of knots. Three cases are shown for number of knots of, 6 and 6. Aroached Maniulator Since M X & U X are known we can find N X, i.e., the olynomial coefficients a, b, c etc. We then lot θ -t andθ -t. From these lots the osition of the end-effector can be located at any value of t. For, if we know the joint angles θ,θ,θ ( θ is user assigned), using Denavit-Hartenberg algorithm we can find the osition of the end-effector in the coordinate sace. The necessary euations are: x=lcosθ Lcosθ Lcosθ (9) y=lsinθ Lsinθ Lsinθ () WOKSPACE Fig. 6 Fig. 7 IV. ESULTS The curves obtained during simulation are shown below: From Es. (9-) the velocity, acceleration and jerk of the end-effector can be calculated using E. (), E. (), E. () resectively V= (x)(y) () ( ) ( ) a = x y () j = ( x) ( y) () III. SIMULATION Using AutoLISP code generated for the urose we have simulated the motion of the maniulator arm tracing the newly designed trajectory in the coordinate sace. Concets of loo and file handling of AutoLISP were utilized for this. Drawing an entity and at the same International Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 677, Volume-, Issue-, 5

5 Analysis of the Effect of Number of Knots in a Trajectory on Motion Characteristics of a Planar Maniulator Fig. 8 : Number of knots Fig. : Number of knots 6 Fig. : Maximum velocity vs. number of knots Fig. 9 : Number of knots 6 Fig. : Maximum jerk vs. number of knots From the above analysis the following observations are made: As we increase the number of knots (i) the trajectory aroaches the target curve (ii) the velocity, acceleration and jerk have a very low and smooth value in the middle region, (iii) at the ends of comletion of motion high fluctuations with increased number of knots and (iv) maximum value of velocity, acceleration and jerk increase with the rise of number of knots. V. CONCLUSION The objective of the work was to study the effect of the number knots on motion characteristics while designing trajectory for a maniulator. In this regard the effect of number of intermediate knots between two ositions within the worksace has been studied and an International Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 677, Volume-, Issue-, 55

6 Analysis of the Effect of Number of Knots in a Trajectory on Motion Characteristics of a Planar Maniulator 8 th order olynomial is considered for the trajectory design. From the above analysis it is evident that it has two major advantages. As we increase the number of knots the trajectory aroaches the target curve. It is evident from comaring the curves of coordinate sace in Figs. (8-) with that of Fig.. It shows how the trajectory aroaches the target curve with increasing accuracy. This may be very suitable for industrial machine tool alications like welding where a contour is to be traced by the maniulator with high accuracy. Another advantage we see from Figs. (8-) is that in site of the end fluctuations the acceleration and jerk have very low values in the middle region. egarding velocity the variation is very smooth in the intermediate regions. However, it is seen that at the ends of motion there are high fluctuations of velocity, acceleration and jerk with the increase of number of knots. The work, therefore, has a scoe of further extension by adoting some otimization techniue to reduce those fluctuations. NOMENCLATUES i. θ,θ,θ : Joint angles. ii. c...i = cos(θ θ...θ i) iii. s...i =sin(θ θ...θ i) iv. L = Length of first link of end-effector. L = Length of second link of end-effector. L = Length of third link of end-effector. v. V = esultant velocity of end-effector. a = esultant acceleration of end-effector. j = esultant jerk of end-effector. EFEENCES [] N. A. Asragathos, Cartesian trajectory generation under bounded osition deviation, Mechanism and Machine Theory (998) [] A. Gasaretto, V. Zanotto, A new method for smooth trajectory lanning of robot maniulators, Mechanism and Machine Theory (7) [] S. F. P. Saramago, M. Ceccarelli, Effect of basic numerical arameters on a ath lanning of robots taking into account actuating energy, Mechanism and Machine Theory 9 () 7 6. [] obert, and P. E. Norton,, Cam Design and Manufacturing Handbook, Industrial Press, N. [5]. Mishra and T. K. Naskar, 6, Synthesis of Ping Finite Otimized Cam Motion Program by B-, Proceedings of International Congress on Comutational Mechanics and Simulation, ICCMS-6, IIT Guwahati, India. [6] M. Mondal, and T. K. Naskar, 9, Introduction of control oints in sline for synthesis of otimized cam motion rogram, Mechanism and Machine Theory, International Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 677, Volume-, Issue-, 56