Robot movement can be predicted by incorporating Forward Kinematics(FK) and trajectory planning techniques. However, the calculations will becomecomplicated and hard to be solved if the number of specific via points is increased.Thus, back-propagation artificial neural network is proposed in this paper to overcomethis drawback due to its ability in learning pattern solutions. A virtual 4-degreeof freedom manipulator is exploited as an example and the theoretical results arecompared with the proposed method

Ang, Chun Kit

Arrifin, Mohd Khairol Anuar Mohd

Mashohor, Syamsiah

Tang, Sai Hong

Agora University Editing House: Journals

INT J COMPUT COMMUN, ISSN 1841-98369(3):253-260, June, 2014.Solving Continuous Trajectory and Forward KinematicsSimultaneously Based on ANNC.K. Ang, S.H. Tang, S. Mashohor, M.K.A.M. ArriﬁnChun Kit Ang*, Sai Hong Tang,Syamsiah Mashohor, Mohd Khairol Anuar Mohd ArriﬁnUniversiti Putra MalaysiaDepartment of Mechanical and Manufacturing Engineering,Faculty of Engineering, 43400 UPM, Serdang, Selangor Darul Ehsan.Email: ack_kit@hotmail.com, saihong@upm.edu.my,syamsiah@upm.edu.my, khairol@upm.edu.my*Corresponding authorAbstract: Robot movement can be predicted by incorporating Forward Kinematics(FK) and trajectory planning techniques. However, the calculations will becomecomplicated and hard to be solved if the number of speciﬁc via points is increased.Thus, back-propagation artiﬁcial neural network is proposed in this paper to overcomethis drawback due to its ability in learning pattern solutions. A virtual 4-degreeof freedom manipulator is exploited as an example and the theoretical results arecompared with the proposed method.Keywords: Artiﬁcial Neural Network (ANN), back propagation neural network,forward kinematics, continuous trajectory.1 IntroductionTrajectory planning can be considered as an important issue in robot industry. Withoutan appropriate trajectory planning, a robot’s motion will become unpredictable and it maycollide with obstacles or go through undesirable points. Thus, an appropriate trajectory planningenables us to determine the required workspace and provides us an opportunity to avoid obstacles.Basically, there are two types of trajectory planning, point to point and continuous trajectory(with via points). However, continuous trajectory planning will be more practical compare topoint to point trajectory because it allows robot to pass through various speciﬁc points and it isuseful for real time robot.Conventionally, there are a few methods that can be used for continuous trajectory planningsuch as Bezier function, Linear Segments with Parabolic Blends (LSPB), and high order polyno-mials. Unfortunately, those methods require extensive calculations when the number of speciﬁcpoints is increased and thus leading to the emergence of a large number of formulas ( [1]- [5]).Hence, it is advisable to develop a method which is able to overcome this drawback.2 PreliminariesForward kinematics (FK) can be used to determined the position and orientation of a robot’shand if all the conﬁgurations are known and the equations can be derived by using Denavit-Hartenberg (DH) method because it is the standard way of modelling the robot motion due toits simplicity of modelling robot links and joints that is applicable for any robot conﬁgurations regardless of its complexity. The total transformation for a robot manipulator from base tohand will beCopyright© 2006-2014 by CCC Publications254 C.K. Ang, S.H. Tang, S. Mashohor, M.K.A.M. ArriﬁnRTH =RT 11 T22T3:::n 1Tn = A1A2A3:::An =0BBBB@nx ox ax pxny oy ay pynz oz az pz0 0 0 11CCCCA (1)and the end eﬀector can be deﬁned as:re(t) = fe((t)) (2)where re 2 Rm in Cartesian space is related to the joint space vector, m is dimensional positionvector of end eﬀector.In fact, what we are actually interested in is the inverse kinematics (IK) because it allowsus to determine the value of each joint in order to place the arm at a desired position andorientation. The inverse kinematic problem is to ﬁnd the joint variable  for any givenre(t).Unfortunately, it is impossible to ﬁnd an analytic solution due to the nonlinearity off((t))inreality. The inverse kinematics problem of manipulators is thus usually solved at the velocitylevel. Diﬀerentiating re(t) = fe((t)) with respect to time yields a linear relation between theCartesian velocity _reand the joint velocity _:Je() _ = _re (3)where Je() 2 Rmn is the end eﬀector Jacobian matrix which is deﬁned as Je() = @f()@ [6-13]. However, the derived IK equations are not necessarily available when the degeneracy andsingularity problems ( [6], [9], [12]- [14])] occurred and thus we need to use some new methods tosolve it such as artiﬁcial neural network, genetic algorithm, neural fuzzy and etceteras ( [7]- [15]).Generally, high order polynomial is used to plan a trajectory with the intention to let therobot’s hand to pass through all the speciﬁc points. However, solving the high order polynomialequations requires extensive calculations. Instead, it is advisable to use combinations of lowerorder polynomials for diﬀerent segments of the trajectory and blend them together to satisfy allrequired boundary conditions ( [1]- [5]). (t) = c0 + c1t+ c2t2 + :::::::+ cn 1tn 1 + cntn (4)For an example, for a 4-3-4 trajectory, a fourth order polynomial is used to plan a trajectorybetween the initial point and the ﬁrst via point, followed by a third order polynomial to plana trajectory between two via points, lastly another fourth order polynomial is implemented toplan the trajectory between the last via point and ﬁnal angle. (t)1 = a0 + a1t+ a2t2 + a3t3 + a4t4 (t)2 = b0 + b1t+ b2t2 + b3t3 (t)3 = c0 + c1t+ c2t2 + c3t3 + c4t4In short, IK allows us to allocate the robot’s hand at desired location and the motion of robot’shand can be determined by incorporating the FK and trajectory techniques. Sadly, the calcula-tions will become tedious and hard to be solved when the number of speciﬁc points is increased.Solving Continuous Trajectory and Forward Kinematics Simultaneously Based on ANN 2553 Theoretical and Simulation ResultsExperiments were performed on a simulated 4-degree of freedom (DOF) robot manipulatorwhich shown in Figue 1, IK was used to determine the starting and ending conﬁgurations forthe robot’s hand while the high order polynomials and FK were used to track the motion of therobot’s hand from beginning to the end in Cartesian space. It was speciﬁed that the robot needto pass through two speciﬁc points. At the same time, velocities and accelerations constraintsneed to be taken into consideration.Figure 1: Simulated 4 DOF manipulatorAssuming that a manipulator needs to spend tf to complete the moving path and a manip-ulator joint needs to reach aat interval time ta then bat interval time tb where ta < tbandta; tb 2 [0; tf ]. One may need to match the position, velocity, and accelerations of the two seg-ments at each point to plan a continuous trajectory. Hence, higher order polynomials in theform of  (t) = C0+C1t+C2t2+ :::+Cn 1tn 1+Cntn are required. However, it is an extensivecalculation process to solve high order polynomial and it is advisable to use the combinations oflower order polynomials for diﬀerent segment of the trajectory such as 4-3-4 trajectory or 3-5-3trajectory.The velocity and acceleration for initial and ﬁnal point are zero. In order to fulﬁll the entireinitial, ﬁnal, and via points requirement, a matrix form can be generated where256 C.K. Ang, S.H. Tang, S. Mashohor, M.K.A.M. Arriﬁntab = tb   taand tc = tf   ta   tb.266666666666666666666666666666641_11220033004_4437777777777777777777777777777775=26666666666666666666666666666641 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 00 0 2 0 0 0 0 0 0 0 0 0 0 01 ta t2a t3a t4a 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 00 1 2ta 3t2a 4t3a 0  1 0 0 0 0 0 0 00 0 2 6ta 12t2a 0 0  2 0 0 0 0 0 00 0 0 0 0 1 tab t2ab t3ab 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 2tab 3t2ab 0  1 0 0 00 0 0 0 0 0 0 2 6tab 0 0  2 0 00 0 0 0 0 0 0 0 0 1 tc t2c t3c t4c0 0 0 0 0 0 0 0 0 0 1 2tc 3t2c 4t3c0 0 0 0 0 0 0 0 0 0 0 2 6tc 12t2c37777777777777777777777777777752666666666666666666666666666664a0a1a2a3a4b0b1b2b3c0c1c2c3c43777777777777777777777777777775Or[] = [M ] [C] (5)By solving those constant values in matrix[C], the joint turning angle at any interval time canbe obtained where (t) = a0 + a1t+ a2t2 + a3t3 + a4t4 0 < t  ta (t) = b0 + b1t+ b2t2 + b3t3 t 2 [ta; tb] ; t = t  ta (t) = c0 + c1t+ c2t2 + c3t3 + c4t4 t 2 [tb; tf ] ; t = t  ta   tbThe equation which is used to represent the robot’s hand at any interval time can be denotedas:rE (ti) =0BBBBBB@cos (1 (ti))l1  cos (2 (ti)) + l2  cos3Pk=2k (ti)+ l3  cos4Pk=2k (ti)sin (1 (ti))l1  cos (2 (ti)) + l2  cos3Pk=2k (ti)+ l3  cos4Pk=2k (ti)l1  sin (2 (ti)) + l2  sin3Pk=2k (ti)+ l3  sin4Pk=2k (ti)1CCCCCCA(6)In fact, all these calculations can be solved simultaneously by back-propagation artiﬁcial neu-ral network. The back-propagation neural network can be activated by presented a trainingset data with inputs x1(p); x2(p):::xn(p)and desired outputsyd;1(p); yd;2(p); :::; yd;n(p). Assumethat the back-propagation network consists of three layers which are input layer, hidden layer,Solving Continuous Trajectory and Forward Kinematics Simultaneously Based on ANN 257and output layer. The actual outputs of the neurons in the hidden layer will be yj(p) =sigmoidnPi=ixi(p) wij(p)  jwhere n is the number of inputs for neuron j in the hidden layer,and sigmoid is the sigmoid activation functionY sigmoid = 1=(1+e X). For the output layers, theactual outputs of the neurons will be yk(p) = sigmoid"mPj=1xjk(p) wjk(p)  k#where m is thenumber of inputs for neuron k in the output layer. An error signal ek(p) = yd;k(p)  yk(p)will begenerated from neuron kat output layer and propagated backward for updating the weights.For updating the weights at the output neurons with learning rate, :Error gradient,k(p) = yk(p) [1  yk(p)] ek(p)ek(p) = yd;k(p)  yk(p)Weight correction, wjk(p) =  yj(p) k(p)Update weight, wjk(p+ 1) = wjk(p) + wjk(p)Similar to output layer, for updating the weights at the hidden neurons with learning rate,:Error gradient,j(p) = yj(p) [1  yj(p)]1Pk=1k(p) wjk(p)Weight correction, wij(p) =  xi(p) j(p)Update weight,wij(p+ 1) = wij(p) + wij(p)The same procedures can be repeated by increasing the iteration puntil the output errorcriterion is satisﬁed.A training data set which consists of inputs and outputs is presented to the networks forlearning purpose. Inputs consists of the ﬁnal angle of each joint f;n and interval time variable tiwhere ti 2 [0; tf ] and tf is the time for the manipulator to complete the moving path. The outputswill be the coordinates of robot’s hand at any interval time with respect to ti,rE(ti) = [px py pz]T .There are two set of results in table 1. The theoretical results which obtained by usingconventional FK and high order polynomial trajectory were compared to the artiﬁcial neuralnetwork (ANN) results. The robot started moving from its initial position (1=2=3=4=0)and passed through two via points before it reached its destinations. It shows that the robot stillable to pass through those speciﬁc points even though the destination point was changed and themoving path could be tracked by using ANN. The theoretical and ANN results were almost thesame and the errors were less than 0:3unit (less than 1%). This can be used as an evidence toproof that artiﬁcial neural network is able to solve FK and continuous trajectory simultaneouslydue to its adaptive learning ability.Since back-propagation neural network is able to track the motion of end eﬀector, it can beused to track the position of each joint in Cartesian space as well. Referring to Figure 1, Joint 1and joint 2 are located at the origin of reference frame, (0,0,0). For tracking the position of joint 3,the training data set inputs and outputs will be [ti 1(ti) 2(ti)]T and rE;3(ti). Correspondingly,for joint 4, the training data set inputs and outputs will be [ti 1(ti) 2(ti) 3(ti)]T and rE;4(ti).Once the joins location of all joints are computed, the manipulator link equations can be derived258 C.K. Ang, S.H. Tang, S. Mashohor, M.K.A.M. Arriﬁnby using parametric equations.Assuming that the location and vector for joint m and joint n at any interval time ti:rE;n(ti) = [px;n py;n pz;n]T ;  !OPn = [an bn cn]T and rE;m(ti) = [px;m py;m pz;m]T ;   !OPm = [am bm cm]T .The link equation that connecting joint n and m will bex px;nan am =y py;nbn bm =z pz;ncn cm = or[x y z]T = [px;n py;n pz;n]T + [an   am bn   bm cn   cm]T For the robot manipulator which is mentioned in this paper, the robot link equations can bedenoted as:Link 1: x px;3a3 a2 =y py;3b3 b2 =z pz;3c3 c2 = 2 0  z  pz;3Link 2:x px;4a4 a3 =y py;4b4 b3 =z pz;4c4 c3 = 3 pz;3  z  pz;4Link 3: x px;5a5 a4 =y py;5b5 b4 =z pz;5c5 c4 = 4 pz;4  z  pz;5Conclusively, the link equation can be derived as long as the coordinates of two successivejoints are provided by using back-propagation neural network and the links parametric equationscan be derived based on the vector of the two successive joints. Once the equation of eachmanipulator link is known, a person is able to predict or compute the entire motion of a robotmanipulator.Besides, obstacle collision can be detected if the link equations are known. Assuming thatthere is a rigid body obstacle enclosed in a sphere which possesses a r radius and center locatesat (xo; yo; zo), the sphere equation will be(x  xo)2 + (y   yo)2 + (z   zo)2 = r2The intersection points between the link and obstacle can be obtained by solving the link andsphere surface equation. For an example, a line that pass through the point (e; f; g)possessing asame direction as vector[a b c]T , the equation for the link will be:x ea =y fb =z gc = or0B@ xyz1CA=0B@ efg1CA+0B@ abc1CA Substituting the link equation into sphere surface equation yields,(a + e  xo)2 + (b + f   yo)2 + (c + g   zo)2 = r2simpliﬁed and yields,A2 + 2B + C = 0where,A = a2 + b2 + c2B = a(e  xo) + b(f   yo) + c(g   zo)C = (e  xo)2 + (f   yo)2 + (g   zo)2   r2Solving Continuous Trajectory and Forward Kinematics Simultaneously Based on ANN 259By examining the discrimination roots, the intersection points between link and sphere surfaceexists when  0,[a(e  xo) + b(f   yo) + c(g   zo)]2  4a2 + b2 + c2 (e  xo)2 + (f   yo)2 + (g   zo)2   r2The intersection point between the manipulator links and obstacle can be determined by solvingthe constant .Table 1: Comparison of Conventional and ANN Techniques in Tracking Robot End EﬀectorMoving Path4 ConclusionsThis paper shows that back-propagation neural network is able to solve trajectory and kine-matic problems simultaneously. The moving path of the robot end eﬀector still able to be trackedeven though the ending point is changed. Besides, this method can be used to track the positionof robot joints which can be used to derive the robot link equations. In comparison to conven-tional continuous trajectory planning methods, ANN is much easier to be applied and providehigh accuracy results.260 C.K. Ang, S.H. Tang, S. Mashohor, M.K.A.M. ArriﬁnBibliography[1] Saeed B. N. (2001). Introduction to Robotics Analysis, System, Application. Prentice Hall,29-172.[2] Ata, A., Myo, T. (2008). Optimal trajectory planning and obstacle avoidance for ﬂexiblemanipulators using generalized pattern research. World J. Modeling and Simulation,4:163–171.[3] Guan, Y., Yokoi, K. , Stasse, O. , Kheddar, A. (2005). A. On robotic trajectory planning usingpolynomial interpolations. Proc. IEEE Int. Conf. on Robotics and Biomimetics, 111-116.[4] Reichenbach, T, Kovacic, Z.(2005). Collision-Free Path Planning in Robot Cells Using Virtual3D Collision Sensors. Cutting Edge Robotics, 683–697.[5] Campos, J., Flores, J.A.R., Montufar, C.P.(2008). Robot Trajectory Planning for Multiple3D Moving Objects Interception: A Polynomial Interpolation Approach. Proc. IEEE Int.Conf. on Electronics, Robotics and Automotive Mechanics, 478–483.[6] Zarkandi S., Vafadar A., Esmaili M. R.(2011). PRRRRRP redundant planar parallel manip-ulator: Kinematics, workspace and singularity analysis. IEEE 5th Int. Conf. on Robotics,Automation and Mechatronics (RAM), DOI: 10.1109/RAMECH.2011.6070457, 61–66.[7] Al-Mashhadany, Y. I. (2010). Inverse Kinematics Problem (IKP) of 6-DOF Manipulator byLRNNs. Int. Conf. on Management and Service Science (MASS), 1 – 5.[8] Zhang, D., Lei, J.(2011). Kinematic analysis of a novel 3-DOF actuation redundant par-allel manipulator using artiﬁcial intelligence approach. Robotics and Computer-IntergratedManufacturing, 27: 157–163.[9] Firmani, F., Podhoprodeski, R. P.(2009). Singularity analysis of planar parallel manipulatosbased on forward kinematic solutions. Mechanism and Machine Theory, 44: 1386–1399.[10] Vesselenyi, T., Dzitac, S., Dzitac, I., Manolescu, M.J. (2007). Fuzzy and Neural Controllersfor a Pneumatic Actuator. Int J Comput Commun, ISSN 1841-9836, 2(4):375-387.[11] Alvandar, S., Nigam, M. J. (2008). Neuro-Fuzzy based approach for inverse kinematicssolution of industrial robot manipulators. Int J Comput Commun, ISSN 1841-9836,3(3):224–234.[12] Mahidzal, D., and Jian-Ding, T. (2011). Forward and Inverse Kinematics Model for RoboticWelding Process Using KR-16KS KUKA Robot. Proc. IEEE Int. Conf. on Modeling, Simu-lation and Applied Optimization, 1–6.[13] Shah, J., Rattan S. S., Nakra, B.C.(2011). Kinematic Analysis of 2-DOF planer robot usingartiﬁcial neural network. World Academy of Science, Engineering and Technology,81: 282–285.[14] Parikh, J.P., Lam, S.S. (2009). Solving the forward kinematics problem in parallel manipu-lators using an iterative artiﬁcial neural network strategy. Int. J. Adv. Manuf. Technol.,40: 595–606.[15] Her, M.-G., Chen, C.Y., Karkoub, M. (2002). Approximating a Robot Inverse KinematicsSolution Using Fuzzy Logic Tuned by Genetic Algorithms. Int. J. Adv. Manuf. Technol.,20: 375–380.

Solving Continuous Trajectory and Forward Kinematics Simultaneously Based on ANN

http://univagora.ro/jour/index.php/ijccc/article/view/112/pdf_108

Solving Continuous Trajectory and Forward Kinematics Simultaneously Based on ANN

Abstract

Similar works

Full text

Available Versions

Agora University Editing House: Journals