optimaltrajectoryplanningofgrindingrobotbasedon...

Research ArticleOptimal Trajectory Planning of Grinding Robot Based onImproved Whale Optimization Algorithm

Ting Wang Zhijie Xin Hongbin Miao Huang Zhang Zhenya Chen and Yunfei Du

School of Mechanical Engineering North University of China Taiyuan 030051 China

Correspondence should be addressed to Ting Wang 2319263087qqcom

Received 30 May 2020 Accepted 31 July 2020 Published 19 August 2020

Academic Editor Andras Szekrenyes

Copyright copy 2020 Ting Wang et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Robot will be used in the grinding industry widely to liberate human beings from harsh environments In the grinding processoptimal trajectory planning will not only improve the processing quality but also improve the machining efficiency -e aims ofthis study are to propose a new algorithm and verify its efficiency in achieving the optimal trajectory planning of the grindingrobot An objective function has been defined terms of both time and jerk Improved whale optimization algorithm (IWOA) isproposed based on whale optimization algorithm (WOA) and differential evolution algorithm (DE) Mutation operation andselection operation of DE are imitated in the part of initialization to process the population initialized by WOA and then thesearch tasks of WOA are performed Motion with a constant velocity of the end-effector is considered during fine grinding -econtinuity of acceleration and velocity will be achieved by minimizing jerk and at the same time smooth robot movement can beobtained Cubic spline interpolation is implemented A six-axis industrial robot is used for this research Results show that optimaltrajectory planning based on IWOA is more efficient than others -is method presented in this paper may have some indirectsignificance in robot business

1 Introduction

-e optimal trajectory planning of a robot is based on theoptimization of parameters and the improvement ofmethods -e most significant optimization criteria are theminimum execute time minimum energy and minimumjerk Trajectory planning in Cartesian space is intuitive andthe movement trajectory and attitude of the end-effector areeasy to observe But this method did not consider problemscaused by kinematic singularities [1ndash4] Trajectory planningin joint space can avoid those problems -e joint trajec-tories are obtained by interpolating functions which satisfykinematic and dynamic constraints -e trajectory based ona system controller is easier to adjust in joint space [5ndash7]

A time-optimal trajectory planning method with con-straints of velocity acceleration and jerk is presented in [8]-is optimization method adopts a hybrid optimizationstrategy and solves the problem from path description ki-nematics of the robot and constraints -e resulting tra-jectory is optimal relative to time but not smooth enough

An analytic method is used to obtain the solution of theminimum jerk trajectory of a point-to-point path A min-imum-timejerk algorithm is presented and it is verified byexperiments in [9] A direct analytical method is proposed in[10] to achieve the linear transformation of the industrialrobotrsquos characteristic equation which is useful for industrialrobot trajectory planning and control According to [11]time-optimal trajectory planning can be considered as anonconvex problem Previous robot trajectory planningmainly focused on time optimization [12 13] In the recentyears multiobjective optimization under kinematic con-straint has been under attention so as to improve the effi-ciency and quality and save energy Meanwhile manyalgorithms used to achieve optimal trajectory planning ofrobot are emerged In order to get the global minimum jerkfor trajectory planning of a 2-dof planar robot a newmethod using GA is proposed in [14] Aiming at trajectoryplanning of a free-floating robot a jerk optimal trajectoryplanning algorithm based on particle swarm optimizationalgorithm is proposed in [15] Joint Angle trajectory is

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 3424313 8 pageshttpsdoiorg10115520203424313

smooth and continuous and the driving torque of the robotjoint angle is minimized while the attitude adjustment of thefree-floating space robot is completed To obtain the optimalsolution for the trajectory of the space robot a multi-objective particle swarm optimization method is used in[16] Invasive Weed Optimization algorithm is used to re-search the optimal solution for a robot arm in [17] Weconsider general asymmetric bounds on velocity accelera-tion and jerk [18] and present a complete trajectory gen-eration algorithm A multiobjective differential evolutionalgorithm proposed in [19] is used in trajectory planning of amobile robot A multiobjective teaching learning-basedoptimization method proposed in [20] is used to achieveoptimal trajectory planning of a welding robot In [21] acombination of the modified potential field and numericalapproach-based Jacobian to achieve path planning of aredundant robot is proposed An improved B-spline curve isproposed which made the angular velocity and angularacceleration smooth and continuous during trajectorymovement and the path of the robot end-effector in theCartesian space is improved while guaranteeing thesmoothness of joint movement [22] Cubic spline is used tointerpolate trajectories to ensure the continuity of jerk in[14] Based on improved cuckoo search algorithm a 3-5-3polynomial interpolation trajectory planning algorithm isproposed in [23] Aiming at an autonomous operation of themobile system Pirnık et al [24] present a designed algo-rithm which is able to integrate inertial sensor data and canimprove the trajectory planning efficiency

-e optimization methods proposed by differentscholars are applied in specific scenes In this paper IWOAis used in grinding robot optimal trajectory planning withcertain objective function and the optimization results areinterpolated by cubic spline -e paper is organized asfollows in Section 2 an optimization problem is introducedand objective function is defined Section 3 describes thedetails of WOA and IWOA In Section 4 grinding path andconstraints of kinematic are given Section 5 shows theresults of some simulation and comparison of the jerk andwork time before and after optimization In Section 6 theconclusion is given and the reasons of optimization resultsare discussed

2 Define Objective Function

-e smoothness of robot motion plays an important role inrobot grinding because it will help reduce vibration andimprove the grinding quality In order to grind the work-piece surface effectively and obtain uniform material re-moval it is necessary to plan each point on the grindingpath Better continuity of speed and acceleration can preventthe mechanical body from shock overshoot and oscillation[25] By reducing the jerk the stability of the robot canimprove and the continuity of speed and acceleration can beachieved But the execution time will increase significantlyand will affect the production beat -erefore trajectoryplanning is needed to meet the requirements of minimumtime and minimum jerk in some extent by introducingweight parameters KT and KJ Firstly the grinding path is

defined in the Cartesian space and then the inverse kine-matics is solved for the ABB IRB2600 robot to obtain theangles of each joint Considering the constraints of speedacceleration and jerk the objective function is defined asfollows

min F KT 1113944

nminus1

i1li + KJ 1113944

M

J11113944

nminus1

i1

1li

1113946tw

t0Pj (t)1113874 1113875

2dt

1113971

(1)

with

Pjprime(t)

11138681113868111386811138681113868

11138681113868111386811138681113868leVmj

PjPrime(t)

11138681113868111386811138681113868

11138681113868111386811138681113868leAmj

Pj (t)11138681113868111386811138681113868

11138681113868111386811138681113868le Jmj

(2)

where j 1 2 M is the number of joints andi 1 2 n is the number of points in the grinding path-e more the points extracted on the curve the closer it isto the contour curve of the workpiece But too many pathpoints will affect the grinding efficiency of the robot sodefining an appropriate number of path points is impor-tant KT and KJ are weight coefficients li is the compu-tational time interval between two continuous points onthe joint path tw is the whole grinding time |Pj

prime(t)| |PjPrime(t)|

and |Pj (t)| are velocity acceleration and jerk of the jthjoint respectively Vmj Amj and Jmj are the upperbounds of velocity acceleration and jerk of the jth jointrespectively

-e first part of (1) represents the running time of therobot and the second part is an expression about jerk Jerk isvectors so to introduce it into the objective function takingits square for operation cubic spline interpolation isselected

li max

j

Pji+1 minus Pji

11138681113868111386811138681113868

11138681113868111386811138681113868

Pprime i 1 2 n j 1 2 M

(3)

P

Pji+1Prime minus Pji

Prime

li (4)

where i 1 2 n and j 1 2 M According to thepositions in joint space time interval and the constraints ofthe velocity and acceleration the robot acceleration matrix isobtained and used to solve (3)

3 Improved WOA

31 WOA WOA is an efficiency nature-inspired meta-heuristic optimization algorithm which mimics the socialbehavior of humpback whales -e efficiency of the WOA isevaluated by solving 29 mathematical optimization prob-lems and six structural optimization problems and opti-mization results demonstrate that WOA is very competitivecompared to the state-of-the-art optimization methods [26]WOA is used to solve many industrial problems and il-lustrated in the literature [27ndash29]

2 Mathematical Problems in Engineering

Step 1 the population is initializedStep 2 the fitness value of each search agent is cal-culated and the current best individual is selectedStep 3 the strategy of shrinking circle and up withspiral-shape is selectedStep 4 it is checked whether each search agent exceedsthe search boundary the fitness value of each searchagent is calculated and the optimal position is updatedStep 5 the maximum iteration is reached and the finalresults are obtained

Details about Step 3 are as follows

311 Exploitation Humpback whales move around theirprey in a shrinking circle while following a spiraling path Tomodel this behavior we assume that there is a 50 chance ofchoosing between shrinking encircling mechanism and aspiral model to update the whalersquos position -e mathe-matical model is as follows

Xrarr

(t + 1) Xlowastrarr(t) minus A

rarrmiddot Drarr

ifplt 05

Dprimerarr

middot eblmiddot cos(2πl) + Xlowastrarr(t) ifpge 05

⎧⎪⎨

⎪⎩(5)

where p is a random number in [01] t is the current it-

eration Ararr

and Drarr

are coefficient vectors Xlowastrarris the position

of the best solution so far and Xrarr

is the position vector Xlowast is

a better solution that will be updated in each iteration Dprimerarr

|Xlowastrarr(t) minus X

rarr(t)| indicates the distance of the ith whale to the

prey b is a constant for defining the shape of logarithmicspiral and l is a random number in [minus1 1]

312 Encircling Prey WOA assumes that the current so-lution is the target prey First the best search agent is de-fined and then the other search agents will attempt toupdate their positions towards the best search agent -isbehavior is expressed in the following equations

Drarr

Crarr

middot Xlowastrarr(t) minus X

rarr(t)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (6)

Xrarr


rarrmiddot Drarr

(7)

-e vectors Ararr

and Crarr

are calculated as follows

Ararr

2 ararr

middot rrarr

minus ararr

(8)

Crarr

2 middot rrarr

(9)

where ararr is linearly decreased from 2 to 0 and r

rarr is a randomvector in [0 1]

313 Prey Search Humpback whales search randomlybased on each otherrsquos position -erefore a random valuegreater than 1 or less than minus1 is used to force the search agentaway from a reference individual -e position of the searchagent is updated according to the random selected search

agent -e mechanism and the |Ararr

|gt 1 allow WOA achieveglobal search -e mathematical model is as follows

Drarr

Crarr

middot Xrandrarr

minus Xrarr11138681113868111386811138681113868

11138681113868111386811138681113868 (10)

Xrarr

(t + 1) Xrandrarr

minus Ararr

middot Drarr

(11)

where Xrandrarr

is a random position vector chosen from thecurrent population

32 IWOA DE has been proved to be an effective globalsearch algorithm in solving complex global optimizationproblems DE and WOA are combined to propose a newalgorithm (IWOA) which has better ability of global op-timization and great robust Figure 1 is P-code of IWOA

By imitatating the operation of mutation and selection inDE the initialize program of WOA is improved -is newinitialize program makes population more representativeand easier to obtain a global optimal solution Populationsize is considered as the number of whales Gm is themaximum number of iterations mutation factor F is cal-culated by (11) and F0 is the initial value of F

F F0 middot 2exp1 minus Gm

Gm1113874 1113875 (12)

321 Mutation Operation For the whale position XhG

(h 1 2 N) a new position VhG+1 can be generated bythe following formula

VhG+1 Xr1G + F middot Xr2G minus Xr3G1113872 1113873 (13)

where N is the number of whales and r1 and r2 are differentintegers randomly selected in [1 N] and instead of index h-e mutation factor is a real constant on [0 2] and theamplification of differential vector Xr2G minus Xr3G iscontrolled

322 Selection Operation In order to determine whethervector VhG+1 can become new positions for whales in theiteration of G+ 1 it will be compared with XhG and if thefitness value of the former is better than the latter XiG willbe replaced by VhG+1 in the G+ 1 iteration otherwise XhG

will be retained -en positions of whales are updated

4 Implement

In the process of robot fine grinding keeping the grindingspeed as a constant contributes to control the material re-moval rate so as to effectively control the grinding quality-erefore the grinding speed is introduced as the inter-mediate variable and optimized by IWOA-en the optimalmotion parameters of each joint of the robot are obtained-e constraints of speed acceleration and jerk are shown inTable 1

-e first grinding path is divided into 50 trajectorypoints as shown in Figure 2

Mathematical Problems in Engineering 3

-en the joint displacements of six axis for the ABBIRB2600 grinding robot are obtained by inverse kinematicssolving -e value of KT and KJ are chosen after trial ofvarying the values of each from 0 to 1 and set as 04 and 06respectively

-e joint positions are used as the input of the IWOAand MATLAB 2018b software is used to complete the op-timization process ABB RobotStudio software is used forsimulation

5 Results

-e jerk values before and after optimization with respect tothe intermittent points on the grinding path are illustrated inFigures 3(a)ndash3(f) In Figure 3(g) every jointrsquos position andorientation are given -e comparison of whole grindingtime before and after optimization with four kinds of al-gorithm is shown in Table 2

From Figure 3 it can be observed that jerk after opti-mization is significantly reduced compared with jerk beforeoptimization and fluctuated around a constant value -emaximum value of jerk before optimization for every joint is250 degs3 250 degs3 250 degs3 250 degs3 375 degs3and 3375 degs3 -e maximum value of jerk after opti-mization for every joint is 0144 degs3 0034 degs30066 degs3 4136 degs3 0039 degs3 and 0042 degs3 Nomatter before or after optimization the fluctuation range ofjoint 4 is significantly larger than others

According to Table 2 it can be observed that grindingtime after optimization is increased -e growing rate byusing GA is 11195 using DE is 9597 using WOA is7126 and using IWOA is 6867 Also the growing rate isthe smallest by using IWOA

6 Discussion and Conclusions

In this paper the objective function including an expressionof jerk and time of total work for every joint IWOA isproposed by combining DE andWOA and IWOA is used toget optimal trajectory of the grinding robot After optimi-zation the goal of jerk being smaller is achieved and time isnot increased significantly

Jerk is reduced significantly by using IWOA Afteroptimization jerk of every joint has reduced significantly Byreducing jerk the movement path is smoother and the speed

Initialize the whales population Xi (i = I 2 n)Perform mutation operations by the eq (12)Perform selection operationsUpdate the whales population XiCalculate the fitness of each search agentXlowast = the best search agentwhile (t lt maximum number of iterations)

for each search agentUpdate a A C I and p

if1 (p lt 05)if2 (|A| lt 1)

Update the position of the current search agent by the eq (6)else if2 (|A| ge 1)Select a random search agent (Xrand)Update the position of the current search agent by the eq (10)end if2

else if1(p ge 05)Update the position of the current search by the eq (4)

end if1end forCheck if any search agent goes beyond the search space and amend itCalculate the fitness of each search agentUpdate Xlowast if there is a better solutiont = t + 1

end whilereturn Xlowast

Figure 1 Pseudocode of IWOA

Table 1 Kinematic limits of a six-axis robot

Joint Velocity (degs) Acceleration (degs2) Jerk (degs3)1 175 100 1202 175 100 903 175 100 1254 360 100 2005 360 100 1806 500 100 230

yndashaxis

ndash60

ndash40

ndash20

0

20

40

60

ndash60 ndash40 ndash20 0 20 40 60 80ndash80xndashaxis

Figure 2 Defined trajectory points in the Cartesian coordinatesystem


jerk1

BeforeAfter

ndash3

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(a)

jerk2

BeforeAfter

5 10 15 20 25 30 35 40 45 500x

ndash3

ndash2

ndash1

0

1

2

3

(b)

jerk3

BeforeAfter

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(c)

jerk4

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

150

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(d)

Figure 3 Continued


and acceleration are more continuous which can improvethe grinding quality to some extent -e jerk fluctuationrange of joint 4 is significantly larger than that of othersbecause joint 4 has the biggest range of motion so as toensure the end-effector passes every point on the grindingpath

IWOA is effective in performing an optimal trajectoryplanning of a robot Compared with other three algorithmsthe growing rate of time is the smallest by using IWOA -ereason is IWOA performa mutation and selection operationfor the population initialized by WOA Mutation and se-lection operation of IWOA mimic the operation of DE

jerk5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(e)jerk6

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(f )

(1)

(2)

(3)(4)(5)

(6)

(g)

Figure 3 Jerk values of joints with respect to each trajectory point before and after optimization (a) Jerk values of joint 1 (b) jerk values ofjoint 2 (c) jerk values of joint 3 (d) jerk values of joint 4 (e) jerk values of joint 5 (f ) jerk values of joint 6 and (g) joints of the robot

Table 2 -e comparison of whole grinding time before and after optimization with algorithms

Algorithm Grinding time before (s) Grinding time after (s) Growing rate ()GA 2008 4256 11195DE 2008 3955 9597WOA 2008 3439 7126IWOA 2008 3387 6867


-en the initial population becomes more representativeand the algorithm is more robust

-is new method of optimal trajectory planning basedon IWOAwill be useful in the industry of robot grinding forexample the grinding of welding line for automobile and itsaccessories and burrs for hardware products

Data Availability

All data are defined according to the needs of research andresults are achieved by software

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was funded by the Central Government GuidedLocal Science and Technology Development Projects ofChina under grant no YDZX20191400002765 and KeyResearch and Development Projects of Shanxi Provinceunder grant no 201903D421006

References

[1] C Zheng Y Su and P C Muller ldquoSimple online smoothtrajectory generations for industrial systemsrdquo Mechatronicsvol 19 no 4 pp 571ndash576 2009

[2] S Zhang A Maria Zanchettin R Villa and S Dai ldquoReal-timetrajectory planning based on joint-decoupled optimization inhuman-robot interactionrdquo Mechanism and Machine -eoryvol 144 2020

[3] G Xuan and Y Shao ldquoReverse-driving trajectory planningand simulation of joint robotrdquo IFAC-PapersOnLine vol 51no 17 pp 384ndash388 2018

[4] S Chiaverini B Siciliano and O Egeland ldquoExperimentalresults on controlling a 6-DOF robot manipulator in theneighborhood of kinematic singularitiesrdquo in ExperimentalRobotics III Lecture Notes in Control and Information Sci-ences T Yoshikawa and F Miyazaki Eds vol 200 pp 1ndash13Springer Berlin Germany 1994

[5] H Wang H Wang J Huang B Zhao and L Quan ldquoSmoothpoint-to-point trajectory planning for industrial robots withkinematical constraints based on high-order polynomialcurverdquo Mechanism and Machine -eory vol 139 pp 284ndash293 2019

[6] G Chen B Yuan Q Jia Y Fu and J Tan ldquoTrajectory op-timization for inhibiting the joint parameter jump of a spacemanipulator with a load-carrying taskrdquo Mechanism andMachine -eory vol 140 pp 59ndash82 2019

[7] X Liu C Qiu Q Zeng and A Li ldquoKinematics Analysis andTrajectory Planning of collaborative welding robot withmultiple manipulatorsrdquo Procedia CIRP vol 81 pp 1034ndash1039 2019

[8] J Dong P M Ferreira and J A Stori ldquoFeed-rate optimi-zation with jerk constraints for generating minimum-timetrajectoriesrdquo International Journal of Machine Tools ampManufacture vol 47 no 12-13 pp 1941ndash1955 2007

[9] V Zanotto A Gasparetto A Lanzutti P Boscariol andR Vidoni ldquoExperimental validation of minimum time-jerkalgorithms for industrial robotsrdquo Journal of Intelligent ampRobotic Systems vol 64 no 2 pp 197ndash219 2011

[10] P Bozek Z Ivandic L Alexander V Lyalin and V TarasovldquoSolutions to the characteristic equation for industrial robotrsquoselliptic trajectoriesrdquo Tehnicki Vjesnik vol 23 no 4pp 1017ndash1023 2016

[11] A Palleschi R Mengacci F Angelini et al ldquoTime-optimaltrajectory planning for flexible joint robotsrdquo IEEE Roboticsand Automation Letters vol 5 no 2 pp 938ndash945 2020

[12] H Liu X Lai and W Wu ldquoTime-optimal and jerk-contin-uous trajectory planning for robot manipulators with kine-matic constraintsrdquo Robotics and Computer-IntegratedManufacturing vol 29 no 2 pp 309ndash317 2013

[13] F J Abu-Dakka I F Assad R M Alkhdour andM Abderahim ldquoStatistical evaluation of an evolutionary al-gorithm for minimum time trajectory planning problem forindustrial robotsrdquo -e International Journal of AdvancedManufacturing Technology vol 89 no 1ndash4 pp 389ndash406 2017

[14] P Aurelio and V Antonio ldquoGlobal minimum-jerk trajectoryplanning of robot manipulatorsrdquo IEEE Transactions on In-dustrial Electronics vol 47 pp 140ndash149 2000

[15] Q Hu Y Wang and Z Shi ldquoMinimum torque trajectoryplanning algorithm for free-floating space robotrdquo Journal ofHarbin Institute of Technology vol 43 no 11 pp 20ndash24 2011

[16] P Huang K Chen and J Yuan ldquoMotion trajectory planningof space manipulator for joint jerk minimizationrdquo in Pro-ceedings of the 2007 IEEE International Conference onMechatronics and Automation pp 3543ndash3548 New YorkNY USA 2007

[17] S Abhronil C Tathagata and K Amit ldquoEnergy efficienttrajectory planning by a robot arm using invasive weed op-timization techniquerdquo -ird World Congress on Nature AndBiologically Inspired Computing vol 43 pp 311ndash316 2011

[18] S Daniel and D Kevin ldquoSmooth cubic polynomial trajectoriesfor human-robot interactionsrdquo Journal of Intelligent amp Ro-botic Systems vol 95 no 3-4 pp 851ndash869 2019

[19] V Sathiya and M Chinnadurai ldquoEvolutionary algorithms-based multi-objective optimal mobile robot trajectory plan-ningrdquo Robotica vol 37 no 8 pp 1363ndash1382 2019

[20] R Amruta B Deepak and B Biswal Bibhuti ldquoOptimaltrajectory generation of an industrial welding robot withkinematic and dynamic constraintsrdquo Industrial Robot--eInternational Journal of Robotics Research and Applicationvol 47 no 1 pp 68ndash75 2020

[21] S-O Park M C Lee and J Kim ldquoTrajectory planning withcollision avoidance for redundant robots using jacobian andartificial potential field-based real-time inverse kinematicsrdquoInternational Journal of Control Automation and Systemsvol 18 no 8 pp 2095ndash2107 2020

[22] Y Li T Huang and D G Chetwynd ldquoAn approach forsmooth trajectory planning of high-speed pick-and-placeparallel robots using quintic B-splinesrdquo Mechanism andMachine -eory vol 126 pp 479ndash490 2018

[23] W Wang Q Tao Y Cao X Wang and X Zhang ldquoRobottime-optimal trajectory planning based on improved cuckoosearch algorithmrdquo IEEE Access vol 8 pp 86923ndash86933 2020

[24] R Pirnık M Hrubos D Nemec T Mravec and P BozekldquoIntegration of inertial sensor data into control of the mobileplatformrdquo in Proceedings of the 2015 Federated Conference onSoftware Development and Object Technologies Springer In-ternational Publishing Berlin Germany pp 271ndash282 2015

[25] Y Sun Study on the Key Problems of Solid Rocket EnginesInner-Wall Insulation Grinding Robot Northeastern Uni-versity Boston MAUSA 2009


[26] M Seyedali and L Andrew ldquo-e whale optimization algo-rithmrdquo Advances in Engineering Software vol 95 pp 51ndash672016

[27] M A Elhosseini A Y Haikal M Badawy and N KhashanldquoBiped robot stability based on an A-C parametric WhaleOptimization Algorithmrdquo Journal of Computational Sciencevol 31 pp 17ndash32 2019

[28] H Chen Y Xu M Wang and X Zhao ldquoA balanced whaleoptimization algorithm for constrained engineering designproblemsrdquo Applied Mathematical Modelling vol 71 pp 45ndash59 2019

[29] M M Ahmed A A Mahmoud and Y A Almoataz ldquoWhaleoptimization algorithm to tune PID and PIDA controllers onAVR systemrdquo Ain Shams Engineering Journal vol 71pp 1ndash13 2019


smooth and continuous and the driving torque of the robotjoint angle is minimized while the attitude adjustment of thefree-floating space robot is completed To obtain the optimalsolution for the trajectory of the space robot a multi-objective particle swarm optimization method is used in[16] Invasive Weed Optimization algorithm is used to re-search the optimal solution for a robot arm in [17] Weconsider general asymmetric bounds on velocity accelera-tion and jerk [18] and present a complete trajectory gen-eration algorithm A multiobjective differential evolutionalgorithm proposed in [19] is used in trajectory planning of amobile robot A multiobjective teaching learning-basedoptimization method proposed in [20] is used to achieveoptimal trajectory planning of a welding robot In [21] acombination of the modified potential field and numericalapproach-based Jacobian to achieve path planning of aredundant robot is proposed An improved B-spline curve isproposed which made the angular velocity and angularacceleration smooth and continuous during trajectorymovement and the path of the robot end-effector in theCartesian space is improved while guaranteeing thesmoothness of joint movement [22] Cubic spline is used tointerpolate trajectories to ensure the continuity of jerk in[14] Based on improved cuckoo search algorithm a 3-5-3polynomial interpolation trajectory planning algorithm isproposed in [23] Aiming at an autonomous operation of themobile system Pirnık et al [24] present a designed algo-rithm which is able to integrate inertial sensor data and canimprove the trajectory planning efficiency

-e optimization methods proposed by differentscholars are applied in specific scenes In this paper IWOAis used in grinding robot optimal trajectory planning withcertain objective function and the optimization results areinterpolated by cubic spline -e paper is organized asfollows in Section 2 an optimization problem is introducedand objective function is defined Section 3 describes thedetails of WOA and IWOA In Section 4 grinding path andconstraints of kinematic are given Section 5 shows theresults of some simulation and comparison of the jerk andwork time before and after optimization In Section 6 theconclusion is given and the reasons of optimization resultsare discussed

2 Define Objective Function

-e smoothness of robot motion plays an important role inrobot grinding because it will help reduce vibration andimprove the grinding quality In order to grind the work-piece surface effectively and obtain uniform material re-moval it is necessary to plan each point on the grindingpath Better continuity of speed and acceleration can preventthe mechanical body from shock overshoot and oscillation[25] By reducing the jerk the stability of the robot canimprove and the continuity of speed and acceleration can beachieved But the execution time will increase significantlyand will affect the production beat -erefore trajectoryplanning is needed to meet the requirements of minimumtime and minimum jerk in some extent by introducingweight parameters KT and KJ Firstly the grinding path is

defined in the Cartesian space and then the inverse kine-matics is solved for the ABB IRB2600 robot to obtain theangles of each joint Considering the constraints of speedacceleration and jerk the objective function is defined asfollows

min F KT 1113944

nminus1

i1li + KJ 1113944

M

J11113944

nminus1

i1

1li

1113946tw

t0Pj (t)1113874 1113875

2dt

1113971

(1)

with

Pjprime(t)

11138681113868111386811138681113868

11138681113868111386811138681113868leVmj

PjPrime(t)

11138681113868111386811138681113868

11138681113868111386811138681113868leAmj

Pj (t)11138681113868111386811138681113868

11138681113868111386811138681113868le Jmj

(2)

where j 1 2 M is the number of joints andi 1 2 n is the number of points in the grinding path-e more the points extracted on the curve the closer it isto the contour curve of the workpiece But too many pathpoints will affect the grinding efficiency of the robot sodefining an appropriate number of path points is impor-tant KT and KJ are weight coefficients li is the compu-tational time interval between two continuous points onthe joint path tw is the whole grinding time |Pj

prime(t)| |PjPrime(t)|

and |Pj (t)| are velocity acceleration and jerk of the jthjoint respectively Vmj Amj and Jmj are the upperbounds of velocity acceleration and jerk of the jth jointrespectively

-e first part of (1) represents the running time of therobot and the second part is an expression about jerk Jerk isvectors so to introduce it into the objective function takingits square for operation cubic spline interpolation isselected

li max

j

Pji+1 minus Pji

11138681113868111386811138681113868

11138681113868111386811138681113868

Pprime i 1 2 n j 1 2 M

(3)

P

Pji+1Prime minus Pji

Prime

li (4)

where i 1 2 n and j 1 2 M According to thepositions in joint space time interval and the constraints ofthe velocity and acceleration the robot acceleration matrix isobtained and used to solve (3)

3 Improved WOA

31 WOA WOA is an efficiency nature-inspired meta-heuristic optimization algorithm which mimics the socialbehavior of humpback whales -e efficiency of the WOA isevaluated by solving 29 mathematical optimization prob-lems and six structural optimization problems and opti-mization results demonstrate that WOA is very competitivecompared to the state-of-the-art optimization methods [26]WOA is used to solve many industrial problems and il-lustrated in the literature [27ndash29]





Xrarr


rarrmiddot Drarr

ifplt 05

Dprimerarr


⎧⎪⎨

⎪⎩(5)


eration Ararr

and Drarr









Drarr

Crarr


rarr(t)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (6)

Xrarr


rarrmiddot Drarr

(7)

-e vectors Ararr

and Crarr


Ararr

2 ararr

middot rrarr

minus ararr

(8)

Crarr

2 middot rrarr

(9)






Drarr

Crarr

middot Xrandrarr

minus Xrarr11138681113868111386811138681113868

11138681113868111386811138681113868 (10)

Xrarr

(t + 1) Xrandrarr

minus Ararr

middot Drarr

(11)

where Xrandrarr





Gm1113874 1113875 (12)







4 Implement






5 Results









if1 (p lt 05)if2 (|A| lt 1)








yndashaxis

ndash60

ndash40

ndash20

0

20

40

60




jerk1

BeforeAfter

ndash3

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(a)

jerk2

BeforeAfter

5 10 15 20 25 30 35 40 45 500x

ndash3

ndash2

ndash1

0

1

2

3

(b)

jerk3

BeforeAfter

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(c)

jerk4

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

150

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(d)

Figure 3 Continued




jerk5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(e)jerk6

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(f )

(1)

(2)

(3)(4)(5)

(6)

(g)







Data Availability




Acknowledgments


References



































Xrarr


rarrmiddot Drarr

ifplt 05

Dprimerarr


⎧⎪⎨

⎪⎩(5)


eration Ararr

and Drarr









Drarr

Crarr


rarr(t)

111386811138681113868111386811138681113868

111386811138681113868111386811138681113868 (6)

Xrarr


rarrmiddot Drarr

(7)

-e vectors Ararr

and Crarr


Ararr

2 ararr

middot rrarr

minus ararr

(8)

Crarr

2 middot rrarr

(9)






Drarr

Crarr

middot Xrandrarr

minus Xrarr11138681113868111386811138681113868

11138681113868111386811138681113868 (10)

Xrarr

(t + 1) Xrandrarr

minus Ararr

middot Drarr

(11)

where Xrandrarr





Gm1113874 1113875 (12)







4 Implement






5 Results









if1 (p lt 05)if2 (|A| lt 1)








yndashaxis

ndash60

ndash40

ndash20

0

20

40

60




jerk1

BeforeAfter

ndash3

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(a)

jerk2

BeforeAfter

5 10 15 20 25 30 35 40 45 500x

ndash3

ndash2

ndash1

0

1

2

3

(b)

jerk3

BeforeAfter

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(c)

jerk4

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

150

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(d)

Figure 3 Continued




jerk5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(e)jerk6

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(f )

(1)

(2)

(3)(4)(5)

(6)

(g)







Data Availability




Acknowledgments


References


































5 Results









if1 (p lt 05)if2 (|A| lt 1)








yndashaxis

ndash60

ndash40

ndash20

0

20

40

60




jerk1

BeforeAfter

ndash3

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(a)

jerk2

BeforeAfter

5 10 15 20 25 30 35 40 45 500x

ndash3

ndash2

ndash1

0

1

2

3

(b)

jerk3

BeforeAfter

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(c)

jerk4

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

150

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(d)

Figure 3 Continued




jerk5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(e)jerk6

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(f )

(1)

(2)

(3)(4)(5)

(6)

(g)







Data Availability




Acknowledgments


References
































jerk1

BeforeAfter

ndash3

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(a)

jerk2

BeforeAfter

5 10 15 20 25 30 35 40 45 500x

ndash3

ndash2

ndash1

0

1

2

3

(b)

jerk3

BeforeAfter

ndash2

ndash1

0

1

2

3

5 10 15 20 25 30 35 40 45 500x

(c)

jerk4

ndash250

ndash200

ndash150

ndash100

ndash50

0

50

100

150

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(d)

Figure 3 Continued




jerk5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(e)jerk6

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(f )

(1)

(2)

(3)(4)(5)

(6)

(g)







Data Availability




Acknowledgments


References


































jerk5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(e)jerk6

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

5 10 15 20 25 30 35 40 45 500x

BeforeAfter

(f )

(1)

(2)

(3)(4)(5)

(6)

(g)







Data Availability




Acknowledgments


References


































Data Availability




Acknowledgments


References
































optimaltrajectoryplanningofgrindingrobotbasedon...

Documents