Inverse Kinematics Analysis of a Five-Degree of Freedom Educated Type of Alpha Robot Using Genetic Algorithm

Inverse Kinematics Analysis Of a Five-Degree Of Freedom Educated Type Of Alpha Robot Using Genetic Algorithm

Israa Rafie Shareef Mechatronics Engineering Dept. Al Khawarzmi College of Engineering University of Baghdad \Iraq

Abstract

Inverse Kinematic is considered as one of the complicated issues in robots domain according to the nonlinearity , multiple solutions , the need for complete knowledge about robotic environment besides to other factors. Whereas a great development has appeared in using the artificial intelligence methods to replace the classical analyzing methods. In this paper , an approach for using the Genetic Algorithm is suggested to deal with the Inverse Kinematics problem of a five degree of freedom robot . Firstly the forward and inverse kinematics of the robot is introduced , then the inverse kinematic equations are solved by the Genetic Algorithm with the aid of MATLAB programming , the results were compared using three ways , firstly by the classic analytical equations , secondly by the robot simulation program ROBCIM , and lastly by the MATLAB Genetic Algorithm Toolbox . Genetic Algorithm had accomplished good results , and got rid of many limitations which may face other analytical methods . Key-Words : Robotic Manipulator , Inverse Kinematics (I. K.) , Genetic Algorithm (G. A) .

1-Introduction It contains chain of links , with an According to the definition end –effector or gripper carrying the presented by the International needed tool (like welding gun or Standard Organization (ISO) , the drill) , these links are attached by robotic manipulator (named as joints (either prismatic or revolute) industrial robot or as robotic arm) is .while , saying 5-axes means having an automated, programmed ,multi- five independently moving degrees purposes machine which deals with of freedom (DOF) [ 16.] several domains to handle work The robotic kinematic structure deals parts. with the calculation of the end effector position (without outer

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36 effected force) , and to make the I.K. may has multiple solutions due robotic manipulator adopt the to the non linearity in equations and wanted configuration [ 22 ]. the robot configuration's dependence The manipulator has non-linearity [ 23 ]. kinematic equations to be solved [1]. I.K. is considered difficult because The robot kinematic analysis is of : many unknowns equations handled in two ways ; forward or variables , multiple solutions , some direct (F. K.) , and inverse kinematic solution are undefined [12 ]. (I. K.) , where the second one is the Many methods were used to solve most important because it gives the the I.K. problems like jacobian robotic joints' values to reach any inversion , optimization , numerical spatial point . Thus , the I.K. is the and lastly the Artificial Intelligence important factor in following paths , methods used like Neural Networks , and controlling robotic motion [ 9 ]. Fuzzy , and Genetic Algorithms In (F. K.),we have the position of the [14]. end-effector (x,y,z) and its In this paper , we will analyze the orientation (θx , θy ,θz ) = f(q) I.K. of a 5-DOF robotic manipulator, Function of (q) the joints' variables using one of the most important either revolute or prismatic. Artificial Intelligence methods , While in I.K. , the opposite case is which is the Genetic Algorithm. occurred where , q=f −1(x,y,z) and Reaching to this goal , we will start (θx , θy ,θz ). by defining the used robot , knowing For its importance ,I.K. had been its physical characteristics , D-H studied for years , for its roll in (Denavit – Hartenberg) manipulator's controlling , as the representation , transformation robot actuators deal with joint space matrix and analyzing the F.K. of this , while the end-effector deals with robot , besides to finding the I.K. cartesian space , thus , the I.K. is the equations using the classical way to transform from the cartesian approach . to the joints space [11] . Estimating the I.K. equations of this Good development had been robot using the G.A. approach will achieved in I.K. for the recent years be simulated using the MATLAB and many approaches were Genetic Algorithm Toolbox [ 5 ]. presented [ 14 ].

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The robot which will be discussed simulated online and offline by for its I.K. , is a scale model for the computer . famous robot (ALPHA) robot as mentioned in [ 20 ] , that's because of the high cost of the original robot This robot has Base , Shoulder , [8] , we substitute by this model of Elbow , Tool-roll and Tool pitch laboratory robot for the sake of reach joints' motions providing the five approximately to its physical degrees of freedom , besides to a information and limitations , Figure1 grip movement by its two fingers below show the microbot ALPHA in gripper[ 24 ]. its laboratory type . 3 Figure1-b clarifies the link coordinates and D-H parameters of the robot . The most important step in analyzing any robot kinematics , is to find the transformation matrix which relates between the base and the end- effector joints , in order to reach this matrix , we have to establish the Denavit-Hartenberg parameters Figure (1-a) the ALPHA robot which were defined in detail in [20&6 ] , as follows in Table1:

Table1 the robot parameters

Joint θ d a α in in in degree Figure (1-b) Link coordinates of the millimeter millimeter ALPHA robot Base q 1 255 0 - 90 It's a 5-DOF robotic manipulator , Shoulder q 2 0 190 0 articulated arm , delivered with a Elbow q 3 0 190 0 specific software that allows it to be

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Too q 4 0 0 - 90 Applying these parameters in the standard transformation matrix from roll base to tool , we'll get : Tool q 5 115 0 0

pitch

풕풐풐풍 −푪표푠 푞234 푻풃풂풔풆 =

Column 4= 푪표푠 푞1 (190 퐶표푠 푞2 +

where , Column1= 190 퐶표푠 푞23 − 115 푆푖푛 푞234)

푪표푠 푞1 퐶표푠 푞234 퐶표푠 푞5 + 푆푖푛 푞1 푆푖푛 푞5 푺푖푛 푞1 (190 퐶표푠 푞2 + 190 퐶표푠 푞23 ) 푺푖푛 푞1 퐶표푠 푞234 퐶표푠 푞5 − 퐶표푠 푞1 푆푖푛 푞5 − 115 푆푖푛 푞234

ퟐ55 − 190 푆푖푛 푞2 − 190 푆푖푛 푞23 −푺푖푛 푞234 퐶표푠 푞5 − 115 퐶표푠 푞234 0 1 Column2= −푪표푠 푞1 퐶표푠 푞234 푆푖푛 푞5 +

푆푖푛 푞1 퐶표푠 푞5

−푺푖푛 푞1 퐶표푠 푞234 푆푖푛 푞5 − 퐶표푠 푞1 퐶표푠 푞5 Thus the robot –tool-configuration will be : 푺푖푛 푞234 푆푖푛 푞5 푪표푠 푞1 (190 퐶표푠 푞2 + 190 퐶표푠 푞23 − 0 115 푆푖푛 푞234) .. 1 Column 3= −푪표푠 푞1 푆푖푛 푞234 푺푖푛 푞1 (190 퐶표푠 푞2 + 190 퐶표푠 푞23 − −푺푖푛 푞1 푆푖푛 푞234 115 푆푖푛 푞234) .. 2

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ퟐ55 − 190 푆푖푛 푞2 − 190 푆푖푛 푞23 − = 푎22(푐표푠 푞22 + 푠푖푛 푞22) 115 퐶표푠 푞234 .. 3 +푎32(cos 푞232 + sin 푞232)+ 2 a2 a3 (cos q2 cos q23 + sin q2 sin q23) − 퐞xp(푞5) 퐶표푠 푞1 푆푖푛 푞234 … 4 =푎22 + 푎32+2 a2 a3 cos q3 − 퐞xp(푞5) 푆푖푛 푞1 푆푖푛 푞234 … 5 ; 푏2−푎22−푎32 − 퐞xp(푞5) 퐶표푠 푞234 … 6 q 3 = ± arcos 2 푎2 푎3 Now , the Inverse kinematics of this 4 b1 = a2 cos q2 + a3 ( cos q2 cos q3 – robot is : sin q2 sin q3)

b2 = a2 sin q2 + a3 ( sin q2 cos q3 + q 1(the base joint) =atan (w2 , w1) cos q2 sin q3) b 1=cos q1 w1+sin q1 w2-a4 cos q234 b1 = (a2 + a3 cos q3 ) cos q2 – ( a3 sin +d5 sin q234 q3) sin q2 b2 = d1 – a4 sin q234 – d5 cos q234 – b2 = ( a2 + a3 cos q3) sin q2 + ( a3 sin w3 q3) cos q2 taking the expressions of w from tool solving the two simultaneous linear configuration and substitute in b1 , and equations : b2 ; (푎2+푎3 cos 푞3)푏1+(푎3 sin 푞3)푏2 cos q2 = 푏2 b1 = a2 cos q2 + a3 cos q23 q 2 =± arcos 풃ퟏퟐ=푎22 cos 푞22 +2 a2 a3 cos q2 cos (푎2+푎3 cos 푞3)푏1+(푎3 sin 푞3)푏2

q23 +푎32 cos 푞232 푏2 5 or using sin q2 b2=a2 sin q2 + a3 sin q23 (푎2+푎3 cos 푞3)푏2−(푎3 sin 푞3)푏1 = 2 풃ퟐퟐ= 푏 q 2 = atan [(a2 +a3 cos q3) b2 – (q3 sin 푎22 sin 푞22 +2 a2 a3 sin q2 sin q23 q3) b1 , ( a2 + a3 cos q3 ) b1 + (a3 sin +푎32 sin 푞232 q3) b2] 2 2 2 푏 =푏1 + 푏2

q 234 =atan (-(- cos q1 w4 + cos 1 w5 ) , – w6)

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q 4 = q234 – q2 – q3 In GA , processes of "initialization , fitness , selection , cross-over , and q 5 = π ln mutation " would be noticed. The first ((푤4)2+(푤5)2+(푤6)2)^(1/2) generation's parents or individuals will be prepared , then those with best Genetic Algorithm (GA) fitness's function will be combined to deliver new generation of children or GA is a special category of individuals having better fitness values evolutionary algorithms [19] , it is an than their parents . This process will optimization method which mimic the continue till the convergence of the behavior of biological evolution [7], population around some individual's its first appearance was in nineties [15]. values with the best fitness value ever In this paper , a GA is proposed to deal [18]. with the solving of robot's I.K. The next steps will discuss the main equations because of reasons , some of procedures of the G.A. with its them : application for our robot ; 1. GA as an A.I.(Artificial Intelligence) Initialization and encoding 2. Fitness method acts as the robot's mind , 3.Selection 4.Cross over 5. Mutation making decisions based on 1.Initialization: Firstly, randomly environment's data , besides its ability values were generated close to the I.K. to deal with the non linear functions results for each of joints variables (ɵ' s) which the robot has[4]. , these values would represent the Also GA is considered as a technique parents in the first generation . In fact , to find the mostly best of approximated the initialization of joints' values solutions for difficult and complex (chromosomes) must be after the problems like the I.K. of robots [2]. angles encoding step , representing in By using GA to solve the I.K. , we're real numbers or coded as strings. trying to get rid of the classical (like The next steps will discuss for one joint numerical) methods disadvantage of angle in a parallel time , for example requiring iterative processes which ɵ1 and this will be applicable on other may fail to reach a solution . joint angles . GA is selected because of its fast 2. Fitness ( or Evaluation) : Each convergence , besides of providing the chromosome (joint angle) will be real-time response , comparing with the classical methods which are very slow evaluated by the fitness function to with redundant iterations [17]. decide which of them will resolve the With GA , there's no need for value of the end-effector position 푃푥 function's derivatives to choose the , 푃푦 , 푃푧 which are described by the optimum [19], and it does not stuck in equations 1,2 & 3 respectively , those local minimum or optimum [15 , 3 , equations will be the indicator to the 21].

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41 fitness of the studied chromosome as singularity phenomenon , and obstacles they'll be compared for the ability to avoidance if existed . give the tool position, with the original 4. cross over : (reproduction process) , where each two chosen chromosomes position , where for the 푖푡ℎ individual ; will give new offspring , many X=푃푥 − 푃푥(푖) methods are available , the arithmetic Y=푃 − 푃 (푖) crossover is used here . 푦 푦 5.Mutation: where a change is inserted

Z=푃푧 − 푃푧(푖) to the individuals , in this paper, the uniform mutation is used lightly, to Evaluation = √푋2 + 푌2 + 푍2 give the opportunity to get rid of the local minimum as it may takes place , Now, the chromosomes of the current and to ensure convergence through generation will be arranged in generations . [10,13] descending order according to its This work algorithm is clarified by fitness function value , then the best Figure2: ones will be selected by the next step. The MATLAB 2013 Genetic Algorithm GUI (Graphical User 3. selection : from all the individuals , Interface) which is used in this work those of suitable fitness will complete appears as shown in figure (3) , where to the next generation , the optimization choice is taken from here another constrains have to be the artificial intelligence applications' checked like the individuals occur list , within the robot's design limits ,

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Figure 2 The work approach flow chart The MATLAB 2013 Genetic constraints limits would be enrolled as Algorithm GUI (Graphical User the variable bounds . Interface) which is used in this work Values are entered according to the appears as shown in figure (3) , where user data like the population size , the optimization choice is taken from fitness limits and stopping criteria , the artificial intelligence applications' changes are also possible in each step list , Then , the function is assigned as of the genetic algorithm like the it was set previously using the m-file crossover and mutation functions . (here the function is the inverse 9 kinematics equations for each robotic What is really going on inside the GA joint variable and will be called will start by the initialization process respectively ) , also the number of where a randomly set of joint angle variables would be selected which was values are delivered by the program to chosen to be one for having each joint be the solutions set pool for the I.K. variable individually , thus its physical problem (these suggested or randomly

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43 delivered values can be set manually to been done in this work ) , the set values alter the solution from a classical one depend on the I.K. equations for each to an optimized one and that's what had robotic joint .

Figure (3) the MATLAB genetic interface Then the Coding process will change possible like single point , two points the decimal joints values into binary , which are both used , or arithmetic) this will be followed by the fitness If needed , the mutation process is function which is constructed to give available also ( uniform , Gaussian and the inverse equations , the selection other methods) , in this work, the step now is taken to choose the best to mutation is used in a limited way. survive for the next generation (the step has many choices to accomplish These processes continue till reaching with either depending on uniform to the best result , or till reaching to the manner which is used or randomly like maximum number of iterations which roulette method). is delivered previously , or when no noticed change in results will take The crossover step will be between the place (in fact even the stopping criteria best from all to be the parents for the could be limited in the program). new offspring (various ways are

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Next is a detailed example clarifying For the end-effector given position the way of solving the GA for this (337.5 , 129.5 , 162)mm , the base joint work ; variable q1 is found by table 2:

Table (2)/A: an example of GA details solving (1st generation)

First Generation

Chromosome X F(X) G(X) 4 G(X) Account No. of Fitness = F(X) Set function / ∑ (atan2) F(X) 100101100 300 23.428 0.254 1.016 1 2 101000000 320 22.109 0.240 0.96 0 0 100011000 280 24.904 0.270 1.08 1 2 101001010 330 21.501 0.233 0.932 0 0 ∑ F(X) ∑ = Account 91.942 = 2

Table (2)/B: continue of an example of GA details solving (1st generation)

The X Chromosomes Crosso Crossover X remainder prepared to ver result chromosomes crossover type

100101100 300 100 101 100 Two 100 011 100 284 point 100101100 300 100 011 000 100 101 000 296

100011000 280 1001 01100 One 1001 11000 312 100011000 280 1000 11000 point 1000 01100 268

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Table (3) an example of GA details solving (2nd generation)

Second Generation

Chromosome X F(X) G(X) 4 G(X) Account No. of Set 100011100 284 24.595 0.254 1.016 1 2 100101000 296 23.710 0.244 0.976 0 0 100111000 312 22.619 0.233 0.932 0 0 100001100 268 25.876 0.267 1.068 1 2 ∑ F(X) ∑ Account = 96.8 = 2

And so on , with the possibility for the actual value obtaining from the mutation in a reasonable rate within laboratory readings , besides it's within generations in order to accelerate the the range of motion ability of this reach to the fittest values , in this case robotic joint ( -185 to + 153 ) , this case of study Genetic Algorithm will go on study is shown in figure 4 till reaching to X= 338 , this value with a considering constant y=129.5 value , q1 = 21 degree which is so near from

Figure4 the robot case study configuration

Next in table (4) is some of the joints motion) which were obtained using values (selected the first three joints as GA for known tool positions . they have mostly the responsibility of

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For the same case ( the second one in joints from this case were picked to be Table4 ) , some plots were selected to showed as examples. The GUI for show the fitness , the range , and the studying the base joint is shown in stopping relations , where the first three Figure5 :

Figure5 the MATLAB interface for the base joint

Its best Fitness is shown in Figure6 F which has the relation between the i generation ( horizontal axis)& the t fitness value (vertical axis) , finding n that best value is 22 while mean value e is 12.78 s s

Generation Figure 6 the best fitness of the base joint along generations While its range is shown in Figure7 which clarifies the best , mean , and

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47 worst values for q1(vertical axis) along generation (horizontal axis)

q 1

Figure9 the MATLAB interface for the Generation base joint Figure7 the best , worst , and mean values its best fitness is clarified in figure (10) of q 1 along generations which has the relation between the generation ( horizontal axis)& the fitness value (vertical axis) , finding And the stopping criteria is shown in that best value is 41.062 while mean Figure8 below where the stall and value is 43.173 generation ( the vertical axis) and how percent it met the stopping criteria (the horizontal axis)

Stall(T) F 13 i Stall(G) t Time n e

Generations s Generation

% of criteria met Figure10 the best fitness of the shoulder Figure8 the stopping criteria (Stall & no. of joint along generations generations ) of q 1 Its range is shown in Figure11 which The GUI for studying the shoulder clarifies the best , mean , and worst joint is shown in Figure9 below

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values for q1(vertical axis) along generation (horizontal axis) as follows

q 1

Figure 13 the MATLAB interface for the base joint

Generation its best fitness is clarified in figure (14) Figure 11 the best , worst , and mean which has the relation between the values of q 1 along generations generation ( horizontal axis)& the The stopping criteria of the shoulder fitness value (vertical axis) , finding joint is shown in Figure12 where the that best value is -87.612 while mean stall and generation ( the vertical axis) value is -79.2 and how percent it met the stopping criteria (the horizontal axis)

Stall(T) 14 F i Stall(G) t Time n e Generations s % of criteria met Generation Figure 12the stopping criteria (Stall & no. Figure 14the best fitness of the elbow joint of generations ) of q 2 along generations The elbow joint interface is shown in Then the range of q3 is obvious in Figure13 Figure 15 below , which clarifies the best , mean , and worst values for

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49 q1(vertical axis) along generation (horizontal axis) Figure16 the stopping criteria (Stall & no. of generations ) of q 3

q And below , Table4 that contains 1 some of the best results which had

been gained during work

In fact , to reach to the most acceptable results , many changes had been made Generation in the available options of the GA Figure 15 the best , worst , and mean characteristics like ; controlling the values of q 3 along generations population type and size , the created function , the initial population , scores

and range ,besides the fitness scaling . The stopping criteria of q3 is shown in The crossover , the mutation and the Figure16 where the stall and generation stopping criteria were controlled also. ( the vertical axis) and how percent it The crossover probability was met the stopping criteria (the horizontal chosen initially to be 0.85 , the axis) mutation probability 0.05

Stall(T)

Stall(G)

Time

Gener ation

% of criteria met Table (4) Inverse kinematics results by analytical , simulated , and GA approaches

Fitness

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End- 풒ퟏ 풒ퟐ 풒ퟑ 풒ퟒ 풒ퟓ 풒ퟏ 풒ퟐ 풒ퟑ 풒ퟒ 풒ퟓ 풒ퟏ 풒ퟐ 풒ퟑ 풒ퟒ 풒ퟓ effecto r positio Analytical Simulated GA n approach Approach approach (x,y,z)

182.5 , 0 , 0 87 87 0 90 90 0

130

88.71 87.94

131.57 132.05

127.31 128.21

401.5 - -

337.5 , 129.5 , -80 87 87 40 -83 90 90 21 38

78.96

21.14 40.05 21.14 88.71 87.94 162 -

270.7 , 0 , 0 -90 0 0 -95 80 0 77 0

100

3.12

91.37

76.22

- 449.7 107.03 105.64

208.8 , 208.7 , 45 -94 50 0 45 79 -90 50 0 0

93.5

77.8 75.2

- 316.8 39.88 48.16

195.8 , 113.05, -90 15 30 60 -90 13 10 56

93.5

33.6

- 215.3 58.43 11.27 36.03 10.05 12.45

169.6 , 36.05 , 40 -95 12 40 -95 10 25

91.4

- 140.64 16.15 14.07 28.88 14.85 38.02 12.48 29.03 339.65

, 123.62 24 40 87 87 20 40 -85 90 90 38

81.9

23.4

78.72

88.71 87.94 , - 161.98

219.4 , 0 , 0 87 0 80 90 0

130 130

100

76.3

- -

77.03 87.94

132.04

105.04 104.03

273.9 -

Discussion As known , the I.K. problems have nonlinear equations , besides they depend on robot configurations ,

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51 thus , multiple solutions may where the values of joints' angles delivered ; also , there's the and end-effector position are important GA characteristic of available ) , so , the fitness values presenting un limited solutions for along the generation's propagation is un limited random input parents or seeds . Then , the results mentioned clear. in this work were selected from the Figures (7) , (11) and (15) of the best of multiple tries. range where the best , worst and From a sight to the figures above , mean scores of the fitness we notice for the stopping criteria individuals during all generations are plots shown in figures (8) , (12) and obvious , and as gradually happened (16 ) ,which was controlled either 17, will be better for results than abrupt according to a maximum number of changes , meaning that the generations ( selected randomly by enhancement had taken place along MATLAB or set by user ) , a limited generations starting from randomly time of calculating , or according to parents individuals and ending by the fitness closeness reaching , or as the best children individuals and that a result of the stall in the algorithm point was satisfied in these figures . execution (breaking down or collapsing just like what happen when fitness values become so far from ideal values , giving a high rate Conclusion of error ). In this paper , the no. of  The Robot Inverse Kinematic generations and the stall were has a growing importance , its 16 depended . classical solutions face some From figures (6) , (10) and (14) problems like complexity , fail which shows the fitness of the first , and slowness , thus , the three joints respectively and which is Genetic Algorithm represents the same is subjected for the last two a good alternator . joints , different numbers of  GA doesn't require a complete population were used to get the good knowledge about the robot values that as near as possible from work environment . the ideal values (that known  This study had been applied previously from real configurations on the famous Alpha robot that taken by the robotic arm with using its mini scale model , the aid of its simulation program analyzing its complete

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تحليللل الحركلللة المعكوسلللة إلنسلللا آلللل الفلللا بنموذجللل التعليمللل ذو خمللل درجات لحرية الحركة باستخدام الخوارزمية الوراثية

إسراء رافع شريف قسم هندسة الميكاترونيك كلية الهندسة-الخوارزم جامعة بغداد / العراق

الخالصة

تعتبر الحركة العكسية كواحدة من القضايا المعقدة ف مجال عم اإلنسا اآلل و ذلك نتيجة الالخطية و تعدد الحلول و الحاجة للمعرفة الكاملة عن بيئة عم اإلنسا اآلل إلى جوانب عوام أخرى . و حيث ظهر تطور كبير ف استخدام طرق الذكاء الصناع لتستبدل طرق التحلي التقليدية , فف هذا البحث تم اقتراح منهاج عم يستخدم الخوارزمية الوراثية للتعام مع مشكلة التحلي العكس لإلنسا اآلل ذو الخم درجات لحرية الحركة . أوال تم تقديم التحلي األمام و العكس لهذا اإلنسا اآلل , ثم تم ح معادالت التحلي العكس باستخدام الخوارزمية الوراثية و برمجتها بواسطة برنامج MATLAB . النتائج قورنت بثالث طرق : األولى بواسطة معادالت التحلي التقليدية , الثانية بواسطة برنامج محاكاة اإلنسا اآلل ROBOCIM , و األخيرة بواسطة صندوق األدوات للخوارزمية الوراثية ف برنامج MATLAB . الخوارزمية الوراثية حققت نتائج جيدة و تخلصت من العديد من المعوقات الت تواج طرق التحلي األخرى .

إسراء رافع شريف مجلة اتحاد الجامعات العربية للدراسات والبحوث الهندسية العدد 1 مجلد 23 عام 2016