Forward Kinematics of the 3RPR planar ParallelManipulators Using Real Coded Genetic Algorithms

Luc RollandMechanical Engineering dept.

Middle East Technical University, North Cyprus CampusKalkanli, Turkish Republic of North CyprusLuc.Rolland@gmail.com or luc@metu.edu.tr

Rohitash ChandraComputer Science dept.

Middle East Technical University, North Cyprus CampusKalkanli, Turkish Republic of North Cyprus

Rohitash c@yahoo.com

Abstract—This article examines Genetic Algorithms to solvethe forward kinematics problem applied to planar parallel ma-nipulators. Most of these manipulators can be modeled by thetripod 3-RPR.

The conversion of an equation system solving problem into anoptimization one is investigated. Parallel manipulator kinematicsare formulated using the explicit Inverse Kinematics Model(IKM). From the displacement based equation systems, theobjective function is formulated specifically for the FKP usingone squared error performance criteria. The proposed approachimplements an elitist selection process where a new mutationoperator for Real-Coded GA is analyzed. Experiments on atypical manipulator example shows the fast convergence of theproposed method.

Index Terms—Parallel robot, planar manipulator, forwardkinematics, displacement based model, genetic algorithms, elitistselection, real root isolation.

I. INTRODUCTION

3 RPR

A1

A2

A3

L1

L2

L3

B1

B3

B2

Yc

Xc

Xo

Yo

O

C

Figure 1. The general planar manipulator and the typical 3-RPR tripod

Planar parallel manipulators are characterized by one base,one mobile platform and 3 kinematics chains which lie in oneplane, namely the 3-RPR, [2]. The manipulator end-effectordisplacements are restricted to that same plane, fig. (1). Thedesign hypotheses state that the bodies are infinitely rigid andthe joints do not yield friction nor play.

Firstly, Primerose and Freudenstein have counted 8 as-sembly modes (manipulator postures) for the 3-RPR and 3-RRR, [3]. Later, Hunt geometrically demonstrated that the 3-RPR yields 6 assembly modes, [4]. For the general 3-RPR,six complex solutions were found using an algorithm basedon variable elimination, [2]. Fast numerical methods usuallyimplement Newton’s method but it is sometimes plaguedby Jacobian inversion problems and numerical instabilities.Moreover, it can only find one solution. Resultant or dialyticelimination methods have been applied but they might addspurious solutions, [5]. Moreover, homotopy methods mightalso be advocated but they are prone to miss some solutions,[5]. Hence, this justified the implementation of another methodwhich could find solutions numerically.

Genetic algorithms were introduced by Holland, [6] andthey have evolved significantly in order to suit real-worldoptimization challenges faced by engineers. In robotics, evo-lutionary algorithms have been applied for solving the FKP ofparallel manipulators. Boudreau and Turkkan proposed a real-coded genetic algorithm (RCGA), also called floating pointgenetic algorithm since real numbers cannot be represented ina digital computer, [11]. It integrates crossover and mutationoperators inspired by operators used in binary GA. However,it was reported that the GA method is more time consumingthan Newton-Raphsons method. However, it was shown thatthe domain in which the GA will converge to a solution islarger allowing to initiate the process with a more distant initialguess. More recently, the FKP solving of the spherical andspatial manipulators has also been approached with geneticalgorithms, the 3RRR by [12] and the the Gough platform by[13].

In this work, RCGA is implemented with Wright’s heuristiccrossover operator, [15]. The Pivot Mutation operator is intro-duced to simply add a small real random number to selectedgenes in the chromosome. The RCGA uses roulette wheelselection combined with the elitist strategy in order to avoidoscillation during the search. We will also introduce an elitiststrategy for the selection process and study its impact of onresolution performances in terms of response times. Using theexact results coming from computer algebra, [16], it will bepossible to verify the results.

The article is divided into three parts. The first part ad-

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dresses the issue of the kinematics modeling and its conversionto an optimization problem. The second section reviews GAtheory and details the selected elitist strategy. The third partpresents the resolution results obtained on one typical plan.

II. THE FORWARD KINEMATICS PROBLEM FORMULATION

A. The kinematics of parallel manipulators

X

OA , CBRf Rm

...

Manipulator configuration

Joint coordinates

coordinatesGeneralized

MODEL

F=0rho3

rho1

Figure 2. Kinematics model

Any manipulator is characterized by its mechanical config-uration parameters and the posture variables. The configurationparameters are thus OA|Rf

, the base attachment point coordi-nates in Rf (the base reference frame), and CB|Rm

, the mobileplatform attachment point coordinates in Rm (the mobileplatform reference frame). The kinematics model variables arethe joint coordinates and end-effector generalized coordinates.The joint variables are described as li, the prismatic jointor linear actuator positions. The generalized coordinates areexpressed as

−→

X , the end-effector position and orientation.The kinematics model is an implicit relation between

the configuration parameters and the posture variables,F (

−→

X , ρ,OA|Rf,CB|Rm

) = 0 where ρ = L = {l1, l2, l3}.This article shall only concentrate on the forward kinematics

problem (FKP ), fig. 2. Usually the inverse kinematics problemis required to model the FKP and is defined as: given thegeneralized coordinates of the manipulator end-effector, findthe joint positions.

Accordingly, the forward kinematics problem is defined as:given the joint positions, find the generalized coordinates ofthe manipulator end-effector.

In the majority of parallel manipulator cases, the FKP is adifficult problem, [17].

B. Vectorial formulation of the basic kinematics model

Containing as many equations as variables, vectorial for-mulation constructs an equation system [18], fig. (3), as aclosed vector cycle between the following points: Ai and Bi,kinematics chain attachment points, O the fixed base referenceframe and C the mobile platform reference frame. For each

kinematics chain, an implicit function−→

AiBi = U1(X) canbe written between joint positions Ai and Bi. Each vector

1

1

1

B

A

L

O

C

Figure 3. The vectorial formulation for one kinematics chain

−→

AiBi is expressed knowing the joint coordinates−→

Li and X

giving function U2(X,−→

L ). The following equality has to be

solved: U1(X) = U2(X,−→

L ). The distance between Ai andBi is set to Li. Thus, the end-effector position X or C can

be derived by one platform displacement−→

OC and then oneplatform general rotation expressed by the rotation matrixR. Vectorial formulation 2 evolves as a displacement basedequation system using the following relation :

−→

AiBi =−→

OC + R−→

CBi −−→

OAi (1)

For each distinct platform point−→

OBi|Rfwith i = 1, ..., 3,

each kinematics chain can be expressed using the distancenorm constraint, [19]:

L2i = ||AiBi||

2 (2)

C. The inverse kinematics problem

A review of planar parallel manipulators shows us that themajority of proposals fall into the four following classes: 3-RPR, 3-RRP, 3-PRR and 3-RRR, [16]. We shall only studythe 3-RPR. For each kinematics chain, the RPR manipulatoris constituted by a prismatic actuator located between two balljoints fixed on the base and the platform. Lets have Li = li.Taking the 1 equation and substituting it in equation 2, oneobtains the following equation :

l2i = ||−→

OC + R−→

CBi −−→

OAi||2 , i = 1 . . . 3 (3)

The rotation matrix R is expressed in terms of one rotationparameter.

D. The Forward Kinematics Problem

The IKP expression gives an equation system comprisingthe first three equations in terms of the classical variables :xc, yc, θ, equation (3). This system contains trigonometricfunctions which can be handled by the numerical solversimplemented in GA.

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E. The conversion to an optimization problem

GA implements a selection process which takes populationsof solutions guesses and retains solutions according to a certainset of rules.

Being only applicable to optimization problems, the GAprocess handles a performance or cost function to be maxi-mized or minimized. Therefore, we need to effectively convertthe problem which solves a system of equations into an op-timization problem. Hence, only the inverse kinematic modelis required from which we can easily derived a cost function,also called fitness function. This function will be calculatedon each solution estimate for the FKP and it will representthe total error on each leg lengths. Let lgi be the leg lengthof kinematics chain i which is given as input of the problem.Then, the objective or fitness function is set to :

3∑i=1

(li − lgi)2 (4)

If we set Hi = l2i coming from equation (3), then the fitnessfunction becomes :

3∑i=1

(sqrt(Hi) − lgi)2 (5)

This function is then to be minimized by our selectedoptimization technique to be covered in the next section.

III. SOLVING WITH GENETIC ALGORITHMS

A. Real Coded Genetic Algorithms

The basic idea used in GA optimization is as follows:initially, a number of possible solutions (chromosomes)make up a population. Over time, the GA evaluates eachchromosome according to its fitness and employs geneticoperators like selection, crossover and mutation for producingnew solutions. These new solutions are called offspringswhich are added into the new population. The process isrepeated until the algorithm obtains the best solution. Theselection of parent chromosomes from the population is doneusing the selection operator. The optimization procedure of astandard RCGA is shown in Algorithm 1.

The major advantages of real-coded GA over standardbinary-coded GA are 1) their maintainability of precisionwhich is usually lost in binary representation of a real numberand 2) the reduction in the size of the chromosome whichdirectly reduces the computation time. Furthermore, real-valued encoding gives a close conceptual approximation ofreal weight values for real world application problems. Thecomputational and optimization power of RCGA has also beendemonstrated in several theoretical studies [7], [22], [23]. Thechoice of the appropriate genetic operator is important asit directly influences the convergence of the GA. However,different forms of the main genetic operators i.e. selection,crossover and mutation are needed according to the type of theGA and the nature of the optimization problem. An overviewof each genetic operator is discussed below:

Algorithm 1 Real Coded Genetic AlgorithmInitialize PopulationEvaluate fitnesswhile !termination do

while i < PopulationSize do1) selection2) crossover3) mutationi + +

end whileUpdate population

end whileGet the best solution

Selection: The selection operator plays an important role inthe GA as it ensures that qualities from the fittest chromosomesalways survive in future generations. Some common selectionstrategies are rank selection [25], roulette wheel selection [26]and the elitist strategy [24]. In rank selection, the selection isdone according to the fitness rank of the individual; the most-fit individuals have priority over lesser ones. In roulette wheelselection, selection is done according to the wheel which givespriority to those individuals with greater fitness. However, inthis way, lesser fit chromosomes are also chosen as they maycontain useful genetic material for the future. In the elitiststrategy, some of the fittest individuals are always retainedin the new population to make sure that the best performingchromosomes always survive.

Crossover: The crossover operator exchanges genetic ma-terial from selected parents and forms either a single ormultiple offsprings. Some common crossover operators forreal-coded GA’s are flat crossover [22], simple crossover[7], [27], arithmetic crossover [27], blend crossover [29],linear breeder crossover [30] and Wright’s heuristic crossover.Wright’s heuristic crossover has also shown superior perfor-mance compared to binary GAs for a set of optimizationproblems [15].

Mutation: In the GA optimization process, the mutationoperator provides random diversity in the population. This isimportant when the GA gets trapped in a local minima. Thisusually happens when the whole population is made up ofsimilar solutions. The effect of mutation rates, the strengthof mutation and its impact in building better solutions hasbeen studied in [31]–[33]. Some common mutation operatorsare random uniform mutation and Michalewicz non-uniformmutation [27], [28]. In uniform mutation, a random number inthe range of [a, b] is added to a selected gene where a and b arethe highest and lowest values in the chromosome, respectively.In non-uniform mutation, the strength of mutation is decreasedas the number of generations increases.

Other mutation operators include Muhlenbeins BGA mu-tation [34] which creates new chromosomes in the neighbor-hood restricted by some probability, and Voigt discrete andcontinuous modal mutation [35] which works in a similar

383

way. Furthermore, Gaussian mutation works by computing themean (set to zero initially) and the standard deviation whichdetermines the strength of mutation [36], [37]. The pointeddirection mutation (PoD) [38] and momentum notion mutation[42] employ a set of co-evolving points which set directionsthat govern the resulting chromosome. The PCA-mutation [39]is based on the principal component analysis while the waveletmutation [40] is based on wavelet theory. The polynomialmutation has demonstrated effectiveness in a set of constrainedoptimization problems, [41]. Power mutation [44] has alsobeen proposed which has demonstrated effiteveneess whencompared to non-uniform mutation and Makinen, Periaux andToivanen mutation [43].

In the past, a number of techniques have been proposedto improve the performance of GAs. Fan et.al. introducedthree improved genetic algorithms which use 1) a dual fitnessfunction to adapt the mutation probability, 2) a new evolution-ary directional operator, and 3) probabilistic binary search toreveal a new offspring, respectively. All these approaches aimin building better offsprings in relation to traditional methods[46]. The crossover and mutation operators are improved fromthe introduction of the local gradient based operators suchas gradient optimizer [45] and evolutionary gradient operators[47] . Other modifications include a new stopping rule basedon asymptotic considerations and mutation scheme based onParticle Swarm Optimization [48].

B. Pivot Mutation

The Pivot Mutation operator selects a pivot point in thechromosome and all the genes after the selected pivot aremutated by adding a small real-random numbers, respectively.For instance, given a chromosome x = (x1, x2, x3, ...xn), theresulting pivoted chromosome becomes y = (x1, x2, y3, ...yn),where ,yi = xi + r given that r is a small real random numberin the interval [a, b], where a and b are small negative andpositive real numbers chosen by the user, respectively.

C. Other genetic operators used in this study

Wrights heuristic crossover operator has shown superiorperformance in comparison with other crossover operators fora set of optimization problems [15], therefore, it is used inthis study. The Wrights heuristic operator produces a singleoffspring given two parents. For a pair of parents x1 =(x1

1, x12, x

13, ...x

1n) and x2 = (x2

1, x22, x

23, ...x

2n) , an offspring

is produced as follows:

yi = r(x1i − x2

i ) + x1i

where r is a real random number belonging to [0,1] and x1 isthe parent with the best fitness. A new ofspring is producedwith a new r until a chromosome with a better fitness iscreated.

The uniform mutation operator developed by Michalewiczadds a small random number [a, b] to a selected gene,where a and b are the greatest and the least values foundin the chromosomes, respectively. The non-uniform mutationoperator [28] is one of the widely used operators in real-coded

GA’s. In this method, from a point xt = (xt1, x

t2, x

t3, ...x

tn), the

muted point xt+1 = (xt+1

1 , xt+1

2 , xt+1

3 , ...xt+1n ) is created as

follows:

xt+1 =

{xt

i + ∆(t, xui − xt

i), if r ≤ 0.5xt

i − ∆(t, xui − xt

i), otherwise

where t is the current generation number and r is auniformly distributed random number in the interval [0, 1].xl

i and xui are the lower and upper bounds of the selected

chromosome, respectively. The function ∆(t, y) given belowtakes value in the interval [0, y].

∆(t, y) = y(1 − u(1− t

T )b)

where u is a uniformly distributed random number in theinterval [0, 1], T is the maximum number of generations andb is a number which determines the strength of the mutationoperator. This mutation operator performs global search duringthe initial search and local search in the later generations.

IV. RESULTS AND ANALYSIS

A. Modeling the planar 3RPR

In this section, we shall examine one example of the FKPresolution on a typical 3RPR manipulator configurations.

The manipulator configuration is written in a text file, noted3RPR8. The file includes the manipulator base coordinatesof the joint center positions OA|Rf

in the base referenceframe Rf and the mobile platform coordinates of the jointcenter positions CB|Rm

in the platform reference frame Rm,the minimum bar lengths if applicable. The correspondingconfiguration examples are shown in table I.

Config A1(x) A1(y) A2(x) A2(y) A3(x) A3(y)3RPR8 0 0 200 0 0 200

Id B1(x) B1(y) B2(x) B2(y) B3(x) B3(y)3RPR8 0 0 50 0 40 40

Table IPLANAR PARALLEL MANIPULATOR CONFIGURATION TABLE

The active joint variables are then specified. For the 3RPR8,the joint variables are the kinematics chain lengths. To com-plete the FKP input data in the aforementioned examples,the square of the joint variables are set respectively to L :=[100, 120, 150].

B. Algebraic system exact solution

In order to verify the method implementing GA, we revertto an exact algebraic method which can compute the certifiedexact results, [16]. For the aforementionned planar parallelmanipulator, the exact results are shown with four exact digitsafter the point:

sol1 = (97.9911, 19.9193, 85.0154)sol2 = (52.8654, 84.9536, -33.4487)

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C. Genetic Algorithm results

The computations were achieved on a 3 GHz Pentium IVpersonal computer equipped with 512k bytes of random accessmemory. The operating system was Mandrake LINUX version9.2.

In this section, the performance of 3 different combinationsof genetic operators in RCGA’s is evaluated. In all exper-iments, the roulette wheel selection is used in conjunctionwith the elitist strategy method. This ensures that the fittestchromosome is retained in future generations. This prevents theGA from oscilating or becoming trapped in a local mimimumwhile searching for the global optimum solution. Note thatif the population size is P , then P selections are done inorder to make P offsprings for the new population. Thefollowing combinations were used in order to get the bestcombination of genetic operators in order to evaluate theperformance of the proposed Pivot Mutation operator. Note thatthe respective mutation and crossover rates were determinedin trial experimental runs.

1. Hx Non-uniform: The wrights heuristic crossover com-bined with Michaelwicz nonuniform mutation. The crossoverand mutation rate are 0.8 and 0.05, respectively.

2. Hx Crossover: In this strategy, the Wright’s heuristiccrossover operator dominates the GA. There is no mutation.The crossover rate is 0.8.

3. Hx Pivot: The wrights heuristic crossover operator iscombined with Pivot Mutation. The crossover and mutationrates are 0.8 and 0.1, respectively.

The fitness function given by the forward kinamatic of 3leg parallel manipulators is given in equation (n) is used. Allexperiments initialise the population with real numbers in therange of [−40, 40]. The population size of 40 and chromosomesize of 3, representing the position’s x, y and θ were used. Theconstant leg lengths for leg1, leg2 and leg3 were 100, 120and 150, respectively. Figure 4 shows the convergence of atypical experimental run for each GA optimization technique.For each method, a total of 50 experimental runs were done.The results are summarised in table II. The optimization resultsfor coordinates x, y, and orientation θ with its respective fitnessis given in table III. The given forward kinematics problem has2 unique solutions. Note that the experiments were done ona 2 Giga Hertz Linux dual core machine and the CPU timeis given in milliseconds. The CPU time was calculated usingthe number of clock ticks taken for the GA in converging to asolution. The CPU executes 29 clock ticks in 1 second (2 Giga-Hz CPU). Therefore, the time in milliseconds was calculatedby T (ms) = (ClockT icks ∗ (1/29)) ∗ 1000.

No. Generations Fitness CPU Time

Hx Non-Uniform 29±4 0.75±0.06 1.49±0.24Hx Crossover 40±7 0.77±0.06 2.11±0.39

Hx Pivot 17±6 0.61±0.08 0.86±0.10

Table IITHE MEAN AND 95 PERCENT CONFIDENCE INTERVAL FOR 50

EXPERIMENTAL RUNS

Figure 4. A typical experimental run showing the convergence of the threemethods

Solution x y θ(deg) Fitness

Hx Non-Uniform 1 53.2047 85.6626 -34.0811 0.7659582 98.7197 20.2729 85.5912 0.627518

Hx Crossover 1 52.4156 85.0839 -33.3571 0.5274142 98.2234 19.3598 85.0078 0.640394

Hx Pivot 1 52.4071 85.4213 -34.1143 0.574972 97.7789 20.0296 84.4414 0.190192

Table IIIOPTIMAL x, y AND θ VALUES FROM THE BEST EXPERIMENTAL RUN FOR

EACH METHOD.

The precision of the GA solutions depends on the numberof generations allowed, and are thus proportional to time.

D. Discussion on model solving

V. CONCLUSION

In this paper, one complete method to solve the parallelmanipulator FKP has been established and explained basedon genetic algorithms. The method was applied to the planar3RPR parallel manipulator. At first, the manipulator kinemat-ics formulation was studied. The inverse kinematics equationsystem was written applying the classical displacement-basedmodel. Then, the objective function was written as an errorsum on all kinematics chain lengths.

Real-coded genetic algorithms provide multiple solutionsto optimization problem given multiple experimental runs withdifferent initial search space. The two solutions reported in thiswork were found in different populations given by multipleexperimental runs, since the entire population is made up ofsimilar solutions towards convergence. This observation meansthat the best solutions survive over time. The proposed ’PivotMutation’ operator has demonstrated its strength when com-pared to standard methods. The two real solutions containedtwo position parameters found within 0.5 mm accuracy andangles within 0.6 degrees. For the first time, the GA methodresults were verified against the exact solutions.

In future work, the implementation of the RCGA procedureinto a real-time robotic system can be foreseen due to its fastconvergence, accuracy and robustness the search procedure.

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