optimisation of parallel manipulators

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Optimization Algorithms for Kinematically OptimalDesign of Parallel Manipulators

Yunjiang Lou, Senior Member, IEEE, Yongsheng Zhang, Ruining Huang, Xin Chen, and Zexiang Li, Fellow, IEEE

Abstract—Optimal design is an inevitable step for parallel ma-nipulators. The formulated optimal design problems are generallyconstrained, nonlinear, multimodal, and even without closed-formanalytical expressions. Numerical optimization algorithms arethus applied to solve the problems. However, the optimization algo-rithms are usually chosen ad arbitrium. This paper aims to providea guideline to choose algorithms for optimal design problems.Typical algorithms, the sequential quadratic programming (SQP)with multiple initial points, the controlled random search (CRS),the genetic algorithm (GA), the differential evolution (DE), andthe particle swarm optimization (PSO), are investigated in detailfor their convergence performances by using two canonical designexamples, the Delta robot and the Gough–Stewart platform. It isshown that SQP with multiple initial points can be efficient forsimple design problems, while DE and PSO perform effectivelyand steadily for all design problems. CRS can be used to generategood initial points since it exhibits excellent convergence evolutionin the starting period.

Note to Practitioners—Numerical optimization algorithms aregenerally inevitable in solving optimal design problems of parallelmanipulators. Various algorithms have been applied in literatureand in engineering. This paper provides a thorough comparison onconvergence performance of typical optimization algorithms, SQPwith multiple initial points, CRS, GA, DE, and PSO. Two parallelmanipulators, the Delta robot and the Gough–Stewart platform,are used as design examples by maximizing the effective regularworkspace. Computation shows that DE and PSO are good choicesfor complicated optimal design problems, while SQP with multipleinitial points is superior for simple problems. CRS performs excel-lently in the starting period. It can be used to generate good initialpoints.

Index Terms—Controlled random search (CRS), differentialevolution (DE), genetic algorithm (GA), optimal design, op-timization algorithms, parallel manipulators, particle swarmoptimization (PSO), sequential quadratic programming (SQP).

Manuscript received December 17, 2012; accepted April 14, 2013. This paperwas recommended for publication by Associate Editor T. D. Murphey and Ed-itor K. Lynch upon evaluation of the reviewers’ comments. This work was sup-ported in part by the National Natural Science Foundation of China under Grant51075085 and Grant U1134004 and in part by the Introduction of InnovativeR&D Team Program of Guangdong Province under Grant 2009010051.Y. Lou, Y. Zhang, and R. Huang are with the School of Mechantronics En-

gineering and Automation, Harbin Institute of Technology Shenzhen GraduateSchool, and the Shenzhen Key Lab for Advanced Motion Control and ModernAutomation Equipments, Shenzhen 518055, China (e-mail: [email protected]).X. Chen is with the School of Mechatronics Engineering, Guang Dong Uni-

versity of Technology, Guangzhou 510006, China (e-mail: [email protected]).Z. Li is with the Department of Electronic and Computer Engineering, Hong

Kong University of Science and Technology, Hong Kong, China, and is alsowith the DG-HUST Manufacturing Engineering Institute, Dongguan 523808,China (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TASE.2013.2259817

I. INTRODUCTION

P ARALLEL manipulators consist of multiple subchainsconnecting to a base and a moving platform, which

form one or multiple closed chains. The unique structuredistinguishes themselves from serial manipulators and leads tofeatures of lower inertia, lower moving mass and thus potentialof high speed and high acceleration. They have been success-fully applied in various applications. However, the distinctstructural characteristic of parallel manipulators makes theirkinematics highly nonlinear, which are even more complicatedthan those of serial ones. (i) For a higher degree-of-freedom(DoF) parallel manipulator, the determination of workspaceis usually not an easy job. (ii) The mathematic dependenceof kinetostatic performances like manipulability, stiffness,velocity/force transmission, and accuracy on geometric param-eters is generally implicit and complicated. The performancesdistribute non-uniformly in the workspace and may vary drasti-cally from one configuration to another. The optimal design, aprocess to find the best suitable design parameters with respectto certain performance indices, becomes an inevitable stepwhen designing a parallel manipulator. A parallel manipulatorwith optimal kinematic parameters may possess dramaticallybetter performance compared with one without an optimaldesign process.Since 1980s, there have been numerous studies on the kine-

matically optimal design of parallel manipulators. Various per-formance indices were proposed to characterize properties of aparallel manipulator and were then used to formulate optimaldesign problems. From the viewpoint of workspace, the per-formance indices can be fundamentally divided into two cate-gories, (i) indices on workspace geometric properties, i.e., shapeand volume; and (ii) indices on workspace quality (or kineto-static performances of a parallel manipulator), e.g., dexterity[1], manipulability [2], singularity [3]–[5], force/velocity trans-mission factor [6], [7], static stiffness [8], accuracy [9], [10].Accordingly, the numerous kinematically optimal design prob-lems of parallel manipulators were formulated as optimizationproblems by taking one index or a mixed index as the objective,while some other indices as design constraints.It is indicated that the various proposed optimal design prob-

lems are usually constrained, nonlinear, multimodal optimiza-tion problems without closed-form expressions [11], [12]–[15].It is generally impossible to find an analytical solution. Nu-merical algorithms are preferable to solve the problems. Tra-ditional gradient-based techniques are local optimization algo-rithms. They may have difficulty to locate the global optimumof an optimal design problem. For a simple parallel manipulatorwith few design parameters, exhaustive search may be effective[16], [17]. For a complicated parallel manipulator, however, it is

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2 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING

necessary to develop or find an efficient algorithm to solve theoptimal design problem. In order to amend the localness of gra-dient-based algorithms, some researchers proposed to apply thealgorithms with multiple initial guesses [12], [18] although themethod provides no guarantee on locating the global optimum.Probability-based direct search algorithms, which are global op-timization algorithms, have been widely used to solve optimaldesign problems, e.g., the controlled random search (CRS) [15],the genetic algorithm (GA) [19]–[23], the differential evolution(DE) [24], [25], and the particle swarm optimization (PSO) al-gorithm [26]–[29]. However, there is no guideline on how tochoose an optimization algorithm suitable for individual designproblems. The optimization algorithms were usually chosen adarbitrium. The paper aims to provide a thorough comparisonof optimization algorithms in solving optimal design of par-allel manipulators so that designers can choose an efficient al-gorithm accordingly for their problems. Typical algorithms likeSQP with multiple initial points, CRS, GA, DE, and PSO, areinvestigated. The Delta robot, a 3-DoF purely translational par-allel manipulator, and the Gough–Stewart platform, a 6-DoFfull-mobility parallel manipulator, are taken as design examplesto conduct the comparison.This paper is organized as follows. In Section II, a typical

optimal design problem formulation, maximization of the effec-tive regular workspace, is briefly introduced. In Section III, typ-ical optimization techniques, SQP with multiple initial guesses,CRS, GA, DE, and PSO are introduced. The optimal designs ofthe Delta robot and the Gougn–Stewart platform are carried outand compared in Sections IV and V, respectively. Finally, con-clusions are drawn in Section VI.

II. OPTIMAL DESIGN PROBLEM FORMULATION

In order to carry out comparison of various optimization algo-rithms, a typical design problem formulation is needed. In thispaper, the design problem formulation of maximizing the effec-tive regular workspace is used. This formulation maximizes theworkspace volume with a regular shape (e.g., rectangle, ball,cube, and cylinder), which possesses good performance in ma-nipulability. Link interference and joint limits are consideredand taken as design constraints. A brief description of the formu-lation is provided as follows. Detailed derivation can be foundin [15].Let us consider a normally actuated parallel manipulator,

where the number of actuators is equal to the number of DoFsof the manipulator. For an -DoF normally actuated parallelmanipulators, let and , respectively, be setsof actuated and passive joint variables, and the total numberof DoFs of all the subchains, and the Cartesiancoordinate representing the position and orientation of theend-effector. Given an , inverse kinematics maps can bederived as follows:

(1)

(2)

where is the set of kinematic parameters, e.g., the link lengths,the position of base points of each subchain, the relative arrange-ment of each axis, and the size and shape of the end-effector, etc.

Hereafter, denotes the set of design parameters of in-terest, where is the number of design parameters. The velocityrelation can in general be written as

(3)

where is the kinematic Jacobian matrix, map-ping joint rate to Cartesian velocity .

A. The Objective Function

Let be a regular-shaped workspace for a gen-eral parallel manipulator, where is the translationalworkspace and the orientational workspace. Ameasure for can be derived based on measures for and. Let and be measures for the volume of and ,

respectively, a measure on the overall volume of is given as

(4)

where , , 2 are constants weighting contributions ofand , respectively. They are assigned according to differentpractical requirements. In this paper, we consider the case wherea constant orientation capability is prescribed, i.e., is con-stant. The objective function is then reduced to . The ob-jective is to maximize the translational regular workspace giventhat at each point in the translational regular workspace the ma-nipulator at least possesses an orientational capability of .In order to remove dimension effect, the design parameters

are normalized as follows:

(5)

where is a given constant, usually 1, andare individual kinematic parameters, and . The (5) impliesthat .

B. The Workspace Constraints

By considering both the actuated and the passive joint limits,the inverse kinematics are used to check the workspace contain-ment. A set is reachable means every is reachable.In other words, the inverse kinematic solution corresponding tothe point exists and is within the joint limits

(6)

(7)

where and are, respectively, the lower and upperbound for the th actuator due to actuator limits. Similarly,and are, respectively, the lower and upper bound for theth passive joint due to passive joint limits.That a point is reachable also requires that there is no me-

chanical interference at the configuration. Assume that a ma-nipulator is composed of links. For simplicity, each link is ap-proximated by the minimal cylinder enveloping the link. Let theradius of the cylinder be and denote by the line segmentpassing the th cylinder axis, . Clearly, there is nomechanical interference if the distance between any pair of line


LOU et al.: OPTIMIZATION ALGORITHMS FOR KINEMATICALLY OPTIMAL DESIGN OF PARALLEL MANIPULATORS 3

Fig. 1. The distance between two links.

segments is larger than the sum of corresponding radii. Fig. 1 de-picts the distance between two spatial cylinder-modeled links.The following inequalities ensure that no link interference willoccur at a point :

(8)

for all , , and links and are not neigh-boring, which means there is no joint connecting link and link. Here, is the function computing distance be-tween two line segments and . Note .The set of points satisfying (6)–(8) constitute the workspace

reachable by the resultant parallel manipulator. Therefore, anypoint should satisfy (6)–(8).

C. The Manipulability Constraints

In order to guarantee the regular workspace to be effective,i.e., the manipulator is able to conduct tasks effectively withinthe regular workspace, constraints on the manipulability indexare introduced to characterize quality of the regular workspace.A frequently used measure for manipulability is the inverse con-dition number of the kinematic Jacobian matrix, which is de-fined as

where denotes the inverse condition number function ofmatrices, and and its minimal and maximalsingular value functions, respectively. Thus, . Here,we treat separately orientation and position manipulability byrewriting the differential kinematics (3) as

where and are linear velocity and angular velocity, respec-tively. Thus, and , respectively, give measures forposition and orientation manipulability. To guarantee positionand orientation manipulability, they are applied in design by im-posing the following constraints:

(9)

(10)

where and are, respectively, lower bounds for positionand orientation manipulability, which are constants assigned ac-cording to practical design requirements.

D. Problem Formulation

Combining constraints (4)–(10), the optimal design problemfor maximization of effective regular workspace is formulatedas following.1) Problem 1: Optimal Mechanism Design: Find a set of

optimal design parameters such that

(11)

(12)

(13)

(14)

(15)

(16)

(17)

where , the regular-shaped workspace, and; ; ,

and link and link are not neighboring.Clearly, the optimal design problem 1 is a constrained non-

linear optimization problem. The objective function (11), themanipulability constraints (12) and (13), and the link interfer-ence constraint (16) usually have no explicitly analytical expres-sions with respect to the set of design parameters . Gradientsand Hessians are thus not readily computed. Furthermore, theobjective function in the optimal design problem is generallymultimodal, i.e., there may exist multiple local maxima in thefeasible region. Gradient-based algorithms may be trapped intoa local maximum and thus have difficulty in locating the globaloptimum.

III. TYPICAL OPTIMIZATION ALGORITHMS

In this section, typical optimization algorithms used insolving the optimal design problem, gradient-based algorithmwith multiple initial points, CRS, GA, DE, and PSO are brieflyintroduced.

A. The Gradient Based Algorithms

Widely-used gradient-based algorithms are interior pointmethod, sequential quadratic programming (SQP), active setmethod, and trust region reflective method. SQP represents thestate of the art in nonlinear programming methods and plays animportant role in the optimal design of parallel manipulators.In this algorithm, at each major iteration, an approximationis made for the Hessian of the Lagrangian function using aquasi-Newton updating method. This is then used to generatea quadratic programming (QP) subproblem whose solution isused to form a search direction for a line search procedure.A feasible initial point is needed in this algorithm. The SQPimplementation consists of three main stages, updating the Hes-sian matrix, quadratic programming solution, and line searchand merit function. SQP method is most efficient if the numberof active constraints is nearly as large as the number of vari-ables, that is, if the number of free variables is relatively small.It requires few evaluations of the functions, in comparison withaugmented Lagrangian methods, and can be more robust on



badly scaled problems than the nonlinear interior-point andactive set methods [30]. Hence, SQP method is chosen as therepresentative of gradient-based algorithms to be studied inSections IV and V. In order to locate the global optimum,multiple initial points are provided by uniformly discretizingthe feasible zone. By comparing the optima generated by themultiple SQP problems, the best one is taken as the globaloptimum.

B. The Controlled Random Search

The CRS algorithm is a global optimization technique fea-turing robustness and flexibility. In 1978, Goulcher and Long[31] proposed the method to solve constrained nonlinear opti-mization problems. Later this method was improved and appliedin many chemical plants [32] and parallel manipulators [15].The basic philosophy of the method is to select new points byrandom selection from normal probability distributions centeredat the best previous value

(18)

The (18) describes how the new points in th iteration, ,are generated in the neighborhood of the previous best point

, where is a vector of random variables that aresubject to normal probability distribution with zero mean andunity standard deviation as follows:

is applied to adaptively modify the stan-dard deviation of the normal probability distribution for everyrandom variable in each iteration. It is actually the standarddeviation for the vector of random variables . Therefore,“control” comes by adjustment of the standard deviation of thedistribution, which explains the name of the method. Comparedwith standard optimization techniques, the random variablecan be regarded as a search direction, while the standard

deviation serves as a kind of “step-length,” which is adjustedautomatically during the search in two situations.(a) Each time a successful trial has been made. In this case,

standard deviations are set according to ,, where is a positive quantity describing the

distance between the variable’s current value and thenearest bound of the variable. is a compressionfactor to reduce search interval and maintain searches inthe neighborhood of the best previous point.

(b) After a specified number, typically 100, of consecutivefailure. Failure means that no improvement is made withrespect to the objective function. When this occurs, forinstance, as the optimum is approached, the standard de-viations are reduced by

where is a positive number.

C. The Genetic Algorithm (GA)

The GA was proposed and developed by Holland and suc-cessfully applied by his student, Goldberg, in the 1970s and

1980s based on the principles of genetics and natural selection.It is a global optimization technique, which does not requirederivative information. So far, it has been widely used in func-tion optimization, combinatorial optimization, automatic con-trol, robotics, image processing, and so on. The computationprocedure of the simple GA is as follows [33], [34].1) Initialization. An initial population formed by individualsis randomly generated within the feasible set defined by theconstraints.

2) Individual fitness evaluation. All individuals are evaluatedby a fitness function, the objective function.

3) Selection. Based on the fitness of the individual, individ-uals are selected to survive or reproduce in the next gener-ation according to some given criterion.

4) Crossover. Using the mechanism of biological crossover,a child is produced by choosing two or more parents torecombine their chromosomes.

5) Mutation. Using the mechanism of biological mutation,one or more gene values are altered in a chromosome fromits initial state by a small probability.

6) Termination. One or more criteria are used to terminatethe search. Typical criteria include a solution found thatsatisfies minimum criteria, fixed number of generationsreached, and computation time reached.

D. The Differential Evolution (DE)

DE was originally introduced by Storn and Price [35] andoffers a way of optimizing a problem without using its gradient.Suppose a dimensional problemwith its population size beingNP, DE can then be described as follows [35], [36].1) Initialization. vectors with random positions are gen-erated in the dimensional feasible search-space.

2) Mutation. In generation , a mutant vector is generated foreach target vector , by

(19)

where are randomly chosenintegers and mutually different with .

is an amplification factor of the differentialvariation .

3) Crossover. The trial vectoris formed by

ifotherwise

, where is a random numbergenerator uniformly distributed in ,the crossover constant determined by the user,

a randomly chosen index,which ensures that gets at least one parameter from

.4) Selection. The trial vector is compared to the targetvector according to the objective function evalua-tion. Taking the minimization problems as examples, if thevector produces a smaller objective function valuethan , then is set to ; otherwise, the oldvalue is retained.



5) Termination. Similar to GA, common terminating condi-tions include a sufficiently good fitness or a maximumnumber of iterations (generations).

In the real computation, the code by Buehren was used [37].

E. The Particle Swarm Optimization (pso)

PSO is originally proposed in [38] and was first intended forsimulating social behavior. PSO does not use the gradient of theproblem being optimized either. It was modified and improvedby researchers, e.g., [39]. Let the position and velocity of thparticle be, respectively, and , and and be, respec-tively, the best fitness value the th particle has achieved so farand corresponding location. Assume that the best fitness valueachieved by any particle of the population is called andthe corresponding location is denoted as . The fundamentalprocess for implementing of PSO is described as follows [40],[41].1) Initialization. A population array of particles with positionsand velocities are randomly generated on dimensions inthe search space.

2) Evaluation. Each particle is evaluated by the fitness func-tion (the objective function) in variables.

3) Comparison and selection. The particle’s fitness value iscompared with particle’s . If current value is betterthan , then set value equal to the current value,and the location equal to the current location in-dimensional space. Then, the particle’s fitness value is

compared with the population’s overall previous best. Ifcurrent value is better than , then reset to thecurrent particle’s array index and value.

4) Adjustment. The velocity and position of the particle ischanged by the following equation:

(20)5) Loop to step 2) until a criterion is met, usually a sufficientlygood fitness or a maximum number of iterations.

In the (20), , where is specified byusers. and are two acceleration constants weighting ofthe random acceleration terms and the functions and

are two random number generators uniformly dis-tributed in . They are randomly generated at each iterationand for each particle.In the real computation, the code by Sam was used [42].

F. Termination Criterion and Discussions

In the early years, gradient-based algorithms were the mainapproach to solve optimal design problems of parallel manipu-lators. Since the algorithms could only find local minima, globaloptimization algorithms were gradually introduced to deal withthe multimodal problems.SQP is a deterministic and local optimization technique,

while DE, PSO, GA, and CRS are all probabilistic and globaloptimization algorithms. It is necessary to use a suitable crite-rion to evaluate them. In order to better estimate an algorithm’sability to locate the true global optimum, a trial can be classifiedas a “success” when the best objective function value reachesa predetermined limit known as the value-to-reach, or “VTR”

Fig. 2. Architecture of the rotational Delta robot.

Fig. 3. The th subchain of the rotational Delta robot.

[36]. Trials that do not reach the VTR within a predeterminedmaximum time, , are treated as “failures.” Once thealgorithm reaches the VTR within , the computation willterminate and output the optimization results. With the VTRand as criteria for success, we can estimate the speed andthe probability with which the algorithm locates the basin ofattraction to which the global optimum belongs.

IV. CASE STUDY: THE DELTA ROBOT

The Delta robot, as shown in Fig. 2, is a 3-DoF purely trans-lational parallel manipulator consisting of 3 identicalsubchains, where represents a revolute joint and a planarparallelogram. This architecture was invented by Clavel [43]and is well-known for its high speed. Up to now, it has beenwidely used in many applications.

A. Optimal Design Problem Formulation

The kinematic parameters of the manipulator are depicted inFig. 3, where denotes the length of arms , the lengthof the parallelogram, and , , , 2,3, with and being centers of the base and the moving plat-form, respectively. The detailed inverse kinematics and Jaco-bian matrix can be found in [15] and [43], which are omitted forbrevity. Let , the set of design parameters is obtained,

, according to its kinematics. The manipulator sizeis normalized by normalization of each subchain, .Since the Delta robot is in a rotational symmetry, a cubic

shape is chosen for the regular workspace . Denoting bythe side length of the cubic workspace, we take the objective

to represent the workspace volume.



In this case study, mechanical interference constraints (16)and the manufacturing constraints are implicitly included byimposing the following constraints on the actuated and passivejoints as in [43].1) due to constructional constraints on theparallelogram’s articulations.

2) is imposed in order to avoid inter-ference between the arms and the parallelogram rods whenthe angle is acute and to avoid ambiguities in computation.

3) is chosen for actuation.If for some applications the lower bound of manipulability isgiven as 0.4, by combining all other requirements together, theoptimal design problem is formulated as follows.1) Problem 2: Optimal Design of the Delta Robot: Find a set

of optimal design parameters such that

where , .

Algorithmic Settings

The algorithms, DE, GA, PSO, CRS and SQP, were appliedto solve the optimization problem. The probability-based algo-rithms, DE, GA, PSO, and CRS, were tested five times. For allalgorithms, VTR was given 0.3701, which is a close estimate ofthe global optimum 0.3702. The maximum allowable timewas set 3600 s, i.e., 1 h. For CRS, DE, GA, and PSO, the initialfeasible points were randomly generated in the feasible region.The population sizes used in DE, GA, and PSO were identicallyset 20, 10 times the number of optimization variables. Since al-gorithmic parameters play important roles in the convergenceperformance, they were set according to the algorithm proper-ties and the problem features as follows.• DE. and . Default values were used forother algorithmic parameters.

• GA. In the realization, the elite count was set 2, thecrossover fraction was set 0.8, the migration fraction wasset 0.2, and the interval was taken 20.

• PSO. The acceleration constants and ,and .

• CRS. The compression factors and .• SQP. Since SQP is intrinsically a local optimization tech-nique, a sufficiently large number of initial points should beprovided to locate the global optimum. For a constrainednonlinear optimization problem, it is usually difficult, evenimpossible, to determine which initial point will lead to theglobal optimum by simple observation. Here, a uniformdiscretization of the feasible region was used to generatethe set of initial points. Each discretized point was appliedas an initial point in the algorithm. The generation of ini-tial points starts from a coarse discretization of the feasible

region. If the algorithm cannot locate the global optimumusing the coarse initial points, a finer discretization will beapplied. The process terminates until it reaches the VTR.

B. Computation and Discussion

The above algorithms were coded in the environment ofMatLab® (R2010b). The computation was implementedusing a personal computer with an Intel® Core™2 Duo [email protected] GHz and a RAM of 2 GB.The computation results are listed in Table I.• DE. The number of generations and the computation timerange from 12 to 35 and from 138.2 to 416.1 s, respectively.The corresponding average values are, respectively, 24 and284.3.

• PSO. The number of generations ranges from 17 to 61. Thecomputation time ranges from 184.4 to 655.5 s. The av-erage number of generations and the average computationtime are 38.6 and 405.3 s, respectively.

• GA. The number of generations ranges from 1 to 94. Thecomputation time ranges from 24.86 to 2093 s. The averagenumber of generations and the average computation timeare 45.8 and 1017 s, respectively.

• CRS. In the five trials, CRS succeeded four times to reachthe VTR. In the four successes, the number of iterationsranges from 26 to 35 and the computation time ranges from793.4 seconds to 1851 s. The average number of iterationsis 31 and the average computation time is 1279 s.

• SQP. There are two independent optimization variables,and , in the problem. In the optimization, a 3 3 dis-

cretization (i.e., 9 initial points) reached the VTR. The cor-responding computation time is 3.931 s.

From Table I, the following observations can be obtained.1) Clearly, all algorithms except CRS succeeded in reachingVTR for five trials. Therefore, the algorithms, DE, GA,PSO, and SQP with multiple initial points, perform moresteadily than CRS.

2) In this example, SQP yielded the least computation time,3.931 s, which is much less than the average time used inthe probability-based algorithms. This is because the Deltarobot is a simple mechanism (3-DoF) and the number ofindependent design variables is rather small (only 2, and).

3) For the global optimization algorithms, DE produced theleast average number of generations, 24. It also possessedthe least average computation time. It is worthy notingthat GA reached the VTR with only one generation and24.86 s in one of trials, which is very superior to other trialsand other global optimization algorithms. However, it per-formed in an inconsistent manner. GA also produced thelargest number of generations, 94, and the longest compu-tation time, 2093 s in all trials. Therefore, the simple GAis inconsistent in computation time for different trials.

4) For DE, PSO, and GA, the computation time increasesmonotonically along with the increase of the number ofgenerations. While for CRS, it is not the case.

In the Delta robot example, the computational performances canbe ranked, , where “ ”means “better than” and “ ” means “much better than.”



TABLE ICOMPUTATION RESULTS OF OPTIMAL DESIGN OF THE DELTA ROBOT

Fig. 4. Computation evolution of DE, CRS, GA, and PSO: the Delta robot case.

TABLE IICOMPUTATION RESULTS WITH TERMINATION TIME 400 S

In order to compare the convergence evolution performance,the four probability-based algorithms were tested with an iden-tical initial points, . The commontermination time of 400 s was used to stop the computations.Fig. 4 presents the typical computation evolution of algorithmsand Table II shows the optimal objectives and the correspondingdesign parameters. From Fig. 4, the rank of convergence speedis in the starting period (from 0 to80 s), where “ ” means “approximate.” In the later computa-tion, the convergence became rather slow for all four algorithms.They performed almost equally and finally they all reached aclose neighborhood of the true optimum, see Table II.

V. CASE STUDY: THE GOUGH–STEWART PLATFORM

A Gough–Stewart platform, as shown in Fig. 5, is composedof a fixed base, a moving platform, and six identical legs,where represents a passive spherical joint and is an actuatedprismatic joint. Themoving platform is controlled by driving the

Fig. 5. A schematic for the Gough–Stewart platform.

-joints to extend/retract legs. The Gough–Stewart platform isperhaps the earliest and the most intensively studied 6-DoF par-allel mechanism, which has been widely used in motion simula-tion, 5 axis machining, force/torque sensing, etc. In this paper,we suppose the manipulator has identical legs. The base balljoints are arranged in a pairwise rotational symmetric manner.The joints on the moving platform are also pairwise rotation-ally symmetric.

A. Optimal Design Problem Formulation

From Fig. 5, the geometry of the manipulator is defined byfive parameters, namely:— : the radius of the circle on the base where the jointlie;

— : the radius of the circle on the moving platform wherethe joint lie;

— : half the angle between two far joints on the base;— : half the angle between two far S joints on the movingplatform;

— : the leg length when the manipulator is at its home po-sition, where all actuators are at their half stroke.

Since only a few set of possible strokes are available forcommercial linear actuators, we may choose it beforehandand modify other design parameters to adapt it. We thereforenormalize the stroke by unity. The set of design parameters is

. A constraint on the manipulator size isimposed as



where is a given constant representing the relative size of themanipulator with respect to the stroke. In the simulation, .The effective regular workspace of the Gough–Stewart plat-

form is composed of two portions, the translational one andthe orientational one . Let us consider the Gough–Stewartplatform for machine tool applications requiring good tilting ca-pability which is usually characterized by the tilting angle .A cube with side length in is designated as the transla-tional workspace . The objective function is chosen as, which is constrained by the tilting capability with

in the simulation.The translation and orientation manipulability constraints are

imposed with and .As discussed above, the strokes are normalized by unity, i.e.,

the leg lengths is constrained by

The inequalities above reflect actuated prismatic joint limits.Assume the spherical joints are identical and the range is

, the constraints due to passive ball joint limits aregiven as follows:

where and are the base ball joint axes at the current con-figuration and at the home position, respectively. The vectorsand are the moving platform ball joint axes, respectively, atthe current configuration and at the home position. The func-tions and compute pivot an-gles of base ball joint and moving platform ball joint on the thleg, respectively. We take in the simulation.Constraints due to leg interference are given as

where represents the line segment of the th leg. Here, it isassumed that the radius of the minimal cylinder enveloping aleg is 0.02.Combining the objective and constraints together, the optimal

design problem of a Gough–Stewart platform is formulated asfollows.1) Problem 3: Optimal Design of the Gough–Stewart Plat-

form: Find a set of optimal design parameters such that

(21)

(22)

(23)

(24)

(25)

(26)

(27)

, and .

By observation the objective (21) and the constraints(22)–(27) are nonlinear without explicitly analytical expres-sions with respect to the design parameters .

B. Computation and Discussion

Similar to the Delta robot case, VTR and are usedas performance criteria for the algorithms, and

seconds (24 hours). For CRS, DE, GA, andPSO, the initial feasible points were randomly generated withinthe feasible region. The population sizes used in DE, GA,and PSO were identically set 40, 10 times of the number ofindependent optimization variables. The algorithmic parametersettings are identical to those in the Delta robot example. Usingthe same testing hardware and software in the Delta robot case,we obtained computation results, shown in Table III.• DE. The number of generations and the computation timerange from 6 to 12 and from 7058.9 to 22578 s, respec-tively. The corresponding average values are, respectively,8.8 and 13562.

• PSO. The number of generations ranges from 4 to 23. Thecomputation time ranges from 7477 to 22387 s. The av-erage number of generations and the average computationtime are 12 and 12572 s, respectively.

• GA. The number of generations ranges from 9 to 20. Thecomputation time ranges from 21522 to 58267 s. The av-erage number of generations and the average computationtime are 13.4 and 37107 s, respectively.

• CRS. In the five trials, CRS succeeded three times to reachthe VTR. In the three successes, the number of iterationsranges from 6 to 11 and the computation time ranges from1209 seconds to 25808 s. The average number of iterationsis nine and the average computation time is 14033 s.

• SQP. There are four independent optimization variablesin the problem. In the optimization, a 4 4 4 4 dis-cretization (i.e., 256 initial points) reached the VTR. Thecorresponding computation time is 323280 s.

From Table III, the following observations are obtained.1) The algorithms DE, PSO and GA succeeded in reachingVTR for all five trials. CRS succeeded three times out offive trials. SQPwithmultiple initial points actually failed toreach VTRwithin since its computation time exceeds

. This example also suggests that DE, PSO, and GAperform more steadily than CRS.

2) In this example, SQP yielded the longest computation time,323280 s. It is several ten times of the computation timesused in the probability-based algorithms. This arises be-cause the Gough–Stewart platform is a 6-DoF complexmechanism and the computation of kinematics becomesrather complicated. Further, the number of independent de-sign variables increases to 4. It shows that SQP is not goodat handling such a complicated problem.

3) For the global optimization algorithms, DE presented theleast average number of generations, 8.8, while PSO pos-sessed the least average computation time, 12572 s. But thecomputation times of DE and PSO are almost in the sameorder of magnitude.

4) Although GA succeeded in all five trials, its computationtimes are much longer than those using DE or PSO. The



TABLE IIICOMPUTATION RESULTS OF OPTIMAL DESIGN OF THE GOUGH-STEWART PLATFORM

Fig. 6. Computation evolution of DE, CRS, GA, and PSO: the Gough–Stewartplatform case.

minimal computation time by GA is nearly equal to themaximal one by DE or PSO. It indicates that the searchefficiency of GA is much worse than that of DE and PSO.

5) It is interesting to note that the minimal time by CRS is1209 s, which is almost one order of magnitude less thanthose by other algorithms. However, its computation timeperformance exhibits great inconsistency among differenttrials since the other two successful trials possess morethan ten times longer computation times.

According to the analysis above, the computational perfor-mances can be roughly ranked,

.As done in the Delta robot case, the four probability-based

algorithms were tested for convergence evolution performancegiven the identical termination time, 60,000 s. An identicalinitial points, , wasused. Typical computation evolution of algorithms was pre-sented in Fig. 6. The optimal objectives and the correspondingdesign parameters are shown in Table IV. In the starting period(0–10,000 s), CRS converged the fastest. It can be roughlyranked, . In the intermediateperiod (10,000–40,000 s), DE and PSO surpassed CRS one byone. The objective in GA was improved gradually, however,GA still performed the worst among the algorithms. In thefinal period (40,000–60,000 s), the convergence of CRS, PSO,

TABLE IVCOMPUTATION RESULTS WITH TERMINATION TIME 60000 S

and DE became slow since they reached a small neighbor-hood of the true optimum. GA still ranked the last althoughit got good improvement. Finally, the convergence rank is

.It is also worthy noting that CRS converges very fast in the

starting period while it improves rather slow in the later periods.Comparing the computation performances of the algorithms

in the two examples, it indicates that (1) for a simple designproblem (a simple mechanism and/or simple kinematics), SQPwith multiple initial points can be efficient. (2) For a compli-cated design problem, however, SQP becomes very inefficient.DE and PSO are the best choices to solve such optimizationproblems. CRS exhibits good convergence evolution perfor-mance in the starting period. It can be used to generate goodinitial points for other algorithms.

VI. CONCLUSION

In this paper, the convergence performances of five typicalalgorithms, CRS, GA, PSO, DE, and SQP with multiple initialpoints, are evaluated by optimal design of two parallel ma-nipulators, the Delta robot and the Gough–Stewart platform.Conclusions are obtained based on the problem complexity.(1) For a simple design problem (a simple mechanism and/orsimple kinematics), SQP with multiple initial points performsbest. (2) While for complicated design problems, SQP performsthe worst and DE and PSO are the best choices. Taking intoaccount the efficient convergence of CRS in the starting period,we may combine it with other algorithm, e.g., DE or PSO. CRSis used to generate good initial points while the other algorithmis applied to continue later search.

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Yunjiang Lou (M’06–SM’13) received the B.S. andM.E. degrees in automation from the University ofScience and Technology of China, Hefei, in 1997 and2000, respectively, and the Ph.D. degree in electricaland electronic engineering from the Hong Kong Uni-versity of Science and Technology, Clear Water Bay,Kowloon, Hong Kong, China, in 2006.He is now with the School of Mechatronics En-

gineering and Automation, Harbin Institute of Tech-nology Shenzhen Graduate School, and the ShenzhenKey Lab for Advanced Motion Control and Modern

Automation Equipments, Shenzhen, China. His research interests include anal-ysis and optimal design of parallel manipulators, integrated design of mecha-tronic systems, and motion control.

Yongsheng Zhang received the B.S. degree inmechanical engineering from the Central SouthUniversity, Changsha, China, in 2009 and the M.E.degree from the School of Mechatronics Engineeringand Automation, Shenzhen Graduate School, HarbinInstitute of Technology, China in 2012.He is now with Luoyang Institute of Electro-Op-

tical Equipment, AVIC, Luoyang, China. Hisresearch interests include optimization algorithms,optimal design of parallel manipulators, and pathplanning of robotics.



Ruining Huang (M’12) received the B.S. degrees inmechanical engineering and the M.E. and Ph.D. de-grees in mechanical manufacturing and automationfrom the Harbin Institute of Technology, Harbin,China, in 2000, 2002, and 2006, respectively.He is now with the School of Mechatronics En-

gineering and Automation, Harbin Institute of Tech-nology Shenzhen Graduate School, and the ShenzhenKey Lab for Advanced Motion Control and ModernAutomation Equipments, Shenzhen, China. His re-search interests include analysis and optimal design

of parallel manipulators, and micro EDM technology.

Xin Chen received the B.S. degree in manufac-turing engineering from Changsha Railway Collage,Changsha, China, in 1982, the M.Sc. degree inmanufacturing engineering from the Harbin Instituteof Technology, Harbin, China, in 1988, and thePh.D. degree in mechanical engineering from theHuazhong University of Science and Technology,Wuhan, China, in 1995.He is a Professor with the School of Electro-

mechanical Engineering, Guangdong Universityof Technology, Guangzhou, China. His research

interests include manufacturing industrial informationlizing and collaborativedesign, mechanical design theory and method, and microelectronic packagingtechnology and equipment.

Zexiang Li (M’89–SM’05–F’07) received the B.S.degree (Hon) in electrical engineering and economicsfrom Carnegie Mellon University, Pittsburgh, PA, in1983, and the M.A. degree in mathematics and thePh.D. degree in electrical engineering and computerscience from the University of California, Berkeley,CA, USA, in 1985 and 1989, respectively.He is a Professor with the Department of Elec-

tronic and Computer Engineering, Hong KongUniversity of Science and Technology, Clear WaterBay, Kowloon, Hong Kong. His research interests

include robotics, nonlinear system theory, and manufacturing.

optimisation of parallel manipulators

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