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On the Optimum Design of3-RPR Parallel Mechanisms
Mohammad Hossein Saadatzi∗, Mehdi Tale Masouleh†, Hamid D. Taghirad∗ Clement Gosselin† and Philippe Cardou†∗ Advanced Robotics & Automated Systems (ARAS), Faculty of Electrical & Computer Engineering
K.N. Toosi University of Technology, P.O. Box 16315-1355, Tehran, IranEmail:[email protected], [email protected]
† Laboratoire de robotique, Departement de genie mecanique,Universite Laval - Quebec, QC, Canada - G1V 0A6
Email: [email protected], [email protected]@gmc.ulaval.ca
Abstract—This paper deals with the optimization of 3-RPRplanar parallel mechanisms based on different performanceindices including the kinematic sensitivity, the workspace andthe singularity locus. Since the kinematic sensitivity has beenproposed only recently, more emphasis is placed on how it shouldbe adapted for the optimization of 3-RPR parallel mechanisms.The optimization is implemented in sequence using first a singleobjective technique, differential evolution, and then resortingto a multi-objective optimization concept, the so-called Non-dominated sorting genetic algorithm-II. A Pareto-based multi-objective approach helps to overcome the problem of unit-inconsistent objectives in the optimization algorithm. Moreover,an ∞-norm is used in computation of the kinematic sensitivitywhich has perhaps the clearest physical interpretation.
I. INTRODUCTION
The last two decades have witnessed an important spread inthe use of parallel mechanisms in applications such as motionsimulators, machine tools and even nano-manipulators. Par-allel mechanisms have evolved from rather marginal devicesto become the state of the art in several applications. This isdue to good kinematic and stiffness properties which enablea high accuracy. We must acknowledge, however, that theirworkspace is more constrained than serial mechanisms. Theoptimum design of parallel mechanisms has received specialattention in order to alleviate some of their shortcomings interms of kinematic properties, such as a limited workspaceand complex singularities.
The performances of parallel mechanisms are highly de-pendent on their geometric properties. Numerous examplesshow that a careful design optimization can lead to significantimprovements over the initial design [1]. In practice, thereare always multiple performance requirements to be satisfied.Usually, an index is defined for each requirement, whichindicates to what extent the requirement is satisfied or violated[1]. Numerous performance indices have been proposed tocompare robot architectures based on their kinematic prop-erties. The most notorious kinematic indices, manipulabilityand dexterity, still entail some drawbacks [1]–[3], that aremainly due to problems which prevent the definition of a singleinvariant metric for the special Euclidean group. Recently, in
[2], two distinct metrics were defined and formulated, namelythe maximum rotation sensitivity and the maximum point-displacement sensitivity. These two indices provide tight upperbounds of the end-effector rotation and point-displacementsensitivity under a unit-magnitude array of actuated-joint dis-placements.
Independent investigation of different performance indiceswhich can be achieved by resorting to single objective op-timization is helpful to gain a better understanding of theireffects on the mechanism performance. In this paper, Dif-ferential Evolution (DE) [4], [5] is proposed, which is quitedifferent from other Evolutionary Algorithms (EA) such asgenetic algorithms and evolutionary strategy [6]. The maindistinction is that in the case of DE, distance and directioninformation from the current population are used to guide thesearch process.
Once the optimum value of one objective is obtained,the corresponding optimum values of the design parametersusually yield unsatisfactory results for the other objectives.For instance, as it was revealed in [1], the SSM, a simplifiedGough-Stewart platform with maximal workspace volume fora given stroke of the actuators has similar base and movingplatform and is therefore architecturally singular. In the casewhere more than one performance objective should be con-sidered, a graphical approach based on an atlas representationof performance indices would be really helpful. However, thisapproach may be useful only for a very small set of designparameters.
Based on the methodology proposed in the literature, whennumerous performance objectives are in place the problemcan be made equivalent to a single objective optimizationby combining them in a weighted linear sum. As majordrawbacks, the obtained solutions will depend on the relativevalues of the specified weights and the sum of differentobjectives with inconsistent units may result in a meaninglessconcept for the problem, and, therefore, lead to erroneousconclusions.
The objective way of solving multi-objective problemsrequires a Pareto-compliant ranking method [7], favouring
nondominated solutions. In this Method, no weight is required,and thus Pareto solutions are those for which improvement inone objective can only occur with the worsening of at leastone other objective. Thus, instead of a unique solution to theproblem, the solution to a multi-objective problem is a setof Pareto points. Among these points, the designer shouldchoose those which tend to fit closely with the specific designobjective. In this paper, we use the Nondominated SortingGenetic Algorithm-II (NSGA-II) [8], which is a standardPareto-based multi objective approach.
Defining a set of comprehensive and practical objectiveswhich covers all kinematic properties, such as singularity-free workspace and mechanical interferences, is a delicatetask. Hence, before using intelligent methods, the mecha-nism must completely be investigated, i.e., the workspaceand singularities of the mechanism should be analysed, andproper constraints must be devised. Upon this investigation,intelligent methods can be used to specify the remainingfree parameters selection. Using intelligent methods solely,would result in a lengthy computation process, and most ofthe time, the designs obtained suffer from singularities andmechanical interferences, therefore the designs are practicallyincompetent.
The remainder of this paper is organized as follows. First,the geometric modeling and the first-order kinematic analy-sis of 3-RPR parallel mechanisms are reviewed. Then, theconstant-orientation workspace and singularity configurationsof a 3-RPR parallel mechanism are investigated. The paperpursues the study by illustrating kinematic sensitivity as aperformance index for mechanisms. The single and multi-objective optimization procedures are elaborated, with theemphasis on refining the objective parameters. The single-objective procedure is minimized by resorting to DE conceptwhich lays down the essentials of the multi-objective optimiza-tion. Subsequently, Pareto-based multi-objective optimizationis considered and NSGA-II is used to design the optimalmechanism. Finally, the paper concludes with some remarksthat provide insight into ongoing works.
II. ARCHITECTURE AND FIRST-ORDER KINEMATICREVIEW OF 3-RPR PARALLEL MECHANISMS
A. Architecture Review
Figure 1 illustrates the schematic representation of a 3-RPRparallel mechanism. As depicted in Fig. 1, a planar 3-RPRparallel mechanism with actuated prismatic joints consists ofa fixed triangular base 4A1A2A3 and a mobile triangularplatform 4B1B2B3. The passive revolute joints, located atAi and Bi, are connected by the prismatic actuator of variablelength ρi, i = 1, 2, 3. The unit vector along the ith prismatic-joint direction is denoted by ni = [nix , niy , 0]T . The pose(position of point C and orientation) of the mobile platformis described by two coordinate systems, as shown in Fig. 1.The position of the mobile platform reference-point C withrespect to the fixed frame is represented by p = [x, y, 0]T .The position vector of point Bi in the fixed and mobile framesare denoted bi and b′
i, i = 1, 2, 3, respectively. It should be
λa
λb
ρ1
ρ2
ρ3
A1 ≡ O
A3
A2
φ
B1
B3
B2C ≡ O′
y
x
Fig. 1. Schematic representation of a 3-RPR parallel mechanism.
noted that in this paper the superscript ′ for a vector stands fora representation in the mobile frame O′
x′y′ . Finally, the vectorconnecting Bi to C expressed in the fixed frame is representedby si. The rotation matrix, Q, representing the rotation of theplatform from frame Oxy to frame O′
x′y′ is:
Q =
cos φ − sinφ 0sinφ cos φ 0
0 0 1
, (1)
where φ stands for the rotation angle of the platform aroundthe axis perpendicular to the plane of the mobile platform.
B. First-order Kinematics and Jacobian Matrix
The IKP, FKP, and Jacobian matrix of a 3-RPR parallelmechanism has been extensively studied [1], [9] and it can beshown that the first-order kinematics of the mechanism maybe written as:
ρ = Jx. (2)
In the above, ρ = [ρ1, ρ2, ρ3]T , x = [x, y, φ]T and:
J =
n1x n1y (b1 × n1) · kn2x n2y (b2 × n2) · kn3x n3y (b3 × n3) · k
, (3)
where k stands for the unit vector along the axis perpendicularto the plane of the mobile platform.
III. PRELIMINARY INVESTIGATION FOR THEDIMENSIONAL SYNTHESIS OF 3-RPR PARALLEL
MECHANISMS
The dimensional synthesis of mechanical systems, includingparallel mechanisms, pertains to determining the dimensionsof the mechanism in such a way that it complies as closelyas possible to the performance needed for the required task.Figure 2 illustrates the constant-orientation workspace whichcorresponds to the shaded area, and the singularity locus,which is the solid-line circle [10], of a 3-RPR parallelmechanism for a given orientation. The constant-orientationworkspace is the common area of the vertex spaces generated
-2 -1 0 1 2 3 4
-2
-1
0
1
2
3
4
w1
w2
w3
y
x
Fig. 2. Mechanism workspace for λa = 1.45, λb = .35, and φ = π2
.Dashed circles with ρmin and ρmax, show the limbs’ vertex spaces and thesolid circle is the singularity locus.
by each limb. The vertex space of each limb is the ring centredat ai + bi, where ai is the position vector of point Ai, withρmin and ρmax as the radii for internal circle and externalcircle, respectively. In this paper, it is assumed that all theprismatic actuators have identical strokes defined by ρmin andρmax. Here, the optimization is performed based on the unitminimum length of the actuators, the strokes of actuators rangefrom 1 to 1.9 of their initial leg length, which implies that:
ρmin = 1, ρmax = 1.9. (4)
In [10], an algebraic expression for the singularity locusof a planar 3-RPR parallel mechanism with similar base andplatform is obtained, which consists of a circle passing throughthe centres of the three rings generated by the vertex spaces, asdepicted in Fig. 2. Moreover, in [11] the question of scalabilityfor an optimal 3-RPR parallel mechanism is formulated asfollows, which is the central idea behind our optimizationprocedure: The maximal singularity-free workspace for a3-RPR parallel mechanism is possible for a design havingequilateral triangle base (with unit area) and mobile platform,and for a special ratio of link lengths of base and mobile plat-form. Henceforth, in this paper, a 3-RPR parallel mechanismwith equilateral triangle for the base and mobile platform isconsidered, since it inherently adopts some optimality for itssingularity and workspace. The aforementioned link lengthsare λa and λb, which are respectively the radius of the baseand the mobile platform, as shown in Fig. 1.
IV. KINEMATIC SENSITIVITY PERFORMANCE INDEX
The analysis of the performance sensitivity to uncertaintiesis an important task [12]. The quantification of performancesensitivity has attracted a great of interest and many indices areproposed, such as manipulability and dexterity. As mentionedabove, the most notorious indices, which are manipulabilityand dexterity, still entail some drawbacks [1]–[3] and theresults obtained using them may be questioned. Therefore, inour optimization objectives these two indices are not taken intoaccount, and we resort instead to the kinematic sensitivity ofthe mechanism.
The kinematic sensitivity is defined as the maximum ro-tation sensitivity and the maximum point displacement sen-sitivity. These two indices provide tight upper bounds to theend-effector rotation and point-displacement sensitivity undera unit-magnitude array of actuated-joint displacements. Themaximum magnitude rotation and point displacement under aunit q-norm actuator displacement are respectively:
σr,q ≡ maxq
| φ |q, σp,q ≡ maxq
‖ p ‖q, (5)
These two indices provide separately upper bounds on the am-plitude of rotation and operating-point displacement inducedby a constant overall magnitude of the active-joint displace-ments. Thus, they are referred to as the maximum rotation andpoint-displacement sensitivities, respectively. These pieces ofinformation are thought to be meaningful for optimal designand have been proposed recently in [2]. Among the manypossible choices of q, infinity has perhaps the clearest physicalinterpretation, as explained by Merlet [13]. Directly, from [2],it follows that the magnitude of the rotation sensitivity maybe cast into a linear programming problem, namely:
σr,∞ = max ∆φ, s.t. L∆x − 16 � 06, (6)
and the point-displacement sensitivity is the maximum of thevalues returned by two linear programs, namely:
σr,∞ = maxt1,t2
{t2 = max ∆y, s.t. L∆x − 16 � 06
(7)where L ≡ [JT − JT ]T , � denotes the componentwiseinequality, and 16 ≡ [1 1 · · · 1]T ∈ R6. The linprog functionfrom MATLAB is used in this paper to compute the kinematicsensitivity of the mechanism in different postures.
Since the kinetostatic indices are dependent on the pose ofthe mobile platform, thus the next step consists in extendingthese two indices to all poses which the mechanism canreach. Following the reasoning presented in [14], instead ofconsidering the index I for a specific pose, a global index ζI
is introduced over the manipulator workspace W by:
ζI =
∫W
IdW∫W
dW
. (8)
One should be aware that I tends to infinity at singularposes when it is applied for σp,∞ and σr,∞. Therefore it isunable to distinguish mechanisms which have singular posesin their workspace. To circumvent this problem the reasoningapplied to the condition number in [14] is considered. Point-displacement and rotation sensitivities are bounded from zeroto infinity and hence their inverses are in the same intervaland are not useful. As the minimization of these indicesis of interest, the maximization of the inverse of their unitoffshoot is suggested. By this means, the proposed measurewill be normalized between [0, 1] interval. Consequently, Eqs.
‖∆ρ‖ =1 ‖∆ρ‖ =1
∆x
∆x
∆x
1 6 6t = max ∆x, s.t. L∆x − 1 � 0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5 αa
αb αc
λa − λb = ρmin λa + λb = ρmax
λb = 0.2 l
λb
λa
Fig. 3. Constraints and design space of λa and λb.
(6) and (7) are reformulated as follows to be applicable to ouroptimization purpose:
σ′r,∞ =
11 + σr,∞
, σ′p,∞ =
11 + σp,∞
. (9)
We then have:0 > σ′
r,∞, σ′p,∞ > 1. (10)
V. OPTIMIZATION PROCEDURES
In this section, we first investigate the existence of thenonvanishing singularity-free workspace and the maximizationof the global singularity-free workspace. Then, the single-and multi-objective optimization procedures are presented,including the final results for the optimized 3-RPR parallelmechanism.
A. Existence of a Nonvanishing Singularity-free Workspace forAll Orientations
In order to have a mechanism of practical interest, theoptimization parameters, λa and λb, should be arranged insuch a way that the optimal values result into a nonvanishingworkspace for every angle φ. This leads us to addressingthe existence of a nonvanishing singularity-free workspace forevery orientation of the mechanism under study. Referring toFig. 2, the coordinates of the centre of each vertex space,wi = [wix, wiy]T , for different orientations of the movingplatform could be formulated as:
wi = ai − Qb′i. (11)
For this particular design with equilateral triangle for the baseand mobile platform, the centres of three circles of vertexspaces lie exactly on the singularity circle, shown in Fig. 2.This leads to the coordinates, ws = [wx, wy]T and radius ρs
of the singularity-free workspace, namely:
wx =w1x + w2x + w3x
3, wy =
w1y + w2y + w3y
3, (12)
ρs =√
(wix − wx)2 + (wiy − wy)2 (13)
11.5 0.1
0.6
1.1
αa
0
π
π
2
−
π
2
φ
π
x
y0.5
11.5
0.1
0.6
0
αb
π
π
2
−
π
2
φ
x
yπ
1 1.5 0.6
1.1
φ 0
π
π
2
−
π
2
π
αc
x
y
1 1.5 0.1
0.6
1.1
Optimal workspace
0
π
π
2
−
π
2
φ
π
x
y
Fig. 4. Workspace as a function of φ for (λa, λb) sets which are located invertices αa, αb, αc, and for the mechanism with optimal workspace.
For φ = 0 and φ = π, the nonvanishing singularity-free workspace can be achieved by imposing the followingconstraints:
λa − λb > ρmin, λa + λb < ρmax. (14)
Moreover, because of practical considerations, the mobile plat-form which carries the manipulator tool must be constrained toa reasonable size. Hence, the following constraint is adoptedin this paper:
λb ≥ 0.2. (15)
Considering these constraints, it can be concluded that thepermitted set of values for λa and λb are inside a trianglewhich is defined by Eqs. (15) and (14), as shown in Fig. 3.In this figure, the vertices of the triangular area formed byEqs. (14) and Eq. (15) are labeled αa, αb and αc, respectively.The robot geometries corresponding to each of these pointsyield workspaces with zero areas at φ = 0 and φ = ±π,respectively, as can be seen in Fig. 4.
B. Maximization of the Global Singularity-free Workspace
The global workspace of the mechanism can be regardedas three-dimensional space for (x, y, φ), as shown in Fig. 4.The volume of the workspace can be approximated numer-ically using discrete integration over φ using the followingformulation:
W =∫ π
−π
A(φ)dφ, (16)
where A(φ) is the area of workspace for a given orientationof the mobile platform, φ. The area of the workspace foreach orientation, A(φ), can be computed using the IKP ofthe mechanism. Each pose in the Cartesian space, (x, y, φ),for which the strokes of all limbs (ρmin < ρi < ρmax,for i = 1, 2, 3), are satisfied belongs to the workspace. Inorder to decrease the computational burden of this problem, anestimation of the area which is likely to be a part of workspaceis helpful. For each vertex space a square shape area can bedefined as follows:
iXmin x : wix − ρmax < x < wix + ρmax :i Xmax x, (17)iYmin x : wiy − ρmax < y < wiy + ρmax :i Ymax x. (18)
Thus, conditions for the existence of all vertex spaces can beformulated as follows:
max{iXmin x} < x < min{iXmax x}, i = 1, 2, 3, (19)
max{iYmin x} < y < min{iYmax x}, i = 1, 2, 3. (20)
C. Single-objective Optimization Using Differential Evolution
Usually in EA algorithms, variation from one generation tothe next is achieved by applying crossover and/or mutationoperators. In DE, mutation is applied first to generate a trialvector, which is then used within the crossover operator toproduce one offspring. The DE mutation operator producesa trial vector for each individual of the current population bymutating a target vector with a weighted differential. This trialvector will then be used by the crossover operator to producean offspring. For each parent xi(t), a trial vector ui(t) is firstgenerated as follows: Select a target vector, xi1(t), from thepopulation. Then, randomly select two individuals, xi2 andxi3, from the population. The corresponding trial vector isgiven by:
ui(t) = xi1(t) + β(xi2(t) − xi3(t)), (21)
where β ∈ [0,∞] is the scale factor, controlling the amplifi-cation of the differential variation. The DE crossover operatorimplements a discrete recombination of the trial vector ui(t)with the parent vector xi(t) to produce the offspring x′
i(t).In this paper, DE with (random) best population chromosomeas target vector and binary crossover is used to find theparameters of an optimal mechanism [6]. Binomial crossoverand exponential crossover are typical crossover methods, inwhich, some genes of the parent vector are changed with thegenes of the trial vector to produce the offspring. In orderto determine which individuals will take part in the mutationoperation to produce a trial vector, and to determine which of
Objective⇒ Workspace σ′r,∞ σ′
p,∞Performance⇓
λa 1.4 1.45 1.43λb 0.2 0.45 0.37
Workspace 3.31 1.42 1.79σ′
r,∞ 0.12 0.29 0.24σ′
p,∞ 0.83 0.95 0.93
.
TABLE IRESULTS OF SINGLE-OBJECTIVE OPTIMIZATION OF MENTIONED
MECHANISM PERFORMANCE INDICES.
the parent or the offspring will survive to the next generation,a selection function is used.
Table I and Figures 5 report the results of single-objectiveoptimization for the performance criteria which was describedin the previous sections. As can be observed from Table I, themechanism with an optimum workspace corresponds to λb =0.2 and λa = 1.4. Note that λb has its minimum value, whichis consistent with Eq. (11). Minimum λb is related to the vectorb′
i of minimum length and it results in three vertex spacesbecoming as close as possible to each other. From Fig. 3, itcan be deduced that λb = 0.2 implied that λa ∈ [1.2, 1.7] andthat the mean value of λa is 1.45. Hence, λa approximatelyhas its mean value (λa = 1.4). As previously mentioned, theleft side of the triangle of the space of design parameters isrelated to a zero area workspace in the plane φ = 0 and itsright side is related to a zero area workspace in planes φ = ±π(see Fig. 4). Hence, the global workspace near these edges issmaller than away from them, closer to the triangle centroid.Thus the optimum design point lies close to the mean value ofλa in the space of design parameters Fig. 5(a). The workspaceof the optimum mechanism is illustrated in Fig. 4.
D. Multi-objective Optimization Using NSGA-II
Multi-objective optimization consists in simultaneously op-timizing two or more conflicting objectives with specificconstraints. EA are well-known methods in multi-objectiveoptimization (MOEA). Genetic algorithms such as NSGA-II and the Strength Pareto Evolutionary Approach 2 (SPEA-2) are common methods. The objective approach to solvingmulti-objective problems needs a Pareto-compliant rankingmethod, favouring non-dominated solutions. In a problem withmore than one objective function, there are two possibilities forany two solutions x1 and x2: one dominates the other or nonedominates the other. If both of the following conditions hold,then a solution x1 is said to dominate x2: (i) the solution x1
is not worse than x2 among all objectives and (ii) is strictlybetter than x2 in at least one objective. We now explain astep-by-step approach for NSGA-II [8].
• Combine parent and offspring populations and constructRt = Pt ∪ Qt. Do a non-dominate sorting of Rt andidentify different fronts: Fi, i = 1, 2, · · · .
• Set new population Pt+1 = 0. Set a counter i = 1. Until|Pt+1|+Fi < N , N being the size of Pt, perform Pt+1 =Pt+1 + Fi and i = i + 1.
(a) Optimal workspace atlas (b) Optimal point-displacement sensitivity atlas (c) Optimal rotation sensitivity atlas
Fig. 5. Atlas representation for the results obtained form the single-objective optimization. The marker • stands for the optimum point.
• If |Pt+1|+Fi > N , perform the crowding-sort procedureand include the most widely spread (N−|Pt+1|) solutionsusing the crowding distance value in sorted Fi to Pt+1.
• Create offspring Qt+1 from Pt+1 by using crossover andmutation operators.
After optimization, all possible solutions in the entire solutionspace are obtained without the need to integrate all objectivefunctions into each other. Figure 6 represents the resultingPareto front where one can readily determine the final solu-tions depending on chosen criteria. Choosing a Pareto pointcould be done based on the expected workspace volume thatwe are deemed to get when using unit minimum lengths ofthe actuators. The chosen optimal point from the Pareto frontaccording to this assumption, also results in optimal values ofthe other two objectives.
VI. CONCLUSION
This paper investigated the optimization of the 3-RPRparallel mechanism by means of single- and multi-objectiveoptimizations. The rotation and point-displacement sensitivi-ties were considered as kinetostatic performance indices. Askinematic sensitivity has been proposed only recently, moreemphasis was placed on its integration to an optimizationprocedure. According to the proposed optimization procedure,the design parameters boil down to specifying the radii ofbase and mobile platform, λa and λb. The optimum pa-rameters for each design criterion were obtained using DE.Finally, NSGA-II was used to optimize the mechanism. Themulti-objective optimization technique applied in this studyalso has an important feature: It circumvents the problemof unit-inconsistency between different objectives. Ongoingwork includes the development of a simple guidelines for theoptimization of parallel mechanisms.
ACKNOWLEDGEMENT
The authors would like to acknowledge the financial support ofthe Natural Sciences and Engineering Research Council of Canada(NSERC) as well as the Canada Research Chair program.
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