Synthesis of Compliant Mechanisms for Path Generation using Genetic Algorithm

Transcription

1 Synthesis of Compliant Mechanisms for Path Generation using Genetic Algorithm Anupam Saxena Mechanical Engineering, Indian Institute of Technology, Kanpur , India In this paper is described a procedure to synthesize the optimal topology, shape, and size of compliant continua for a given nonlinear output path. The path is prescribed using a finite number of distinct precision points much in accordance with the synthesis for path generation in traditional kinematics. Geometrically nonlinear analysis is employed to model large displacements of the constituent members. It is also essential to employ nonlinear analysis to allow the output port to negotiate the prescribed path accurately. The topology synthesis problem is addressed in its original binary form in that the corresponding design variables are only allowed to assume values of 0 for no material and 1 for the material present at a site in the design region. Shape and size design variables are modeled using continuous functions. Owing to the discrete nature of topology design variables, since gradient based optimization methods cannot be employed, a genetic algorithm is used that utilizes only the objective values to approach an optimum solution. A notable advantage of a genetic algorithm over its gradient based counterparts is the implicit circumvention of nonconvergence in the large displacement analysis, which is another reason why a genetic algorithm is chosen for optimization. The least squared objective is used to compare the design and desired output responses. To allow a user to specify preference for a precision point, individual multiple least squared objectives, same in number as the precision points are used. The multiple objectives are solved using Nondominated Sorting in Genetic Algorithm (NSGA-II) to yield a set of pareto optimal solutions. Thus, multiple solutions for compliant mechanisms can be obtained such that a mechanism can traverse one or some precision points among those specified more precisely. To traverse the entire path, a solution that minimizes the sum of individual least square objectives may be chosen. Synthesis examples are presented to demonstrate the usefulness of the proposed method that is capable of generating a solution that can be manufactured as is without requiring any interpretation. DOI: / Introduction Contributed by the Mechanisms and Robotics Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 29, revised September 29, Associate Editor: G. K. Ananthasuresh. The advantages of compliant mechanisms are well known 1 3 and the need for their systematic design has led to the development of two approaches over the last decade. The first is the pseudo rigid body model approach 1,3 wherein a compliant mechanism is regarded as an assemblage of rigid links with joint springs. This allows numerous synthesis methods in rigid-body kinematics to be used for the design of compliant mechanisms. The second is the topology synthesis method 4 7 that entails generating an appropriate topology within a design region to achieve a single piece continuum with desired deformation and load bearing characteristics. An early method in topology synthesis of compliant mechanisms was by Ananthasuresh et al. 4 who posed the design objectives as maximizing the output deformation along a specified direction, and maximizing the stiffness or minimizing the strain energy. An improved formulation was proposed by Frecker et al. 5 and Nishiwaki et al. 8 who posed the multicriteria objective as extremizing the ratio of the output displacement and strain energy. Sigmund 6 proposed a similar method of maximizing the mechanical advantage with constraints on volume and input displacements. Saxena and Ananthasuresh 9 proposed an energy based formulation wherein the ratio of output energy proportional to the square of the output displacement and strain energy was maximized. The flexibility and stiffness measures were generalized by Saxena and Ananthasuresh 7 who employed an optimality criteria method for topology design of compliant mechanisms. Saxena and Ananthasuresh 10 also employed local stress constraints to directly pose the failure criteria in topology synthesis. Larsen et al. 11 presented the synthesis of compliant topologies with multiple input and output ports. Hetrick and Kota 12 employed size and shape optimization to improve compliant topologies by maximizing the mechanical efficiency subject to the desired mechanical or geometric advantage requirements. Topology optimization of compliant mechanisms was incorporated with nonmechanical actuation as well. Synthesis with electrothermal actuation, for example, was performed by Jonsmann et al. 13, Yin and Ananthasuresh 14, Mankame and Ananthasuresh 15, Saxena and Saxena 16 and with piezoelectric actuation was accomplished by Canfield and Frecker 17. The aforementioned works employed small deformation analysis for the synthesis of compliant mechanisms that could generate only qualitatively accurate solutions. That is, the resulting topologies could model the output deformation accurately only at the start of actuation. Realizing that the constituent members in a compliant continuum undergo large deformation, Burns and Tortorelli 18, Saxena and Ananthasuresh 19, and Pedersen et al. 20 employed geometrically nonlinear finite element analysis for their synthesis. In addition to accurate modeling with large deformation, topology design of compliant mechanisms for prescribed nonlinear force-deflection characteristics and curved output paths was also made possible. Sigmund 21,22 performed topology synthesis of electro-thermal actuators with nonlinear deformation using multiple ports and materials. Mankame and Ananthasuresh 23 designed compliant mechanisms using contact mechanics. Journal of Mechanical Design Copyright 2005 by ASME JULY 2005, Vol. 127 / 745

2 Motivation Optimization methods in the aforementioned works have been mainly gradient-based with algorithms comprising the optimality criteria method 7,24, Sequential Linear 5 and Quadratic 10,19 Programming and the Method of Moving Asymptotes 6,21,22. Ideally, a topology design problem for a fixed finite element grid should be posed in its original discrete form. That is, whether the chosen material should be assigned to a finite element or not. This requires binary type decision making for design variables like cross sections, elastic moduli associated with the subregions or finite elements. For example, a zero value for a design variable implies that the corresponding subregion is absent while a unit value refers to the presence of a subregion. The first relaxation with a gradient based method is to choose a very small but positive lower bound for design variables to avoid the global stiffness matrix from being singular. Further, the aforesaid binary material assignment is relaxed to a continuous material interpolation function to facilitate the computation of gradients. The resultant topologies may then possess subregions with material partly assigned to them. That is, the corresponding design variables may assume intermediate values between 0 and 1 for which the results are often difficult to interpret and manufacture. Attempts have been made to discourage the design variables from seeking the intermediate values. A popular way with linear plane stress finite elements, called the SIMP Solid Isotropic Material with Penalization interpolation, is to penalize the design variables with a user-specified exponent usually 3 25,26. Additionally, an explicit penalty function with regularized intermediate density control has been proposed by Borrvall and Petersson 27 to get the black and white designs for minimum mean compliance problems. Another issue when implementing large deformation analysis based synthesis with gradient-based optimization is to circumvent nonconvergence or snap-through during the analysis procedure. A gradient-based search is point-to-point in nature with the gradients at an intermediate point solution determining the step size to obtain the subsequent point solution. Nonconvergence in large deformation analysis occurring for an intermediate solution hampers the gradient computation and thus the determination of step size and subsequent solution. In that case, the search may not proceed further and may require user intervention to physically remove the subregion s close to their nonexisting states that cause bifurcation. Sigmund et al. 28 have suggested ways to circumvent this problem. For example, one way may be to disregard the force-balance equations at nodes surrounded by nonexisting finite elements. In this paper, topology, size and shape optimization of large displacement compliant mechanisms is performed for prescribed function generation and path generation like kinematic specifications. The topology optimization problem is posed in its original discrete form in that the layout design variables are strictly allowed to have only 0 or 1 values to represent the void and material states respectively. Because of the discrete nature of such variables, the gradient computation is not possible. Thus, a function based search method, namely, genetic algorithm is employed. As explained later in the paper, with genetic algorithm, nonconvergence or snap-through like issues in large displacement analysis can be implicitly circumvented without permanent loss of any subregion in the design domain. Genetic algorithms have previously been used successfully in topology optimization of stiff structures for example, 29 31, and compliant mechanisms e.g., This paper is organized as follows. First, the topology, size and shape optimization problem for compliant mechanisms is formulated to achieve function and path generation like kinematic requirements. Geometrically nonlinear analysis with frame finite elements used in this work is briefed next. Kirchhoff s shallow arch beam theory is implemented 36 to obtain the element-level equilibrium equations. The search procedure in genetic algorithm is briefed followed by adaptations made in the function evaluation Fig. 1 Problem definition routine to circumvent nonconvergence in the large displacement analysis. Examples with realized prototypes are presented later to illustrate the synthesis procedure. Problem Statement Consider a generic design domain in Fig. 1 with input forces, say, F 1 j and F 2 j, j=0,...,n and fixed boundary region shown as shaded. The intent is to seek an appropriate material connectivity within the design region so that the output port P 0 shown at the initial location traverses a prescribed nonlinear path given by the precision points P 0,P 1,P 2,...,P n. This is analogous to path generation specification in traditional kinematics. A special case is when only one input force, say, F 1 j, j=0,...,n is chosen for actuation and the prescribed points P 0,P 1,P 2,...,P n are all rectilinearly placed. The specification then may be termed as function generation. Consider an arrangement of frame finite elements representing the design region shown as a super ground structure 7. The ground structure is a regular array of cells wherein each cell has four elements representing its rectangular boundary and four elements placed as a cross representing its interior. Although frame elements do not span the entire design region in a relatively coarse mesh, they do capture the bending modes of deformation and seem reasonable for topology and not shape optimization. The accomplishment of path generation specification would require obtaining an appropriate topology, size and shape of a fully compliant mechanism. Ignoring any one of the three attributes may result in a mechanism that traverses a specified path only approximately. To model material connectivity, the elastic moduli, E i i=1,...,n of frame elements may be chosen as design variables. Here, N is the number of frame elements representing the region. For E i =0, the ith frame element is absent from the connectivity and for E i =E 0, where E 0 is the modulus of the chosen material, the corresponding frame element is present. Any value of the elastic modulus other than 0 or E 0 is strictly not allowed, which renders the discrete nature to the optimization problem. To model the size, cross sections of frame elements may be additionally treated as design variables. This is tantamount to choosing the uniform out-of-plane thickness t and the in-plane widths w i as design variables for planar mechanisms. Thirdly, shape optimization may be partially accomplished by varying the nodal positions. That is, frame elements individually would still be linear in shape, however, varying the lengths and orientations of the elements by altering their nodal coordinates would partially provide local shape changes. To ensure nonoverlapping of elements, a bounding region within which a node may be repositioned can be identified. If l c and w c are the length and width of a cell Fig. 2, an intermediate node j can be positioned within the region of size 1 2 l c 1 2 w c such that the initial position of node j is the centeroid of this region. Likewise, a corner node k of a cell may also be located within a region of the same size. 746 / Vol. 127, JULY 2005 Transactions of the ASME

3 Fig. 2 Regions of node placement when nodal positions are treated as design variables A set of design variables may therefore comprise the elastic moduli E i i=1,...,n of frame finite elements that are to be discretely specified 0 or E 0, the in-plane widths w i i =1,...,N that would be varying linearly and bounded within the specified manufacturable limits w l and w u, the uniform out-ofplane thickness t with bounds t l and t u, and the nodal coordinates x j,y j, j=1,...,m that would be positioned within a rectangular region of size 1 2 l c 1 2 w c. Here, M is the total number of nodes. The number of elements representing the design region fineness of a mesh, material or E 0 chosen, and the positions of fixed degrees of freedom would also play a role in determining the optimal material layout, size and orientations of individual elements. Their specification in this work, however, has been left to user s discretion. Having identified the design variables, their nature discrete and continuous and their bounds, we can now formulate the objective. For a candidate solution, let Q 0,Q 1,Q 2,...,Q n be the response of the output port for p+1 actuation forces F k j, j=0,...,n;k=0,...,p.an approach to obtain the desired path may be to minimize the least square metric, as suggested by Saxena and Ananthasuresh 19 and Pedersen et al. 20. That is n Minimize: Q i P i Q i P i i=0 with respect to: E i,w i, i =1,...,N, t and x j,y j, j =1,...,M such that: E i is either 0 or E 0 w l w i w u t l t t u and x j,y j l c,w c x j,y j x j,y j l c,w c P1 where x j,y j 0 is the initial position of node j. In P1, among the precision points, a user may prefer that the output port traverses through one or some of them more precisely as opposed to all. An option is that the above objective can be generalized to the weighted least squared one below. n Minimize: W i Q i P i Q i P i P2 i=0 where W i, i=0,...,n are the user specified weights. For all W i sas 1, one achieves the conventional least square objective in P1. If a user desires that the output port of the mechanism passes through only one precision point, say, P j, W j can be set to 1 while all other weights can be chosen as 0. The weights allow a user the flexibility to assign the order of importance to the precision points, and accordingly, they can be chosen such that higher weights are assigned to more important points. To avoid specifying the weights, terms in the summation in P2 can individually be chosen as separate multiple objectives. In other words, Minimize: Q i P i Q i P i, i =0,...,n P3 The genetic algorithm procedure NSGA-II 39, implemented in this paper, performs the minimization of multiple objectives simultaneously. NSGA-II yields a set of pareto optimal solutions among which there exists a solution that corresponds to the minimum of an objective. Thus, for n precision points or n multiple objectives ignoring the undeformed position of the output port, a user can choose n+1 optimal solutions from the pareto optimal set. Among them, n solutions would respectively correspond to n multiple objectives minimized individually, and one solution would be related to a combined minimum when all objectives are pooled, say, in a linear combination manner as in P2. An intrinsic advantage is that a user gets to choose the desired solution from a set of pareto optimal ones which is usually not the case with the gradient based approaches. Geometrically Nonlinear Analysis Large deformation analysis is essential as it accurately models the response of a flexible continuum subject to large input loads. The strains are usually large and need more terms in the Taylor series expansion to be modeled. Usually for axial strain, x = du dx + 1 d 2 u 2 dx 2 is the definition adhered to. The material still follows the linear relationship, that is, x =E x with x as the axial stress and E is the Young s modulus. Equilibrium equations are derived using the virtual work principle. First, strain and stress measures are chosen and then the energy stored in the continuum is computed and equated to the work done by the external loads. Once a strain measure is chosen, the corresponding work conjugate stress measure is employed to compute the internal energy 36. The product of strain and stress measures is then integrated over the continuum volume to obtain the internal energy. The internal force vector f e at the element level and the tangent stiffness matrix k t are then derived and assembled to form the global internal force vector F int X 0 +U and the tangent stiffness matrix K t X 0 +U. Note that both the internal force vector and tangent stiffness matrix are implicitly dependent on the initial configuration X 0 and the nonlinear displacements U. It should be noted that the element-level equilibrium equations may be derived using either the Total Lagrangian or Eulerian formulation. The assembled equations can be solved using the combined incremental-iterative approach. In this paper, Kirchhoff s shallow arch beam theory is implemented for which the element level equations are well documented in Crisfield 36 and Bathe 37. Details and benchmarking of the geometrically nonlinear analysis using frame elements is done in Saxena and Ananthasuresh 19 wherein results are compared with the commercially available codes and analytical solutions. Genetic Algorithm With reference to P1 P3, it is the discrete-valued elastic moduli as design variables that limit the search procedures to the use of function-based optimization algorithms. This is so as the objective gradients to determine the step size of the discretevalued variables cannot be computed. A genetic algorithm is a candidate function-based search procedure that emulates three operations of nature, viz., reproduction, crossover and mutation in its quest for the best solution using the survival of the fittest strategy. In view of the nonlinear analysis implemented in this paper, an Journal of Mechanical Design JULY 2005, Vol. 127 / 747

4 Fig. 3 Topology, size, and shape synthesis of compliant pliers with fivepoint path generation specification a designvariables, b optimal solution minimizing Q 1 P 1. Q 1 P 1 ; c solution minimizing Q 2 P 2. Q 2 P 2 ; d e solutions minimizing Q 3 P 3. Q 3 P 3 and Q 4 P 4. Q 4 P 4 ; f that minimizing the sum of all individual objectives; g output deformation history for all solutions advantage when using a genetic algorithm in topology synthesis of compliant mechanisms is that it circumvents the nonconvergence problems in nonlinear analysis faced by the gradient-based counter parts. A genetic algorithm works with a population of vectors. In reproduction, vectors are selected based on their fitness values for possible inclusion in the new population. In crossover, based on some probability p c, two vectors in the mating pool are blended to get two new ones for the subsequent generation. The crossover probability is usually chosen very high to encourage the fit vectors to mate. Mutation usually follows crossover wherein, based on a specified probability p m, the vector s constituent values are changed to new ones. The probability p m is chosen very low as high mutation rate destroys fit vectors degenerating the algorithm into a random search. After the new population is generated, fitness values of the vectors are computed and accordingly, better vectors are sent to the mating pool for the generation of subsequent population. Genetic algorithms are in wide use as search procedures and are well documented in literature for example 38. In this paper, Nondominated Search in Genetic Algorithms NSGA-II developed by Deb et al. 39 is implemented. NSGA-II allows the design variables to be modeled both as binary and real coded, or in other words, discrete-valued and continuous variables. In P1 P3, the layout design variables E i can be modeled as E i = i E 0, where i can assume values of either 0 or 1. This can be accomplished by modeling i using a single bit that stores binary values much as in the work by Jakiela et al. 31. The size optimization variables, namely, the in-plane widths w i and the out-of-plane thickness t, and the shape design variables, that is, the nodal positions x i,y i can both be modeled as real coded continuous variables that can be varied within the respective bounds specified in P1. Further, NSGA-II implements a ranking procedure that allows the vectors in the population to coexist at different minima corresponding to multiple objectives. Implementation details of the procedure are provided in 39. The objective s in P1 and P3 is are evaluated for a vector in the population as follows. For i =0, the corresponding finite elements are removed from the mesh while for i =1, the elements are retained. The nodes and elements are renumbered with their respective coordinates, in-plane widths and thickness assigned from the variable vector. Quite often, it may happen that the subregions are disconnected and the global tangent stiffness matrix K t X 0 +U is singular. In that case, a very high dummy fitness value is assigned to the vector. Else, the nonlinear analysis is proceeded with. In case of convergence, the objective s in P1 748 / Vol. 127, JULY 2005 Transactions of the ASME

5 and P3 is are computed and correspondingly, fitness is calculated. Otherwise, in case of bifurcation or snap-through during analysis wherein K t X 0 +U becomes near singular in an intermediate increment or iterative step, the vector is again assigned a high dummy fitness value. Thus, a nonconvergent solution is discouraged from further participation in topology optimization. A genetic algorithm works with a set of solution vectors, and for this set adequately large, there would not be any permanent loss of a subregion in the domain. This is so as a subregion absent in a variable vector would be contained in another vector in the population. This is an advantage of a genetic algorithm over its gradient-based counterparts. The latter are point-to-point search methods that rely on gradient computation to estimate the step size for the subsequent solution. If an intermediate solution is nonconvergent, the gradient computation would not be possible in which case, the search would cease. This may require user intervention to permanently eliminate the elements close to their nonexisting states and start the search. Note, however, that alternative measures have been suggested with gradient-based approaches, for example, an approach suggested by Sigmund et al. 28 is to ignore the force-balance at nodes surrounded by nonexisting finite elements. Another benefit when using a genetic algorithm is that the analysis is more accurate since the elements in their nonexisting states are completely removed before computing the objective. In contrast, with gradient-based approaches, the lower bound for the modulus is represented by a small positive number to keep the stiffness matrix from being singular. Thus, even when the elements are close to their nonexisting states, they do, to some extent, contribute to the global stiffness and thus affect the accuracy of analysis. For a continuum undergoing large deformation, there is a tendency of the constituting members to establish mutual contact with each other during deformation. While mutual contact may be modeled in the analysis and employed to synthesize compliant mechanisms for a variety of curved output paths, in certain cases, it may be discouraged. This is so since the friction involved in element element contact may lead to wear and the primary advantage in using a compliant mechanism may be compromised. Moreover, local friction leading to wear would alter the local size and thus the original deformation functionality of the mechanism over time. To discourage mutual contact, appropriate adaptation can be made in the design algorithm. A way is to check for intersection or overlap between subregions which becomes relatively easy with frame finite elements. For a solution, if at any stage of deformation two line elements intersect, that solution may be kept away from the mating pool. That is, a high dummy fitness value may be assigned like in case of a nonconvergent solution vector. Also, for cases wherein if any node during deformation crosses Table 1 Five-point curved-output path specification for compliant pliers Input force x y 5 N 1 mm 5 mm 10 N 3 mm 12 mm 15 N 6 mm 21 mm 20 N 12.5 mm 30 mm the line of symmetry or appears in regions where it is not allowed, a high artificial fitness may be assigned to the vector. Synthesis Examples The first example on path generation is the synthesis of compliant pliers whose design specifications are shown in Fig. 3 a.a rectangular region of size 150 mm 60 mm is discretized using 117 frame elements arranged in a super ground structure. The lower horizontal edge of the region is fixed. The actuation force acts at the top left corner, and the output port is distanced at mm from the left edge and 30 mm from the bottom edge. The example is solved for five precision-point specification given in Table 1. For this example, the least square objectives for individual points are minimized separately as in P3. ABS plastic is chosen as the design material. Elements are allowed to have the moduli of only 0 or 2000 N mm 2. Shape optimization is performed in addition to topology and size optimization. Nodal positions are allowed to vary within a rectangular region of size 6.25 mm 5 mm which is 1 4th the size suggested in Fig. 2. The in-plane widths and out-of-plane thickness as size design variables are bounded within 2 mm and 5 mm. The population size is taken as 100 and the number of generations used is Though the number of function evaluations is quite large, the procedure takes min 3.7 h on an Intel P GHz processor to generate the results. The solution minimizing the least square objective for the second precision point to is shown in Fig. 3 b. Figures 3 c 3 e depict the solutions that minimize the objectives for the third, fourth, and fifth precision points respectively. The optimal values of these objectives are ,1 10 3, and Figure 3 f depicts a solution for which the sum of all individual objectives is minimized to The out-of-plane thicknesses for individual solutions are mentioned below the respective figures, and the output deformation history for all solutions is depicted in Fig. 3 g. Note the unequal scales for the two axes that are used to distinguish the output paths for the five solutions. The inset at the top right corner Fig. 4 a ABS prototype of the mechanism in Fig. 3 f ; b output port in undeformed position first precision point ; c d output port near third and fifth precision points respectively Journal of Mechanical Design JULY 2005, Vol. 127 / 749

6 Fig. 5 a Schematic of Chebyshev s straight line mechanism b design specifications for topology, size, and shape optimization. represents the paths in same-scale with the thick solid line depicting the path for the solution in Fig. 3 f. An ABS prototype of the mechanism in Fig. 3 f is shown in Fig. 4 a. The prototype is fabricated as is from the optimal solution. Figures 4 b 4 d illustrate the performance of the mechanism suggesting the output deformation behavior similar to that depicted by path f in Fig. 3 g. Next, synthesis of the fully compliant version of Chebyshev s straight line mechanism is presented the rigid-body schematic of which is shown in Fig. 5 a. Figure 5 b shows the design region of size 120 mm 120 mm discretized using 228 elements with loading and boundary conditions. The input port is located on the bottom edge at 20 mm from the bottom left corner while the output port is located at the top horizontal edge at the mid point. The maximum input load is 40 N and the output port is desired to traverse a straight line between points A and B in the figure. The corresponding five precision-point specification is given in Table 2. All, topology, size, and shape syntheses are performed for this example. Polypropylene is chosen as the design material with the flexural modulus of N mm 2. In-plane widths are bounded within 2 mm, 5 mm while the out-of-plane thickness is limited by 2 mm10mm. Nodal positions are varied within local regions of size 5 mm 5 mm. The population size is chosen as 40 while the number of generations is taken as Optimal solutions in Fig. 6 are obtained in 255 min 4.25 h. Solutions that minimize the four objectives pertaining to the respective four precision points are shown in Figs. 6 a 6 d. The objectives are minimized to , , and 0.8. Individual out-of-plane thicknesses are mentioned below the figures. Figure 6 e depicts the solution that minimizes the sum of the four least square objectives. Figure 6 f shows the deformed output profiles of all solutions in Figs. 6 a 6 e. Unequal scales for the two axes are used to distinguish the output paths for the solutions. The inset at the top right corner represents the paths in same-scale with the thick solid line depicting the path for the solution in Fig. 6 e. Figure 7 shows the ABS prototype of the solution in Fig. 6 e tested for performance for rectilinear output path. Table 2 Five-point specification for compliant Chebyshev s mechanism Input force x y 10 N 10 mm 0 mm 20 N 20 mm 0 mm 30 N 30 mm 0 mm 40 N 40 mm 0 mm Fig. 6 Topology, size, and shape synthesis of fully compliant Chebyshev s straight line mechanism with five-point path generation specification; a optimal solution minimizing Q 1 P 1. Q 1 P 1 ; b solution minimizing Q 2 P 2. Q 2 P 2 ; c d solutions minimizing Q 3 P 3. Q 3 P 3 and Q 4 P 4. Q 4 P 4 ; e that minimizing the sum of all individual objectives; f output deformation history for all solutions The third example is of the synthesis of a compliant mechanism for an approximate L-shaped path given in Table 3. For the example, nine precision points are chosen for the output port to traverse. This is because to traverse the output path overall and accurately, the least square objective with a few precision points is not appropriate as suggested by the mechanism in Fig. 3 f. Quite many sampling points may be required to match the target and design output curves. Saggere and Kota 40 have used the deviation between the deformed and target curves as the average Euclidian distance between the two curves at the sampling points. Taietal. 33 have considered this least squared objective wherein the number of sampling points on both curves is equal to the 750 / Vol. 127, JULY 2005 Transactions of the ASME

7 Fig. 8 Synthesis of a compliant mechanism for an approximate L-shaped path; a design region; b optimal solution; c output response Fig. 7 ABS prototype of the mechanism in Fig. 6 e ; a output port in undeformed position; b e output port passing through second, third, fourth, and fifth precision points, respectively number of analysis steps in the nonlinear analysis. For morphing structural shapes, Lu and Kota 34 have addressed curve matching using modified Fourier transformation. The design region is shown in Fig. 8 a which is of size 150 mm 100 mm. The input force acts at the top left corner along the vertically downward direction. The output port is located at mm horizontally and 37.5 mm vertically from the bottom left corner. ABS plastic is chosen as the design material. All, topology, shape, and size optimization are performed for this example. Elastic moduli are varied as 0 or 2000 N mm 2. In-plane widths are varied between 2 mm and 7 mm, and the out-of-plane thickness is bounded between 3 mm and 10 mm. Nodal positions are varied within a rectangular region of size 5 mm 5 mm. The optimal solution is shown in Fig. 8 b that results by minimizing the combined least square objective in P1. The objective is minimized to 0.87 and the thickness is obtained as 3.3 mm. The displacement history of the output port is shown in Fig. 8 c. As can be observed, though the trend is partially captured, the output path is not desirably close to the specified path. This can be attributed to the use of the least squared objective. Alternatively, the min max criterion or Fourier coefficients may be used to compare the target and design output paths to synthesize optimal compliant mechanisms for path generation. Table 3 Approximate L-shaped output path for a compliant mechanism Input force N x mm y mm Input force N x mm y mm Closure A procedure to synthesize compliant mechanisms for prescribed nonlinear output paths is presented in this paper. Topology, size, and shape optimization are considered simultaneously in a generic setting. Topology optimization is modeled using the elastic modulus of frame finite elements. These moduli are either assigned the values of 0 or E 0, the modulus of the specified material. Due to this binary nature of topology optimization, genetic algorithm is implemented as the optimization procedure. An advantage is the implicit circumvention of nonconvergence in the large displacement analysis, which is another reason why genetic algorithm is chosen for optimization. Size optimization is performed using the in-plane widths and uniform out-of-plane thickness as design variables. Optimal shape is achieved by varying the nodal positions. Geometrically nonlinear analysis using Kirchhoff s shallow arch beam theory is performed to compute the least square multiple objectives. A user may desire that the output port of a compliant mechanism traverses one or some precision points as opposed to all. To facilitate this, multiple least square objectives are formulated, one for each precision point. Nondominated Search in Genetic Algorithm or NSGA-II is implemented to solve these multiple objectives. A note to mention is that the number of function evaluations taken to obtain optimal solutions is quite large. The compute time can be reduced by lowering the values of the user-defined population size and number of generations at the cost of accuracy between the design and prescribed output paths. Nevertheless, large compute time required is a concern and future work will entail investigation into obtaining optimal solutions efficiently using genetic algorithms. Another issue is the use of least squared objectives. The resultant output paths do not compare well with the specified paths as suggested by a few examples though the deformation trend is partially captured. Future attempts will also explore the use of better objectives like the min-max criterion or Fourier coefficients. Notwithstanding the large number of function evaluations required in obtaining optimal solutions and the nature of the objective, this paper alludes to many advantages of genetic algorithm to solve layout design problems for large displacement compliant mechanisms. Those noteworthy are a ability to model a topology design problem in its original binary form b circumvention of problems related to nonconvergence or buckling in large deforma- Journal of Mechanical Design JULY 2005, Vol. 127 / 751

8 tion analysis for which case the gradient-based methods usually fail, c ability to address mutual interference between elements during deformation, and d accurate analysis using the mesh update scheme. Further, an optimal solution can be manufactured as is without requiring any interpretation, as shown for two examples in this paper. Acknowledgments The author would like to acknowledge the Department of Science and Technology, India for financial support DST/ME/ Useful discussions with Dr. Nilesh D. Mankame and Professor G. K. Ananthasuresh of the University of Pennsylvania are gratefully acknowledged. Mr. TVK Gupta of the 4I Lab at IIT, Kanpur has helped in fabricating the prototypes. Finally, the author expresses his sincere thanks to Professor K. Deb for the NSGA-II implementation and the anonymous reviewers for suggesting significant improvements in this paper. References 1 Howell, L. L., and Midha, A., 1996, A loop-closure theory for the analysis and synthesis of compliant mechanisms, ASME J. Mech. Des., 118, pp Ananthasuresh, G. K., and Kota, S., 1995, Designing Compliant Mechanisms, Mech. Eng. Am. Soc. Mech. Eng., 117, pp Howell, L. L., 2001, Compliant Mechanisms, Wiley, New York. 4 Ananthasuresh, G. K., Kota, S., Kikuchi, N., 1994, Strategies for Systematic Synthesis of Compliant MEMS, in Proceedings of the 1994 ASME Winter Annual Meeting, Chicago, pp Frecker, M., Ananthasuresh, G. K., Nishiwaki, N., Kikuchi, N., Kota, S., 1997, Topological Synthesis of Compliant Mechanisms using Multi-Criteria Optimization, ASME J. Mech. Des., 119, pp Sigmund, O., 1997, On the Design of Compliant Mechanisms using Topology Optimization, Mech. Struct. Mach., 25, pp Saxena, A., and Ananthasuresh, G. K., 2000, On an optimality property of compliant topologies, Struct. Multidiscip. Optim., 19, pp Nishiwaki, S., Frecker, M. 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