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Form-finding of nonregular tensegrities using a genetic algorithm
Mechanics Research Communications 37 (2010) 85–91
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Form-finding of nonregular tensegrities using a genetic algorithm Xian Xu, Yaozhi Luo * Space Structures Research Center, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 27 May 2009 Received in revised form 15 September 2009 Available online 20 September 2009 Keywords: Form-finding Nonregular tensegrity Genetic algorithm
a b s t r a c t In this paper, the form-finding problem of nonregular tensegrities was converted into a constrained optimization problem. A genetic algorithm was used to solve this problem. Two cases of form-finding were considered. In the first case, the number of members, the rest lengths of members, the elastic moduli of members and the connectivity of members were given, and the only variables are the initial locations of nodes. In the second case, the elastic moduli of members were also treated as variables besides the initial locations of nodes. Typical examples were carried out to verify the proposed method. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction A key step in the design of tensegrity structures is the determination of their geometrical configurations and their initial prestress, known as form-finding. The topic of form-finding has received much attention since tensegrity systems were invented in the beginning of the 20th century (Motro, 1997). The pioneers of tensegrity systems, namely R.B. Fuller, K. Snelson, and D.G. Emmerich, were architects and artists. They used empirical and heuristic methods which usually included a process of trial and error to determine the form of tensegrity systems (Motro, 2003). Recently, researchers from Sheffield University have proposed a novel method for designing tensegrity systems based on the analogy between an air balloon and a tensegrity (Sakantamis and Larsen, 2004). These interactive methods are fit for designers without a strong mathematic background, and capable to handle relatively simple systems. As the applications of tensegrity systems are extending into much more scientific and engineering areas including civil structures (Geiger, 1988; Geiger et al., 1986), space structures (Furuya, 1992; Hanaor, 1993), cell mechanics (Ingber, 1993, 1997, 2003) and robotics (Paul et al., 2006), many mathematicians and engineers have paid attention to the form-finding of tensegrity systems since the 1980s. As a result, analytical methods and numerical algorithms for form-finding tensegrity systems based on mathematical theories, and mechanical principles have been developed (Juan and Tur, 2008; Tibert and Pellegrino, 2003). However, most existing form-finding methods are still limited to regular tensegrities based on polyhedrons and prisms. Though there are a few methods that claim themselves fit for form-finding of non* Corresponding author. Tel.: +86 571 8795 2349; fax: +86 571 8820 8472. E-mail address: [email protected] (Y. Luo). 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.09.003
regular tensegrities, the core idea of them is creating nonregular tensegrities by modifying the corresponding regular tensegrities (Micheletti and Williams, 2007; Vassart and Motro, 1999; Zhang et al., 2006). This idea is only suitable for a small part of nonregular tensegrities. When the form-finding problem of tensegrities is modeled as a constrained optimization problem, some solving methods that are used for constrained optimization problems become suitable for form-finding problems. An example of this is the non-linear programming method for form-finding of tensegrities proposed by Pellegrino (1986). However, the non-linear programming is an algorithm based on inexplicit enumeration, the efficiency of which is relatively low, especially for optimization problems with large solution spaces. For optimization problems with large solution spaces, stochastic search techniques such as genetic algorithms (GA) and the simulated annealing algorithm (SA), are usually applied. Recently, Paul and Valero-Cuevas (2005) proposed an evolutionary algorithm to generate irregular tensegrity structures. And based on this, Rieffel et al. (2009) developed a more efficient and general algorithm by using graphs to represent tensegrity structures. The works of Paul and Valero-Cuevas (2005) and Rieffel et al. (2009) have opened up a brand new path to solving the form-finding problem of nonregular tensegrity systems. The work presented in this paper is following their track, but using different optimization models and a much simpler binary-coded genetic algorithm, together with an interesting finding on multi-stable configurations of tensegrity systems. The paper is laid out as follows. Section 2 proposed a general constrained optimization model for form-finding of nonregular tensegrities. Two cases of form-finding were considered. In the first case, the number of members, the rest lengths of members, the elastic moduli of members and the connectivity of members were
86
X. Xu, Y. Luo / Mechanics Research Communications 37 (2010) 85–91
given, and the only variables are the initial locations of nodes. In the second case, the elastic moduli of members were also treated as variables besides the initial locations of nodes. In Section 3, an improved GA was developed to solve the optimization problem of form-finding. Typical examples were carried out to verify the proposed method in Section 4. And then Section 5 validated the feasibility of the found tensegrity structures by physical models. Finally, Section 6 discussed the advantages and limitations of the proposed method and concluded the paper.
If a more general case where the elasticities of members are also variables in a given range is considered, the corresponding optimization model becomes
2. Optimization model
where KL and KU = the lower bound and the upper bound of the elasticity of members.
Here we assume that (a) all the members are linear-elastic; (b) the number and the connectivity of members are given in advance; and (c) the members are massless and no external loads act on the structure. Hence, the potential energy caused by member strains is
1 E ¼ DLT KDL 2
ð1Þ
where E = potential energy, DL = vector of increments of member lengths, and K = matrix of linear elasticity of members. The increments of member lengths are determined by the nodal coordinates, i.e.
DL ¼ DLðXÞ;
X 2 ðXL ; XU Þ
ð2Þ
where X = vector of nodal coordinates, and XL and XU = the lower bound and the upper bound of nodal coordinates, respectively. When a structural system is in a state of stability, its potential energy should be in a minimum. The stationary condition of Eq. (1) yields the equilibrium equation:
AðXÞtðXÞ ¼ P
ð3Þ
where A = equilibrium matrix, t = vector of internal forces of members, and P = vector of equivalent nodal loads. Since there is no external load acting on the system, P is totally caused by the unbalanced internal forces of members. For a random generated X, it usually has P – 0, i.e., the stable equilibrium of the system cannot be ensured. As a result, the structure will deform and adjust to a new state of stable equilibrium:
AðX0 ÞtðX0 Þ ¼ 0
ð4Þ
0
where X = vector of nodal coordinates corresponding to the state of stable equilibrium. However, it was found that most configurations satisfying the stable equilibrium condition are usually one-dimensional or twodimensional ones which have little impact on engineering applications (Paul and Valero-Cuevas, 2005). Here we are interested in the tensegrities with three-dimensional configurations. Searching of three-dimensional tensegrities can be modeled by the following constrained optimization problem:
Max ðDÞ s:t: AðX0 ÞtðX0 Þ ¼ 0;
ð5Þ
X 2 ðXL ; XU Þ where D = space dimensions of tensegrities. Besides, in practical applications it is not only required that the tensegrity be in threedimensional space but also expected to meet some specific requirements. The optimization model corresponding to this situation is
Max ðf ðX0 ÞÞ s:t:
AðX0 ÞtðX0 Þ ¼ 0; X 2 ðXL ; XU Þ;
ð6Þ
D¼3 where f(X0 ) = is an objective function describing the practical requirements.
Max ðf ðX0 ; KÞÞ s:t: AðX0 ; KÞtðX0 ; KÞ ¼ 0; X 2 ðXL ; XU Þ;
ð7Þ
K 2 ðKL ; KU Þ; D¼3
3. Genetic algorithm In this paper, an genetic algorithm was used to solve the above optimization problem of form-finding. For the sake of simplicity, the basic and well known steps of a simple GA such as coding, selection, crossover and mutation are not discussed here in detail. Emphasis is instead placed on the new features of the GA used. 3.1. Determination of state of stable equilibrium and fitness estimation Before the fitness of an individual can be estimated, the state of stable equilibrium (i.e. X0 ) should be determined. Under any given nodal coordinates X, the corresponding internal forces of members usually cannot be equilibrated by themselves. The assembly deforms under the unbalanced internal forces. As one of the effective methods for form-finding of tensile structures, the dynamic relaxation method was adopted to determine the equilibrium state of a structural system with a randomly given set of nodal coordinates. The dynamic relaxation method consists of two iterative steps: firstly the unbalanced nodal forces are calculated and then the structural configuration is adjusted to reduce the unbalanced forces; and secondly the above procedure is repeated until the unbalanced forces are small enough according to a given numerical criterion. For detailed introduction of the dynamic relaxation method, please refer to the literature Barnes (1999). When the state of stable equilibrium is determined, the number of dimensions and other characteristics of the tensegrity can be easily determined. Then, the fitness of the corresponding individual can be estimated based on the characteristics of the tensegrity. 3.2. Niche A niche process combining a crowding check (De Jong, 1975) and a filter process is implemented in this study. For the initial generation, when fitness assignment is finished, the individuals are arranged in descending order with respect to the fitness and the first N (N 6 M) individuals are saved in a filter. Then, when a new generation is obtained and their fitness is assigned, a niche procedure as follows is conducted. Firstly, the entire new population (M individuals) is combined with the N individuals in the filter to constitute an extended population of M + N individuals. The Hamming distances between every two individuals of the extended population are computed. Secondly, they are compared with a given criterion ed. If the Hamming distance of two individuals is smaller than the criterion, the individual with a lower fitness will be identified by a penalty. In practical programming of this study it is achieved by multiplying the fitness by a penalty multiplier k, i.e.
F min ðx0i ; x0j Þ ¼ kF min ðx0i ; x0j Þ
ð8Þ
Thirdly, the individuals of the extended population are rearranged in descending order with respect to their new fitness and
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the first N individuals replace the old ones in the filter. Finally, the stopping criterion is checked to determine whether the algorithm has converged. If not, the first M individuals of the extended population are selected to constitute the parent population for next generation. Via the above process, the fitness gap between the superior individual and the inferior individual(s) within a distance ed is magnified. As a result, the inferior individuals have a higher probability of being eliminated in the next evolution process. In other words, within a space distance of ed there is only one good individual. Thus, the diversity of the population is ensured and meanwhile the even spread of individuals in the constraint space is achieved. Meanwhile, the best N individuals are recorded by the filter and flow into the parent population at every generation, which avoids the loss of good genotypes and expedites the evolution process.
4. Examples 4.1. Example 1 In this example, the target is to find as many three-dimensional tensegrities as possible under the given number of members, rest lengths of members, elastic moduli of members and connectivity of members. The only variables are the initial locations of nodes. Without loss of generality, two typical assemblies are considered here. One assembly consisting of six identical compression bars and 24 identical tension cables which connect to each other in accordance with a well known regular tensegrity called expanded octahedron tensegrity was considered, as shown in Fig. 1a. The other assembly is based on the zigzag icosahedral tensegrity which consists of 30 identical compression bars and 90 identical tension cables (Fig. 2a). The compression bars are assumed to have a rest length of unit and an elasticity of 1000. The tension cables of the first assembly and the tension cables of the second assembly are assumed to have rest lengths of 0.536 and 0.312, respectively. All the cables are assumed unit elasticity. The allowable range of nodal coordinates is [5, +5]. Each component of the vector of nodal coordinates is represented by a 15-number binary code string. As a result, the total length of a genome string = 15 3 number of nodes. The GA proposed in last section is used to carry out the optimization. Each run of the GA is conducted for 20 generations, using a population size of 20. Proportional selection is employed to probabilistically select genotypes for mutation and crossover. Pairwise one-point crossover operation and uniform mutation operation are adopted. The crossover rate and the mutation rate are empirically set to 0.6 and 0.002. For the 6-bar assembly, three stable self-equilibrium configurations are identified in the final generation, as shown by Fig. 1. The
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corresponding strain energies for the three configurations are 0.071, 0.173 and 0.241, respectively. As expected, the configuration with lowest energy is an expanded octahedron tensegrity (Fig. 1a). The configuration with the highest energy (Fig. 1c) is identical to the higher energy state reported in literature Defossez (2003). While the configuration with secondary energy (Fig. 1b) has not been reported elsewhere, to authors’ knowledge. For the 30-bar assembly, four configurations in stable self-equilibrium are found by the GA, as shown by Fig. 2. The corresponding strain energies of them are 0.090, 2.16, 3.15 and 3.22, respectively. Similar to the case of 6-bar assembly, the configuration with lowest energy is a regular tensegrity (i.e. zigzag icosahedral tensegrity) with which we are familiar. The configuration with secondary energy (Fig. 2b) has been reported (Defossez, 2003). However, the configurations with the highest two energies (Fig. 2c, d) are firstly reported, to authors’ knowledge. Since the three 6-bar configurations are corresponding to the same unprestressed assembly, as well as the four 30-bar configurations. They must be able to mutually transform between each other under certain conditions. Transformations between the configurations with different energy levels will release energy to environment or absorb energy from environment, which indicates that the tensegrity has potential applications in actuators and energy absorbers. 4.2. Example 2 In this example, we consider a more general case where elasticities of cables are also variable. In additional to the assumptions used in example 1, the variable range of the linear elasticities of cables is assumed to be [1, 10] and the rest lengths of cables are assumed to be zero. The target of this example is to find tensegrities whose spans in three-dimensions are as close as possible to each other. Hence, the root mean square of the spans in three-dimensions is used as the objective function:
f ðX0 Þ ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u3 uP 2 u ðdi dÞ ti¼1 3
ð9Þ
is the averwhere di is the span of a structure in ith dimension and d age of spans in three-dimensions. In addition, the constraint on the dimensions of the structure D = 3 was considered by converting the objective function into
gðX0 Þ ¼ D þ
1 1 þ f ðX0 Þ
ð10Þ
By doing this, it can be ensured that the structure with a higher dimension corresponds to a larger value of the objective function.
Fig. 1. Three self-equilibrium configurations of the assembly based on expanded octahedron.
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Fig. 2. Four self-equilibrium configurations of the assembly based on zigzag icosahedral tensegrity.
Meanwhile, for two structures with the same dimension, the structure that has closer spans in three-dimensions gets a larger value of the objective function than the other structure. The fitness of an individual is evaluated by directly setting it equal to the value of objective function. The GA used in this example is the same to that used in example 1 except the terminating criterion. In this example, it stops when the improvement of the best individual is lower than a critical value for 10 continuous generations or it runs for 100 generations. Formfindings on assemblies based on the expanded octahedron tensegrity (Fig. 1a), the hexagonal prism tensegrity (Fig. 4a), and a three-stage cylindrical tensegrity (Fig. 5a) were carried out, respectively. Except for the assembly based on the hexagonal prism tensegrity whose evolution stopped after 94 generations, the evolutions on the other two assemblies stopped after 100 generations. For each assembly, the tensegrities represented by the individuals of the final population are different from each other. They have very free geometrical configurations which are significantly different from the regular tensegrities well known to us. Taking the best tensegrities corresponding to the three assemblies as examples, they all have nonregular shapes and differ from their regular counterparts (Figs. 3–5). The values of the objective function for the three best tensegrities are 3.99995, 3.99994 and 3.99995, respectively. For the regular expanded octahedron tensegrity, it is obvious that its spans in three-dimensions are equal to each other when the coordinate axes are located parallel to the three orthogonal pairs of struts. Hence the value of the
objective function for the regular expanded octahedron tensegrity is 4.0. For the other two cases, it also can be deduced that the circumdiameter of the bottom polygon of a regular prim tensegrity or a regular cylindrical tensegrity can be equal to the height of the tensegrity under a certain set of force density in elements, according to the formulations given in literature Connelly and Terrell (1995). Therefore, the objective function values of the regular hexagonal prism tensegrity and the cylindrical tensegrity are also equal to 4.0. So the values of the objective function for the three nonregular tensegrities are comparable to their regular counterparts, but have free and variable shapes. In a pure viewpoint of optimization, it seems that the optimized nonregular tensegrities are not superior to the regular ones in this example. But it should be noted that the optimality of the found tensegrities heavily depends on the objective function. Here we just choose the objective function given by Eq. (10) to direct a target for the evolution of nonregular tensegrity, and the regular tensegrity becomes one of ideal solutions for the optimization problem by coincidence. If other indexes such as the cover area is taken as a maximum target, nonregular tensegrities which are now superior to the regular ones may be found.
5. Physical models Physical models of the nonregular tensegrities were made using wooden sticks and cotton threads, as shown in Figs. 6 and 7. Due to
X. Xu, Y. Luo / Mechanics Research Communications 37 (2010) 85–91
manual errors and viewpoint of photographs, it is unavoidable that there is a little difference between the theoretical configurations
Fig. 3. The best nonregular tensegrity corresponding to the assembly based on the expanded octahedron.
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and the physical models. But it still can be seen that the physical models retain the basic shapes of the corresponding theoretical configurations, which further verify the proposed method. It should be pointed that conceptual structural members without practical sizes were used in foregoing examples, i.e. crossing and touching of the structural members are not considered. As a result, when practical members with physical sizes are used contact between members may occur (Fig. 6c). A possible way to solve this problem is adjusting the prestress in members or the length of members to modify the configuration of the tensegrity to a state near the initial configuration but without member crossing and touching. Another point should be noted is that the axial forces of members of a nonregular tensegrity obtained are normalized ones and represent a prestress mode that is able to stabilize the structure. It is unnecessary to consider bar buckling and cable slackening at this stage of form-finding. Because we can choose appropriate bars and cables with suitable material and size to avoid the buckling and slackening of them at the stage of model making or structural design for a real tensgrity structure.
Fig. 4. Regular and nonregular tensegrities based on hexagonal prism.
Fig. 5. Regular and nonregular tensegrities based on cylindrical mast.
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Fig. 6. Physical models for the three stable configurations in Fig. 1.
Fig. 7. Physical models for the best tensegrities in example 2.
6. Conclusions and discussions In this paper, the form-finding of nonregular tensegrities was modeled as a constrained optimization problem. An GA was used to solve this problem. Examples were carried out to verify the proposed method. It was shown that tensegrities met the stable equilibrium and meanwhile optimized some practical requirements can be found by the proposed method. Because the GA is a parallel algorithm, a run of it can find more than one tensegrities that meet the stable equilibrium condition. In contrast, traditional form-finding methods are unit commitment algorithms, and at most one tensegrity can be found after each run of them. Moreover, the proposed method is more general than the traditional ones. It imposes fewer constraints on the initial assembly, and certain parameters of tensegrities can be optimized by properly choosing the objective function. As a result, tensegrities with more practical consider-
ations can be found, which advances the engineering applications of tensegrity structures. The GA used in this paper is different from the GA of Paul and Valero-Cuevas (2005) in two aspects: (a) using a much simpler binary encoding instead of a real number encoding, and (b) introducing a niche process based on strain energy to insure the diversity. Because geometrical stability is required for a tensegrity structure. An assembly with a given initial nodal position will jump into the same stable configuration when the nodal coordinates vary in a neighborhood of the given position, after a dynamic relaxation (or kinematic relaxation) process. In other words, tensegrity structures discretely exist in the space, though the search space for the initial nodal position seems continuous. Hence, we believe that a discrete representation is enough to cover the feasible solutions rather than a continuous representation. When the number of nodes increases, the proposed method will become much time
X. Xu, Y. Luo / Mechanics Research Communications 37 (2010) 85–91
cost. It is caused by the inherent shortage of the dynamic relaxation algorithm (kinematic relaxation algorithm) which is not effective with a large number of nodes (Motro et al., 1994). But using a much powerful computer such as a work station in stead of a PC may be able overcome this problem. This paper, as well as previously reported works on form-finding of nonregular tensegrity structures, has little control on the specific geometrical characters of the tensegrity structures obtained. They are only capable to optimize some gross and global geometrical targets such as volume and span, but have a difficult time handling more specific requirements, such as locating some nodes in given positions. Specific requirements like that mentioned above put such a heavy constraint on the problem that there may be no feasible solutions. Extending the search space by including the elastic constants of bars and connectivity of members in the variables will increase the possibility of finding feasible solution. But the efficiency of the algorithm will be affected if a larger search space is adopted. Form-finding of nonregular tensegrity structures that meet more specific requirement will be an interest subject for future work. Acknowledgements This work was supported by grants from NCET (Grant No. 060517) and National Natural Science Foundation of China (Grant No. 50638050). The authors also thanked Mr. Feng Yu, Jingmeng Xu, Zhouneng Zhong and Pengcheng Yang for their great help in making the physical models. References Barnes, M.R., 1999. Form finding and analysis of tension structures by dynamic relaxation. International Journal of Space Structures 14, 89–104. Connelly, R., Terrell, M., 1995. Globally rigid symmetric tensegrities. Structural Topology 21, 59–78. De Jong, K.A., 1975. An Analysis of the Behavior of a Class of Genetic Adaptive Systems. PhD Thesis, University of Michigan, MI. Defossez, M., 2003. Shape memory effect in tensegrity structures. Mechanics Research Communications 30, 311–316.
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Furuya, H., 1992. Concept of deployable tensegrity structures in space application. International Journal of Space Structures 7, 143–151. Geiger, D.H., Stefaniuk, A., Chen, D., 1986. The design and construction of two cable domes for the Korean Olympics, shells, membrances and space frames. In: Proceedings of IASS Symposium, Osaka, Japan, pp. 265–272. Geiger, D.H., 1988. Design details of an elliptical cable dome and a large span cable dome (210 m) under construction in the United States. In: Proceedings of the IASS ASCE. International Symposium on Innovative Applications of Shells and Spatial Forms, Bangalore. Hanaor, A., 1993. Double-layer tensegrity grids as deployable structures. International Journal of Space Structures 8, 135–145. Ingber, D.E., 1993. Cellular tensegrity: defining new rules of biological design that govern thecytoskeleton. Journal of Cell Science 104, 613–627. Ingber, D.E., 1997. Tensegrity: the architectural basis of cellular mechanotransduction. Annual Review of Physiology 59, 575–599. Ingber, D.E., 2003. Tensegrity I. Cell structure and hierarchical systems biology. Journal of Cell Science 116, 1157–1173. Juan, S.H., Tur, J.M.M., 2008. Tensegrity frameworks: static analysis review. Mechanism and Machine Theory 43, 859–881. Micheletti, A., Williams, W.O., 2007. A marching procedure for form-finding of tensegrity structures. Journal of Mechanics of Materials and Structures 2, 857– 882. Motro, R., Belkacem, S., Vassart, N., 1994. Form finding numerical methods for tensegrity systems. In: Proceedings of IASS-ASCE International Symposium on Spatial, Lattice and Tension Structures, Atlanta, USA. Motro, R., 1997. Tensegrity systems and structural research. In: Proceedings of the International Association of Shell and Spatial Structures Colloquium, Madrid, pp. 161–173. Motro, R., 2003. Tensegrity: Structural Systems for the Future. Kogan Page Science, London. Paul, C., Valero-Cuevas, F.J., 2005. Evolutionary Form-forming of Tensegrity Structures, Genetic and Evolutionary Computation Conference, Washington, DC, USA, pp. 3–10. Paul, C., Lipson, H., Valero-Cuevas, F., 2006. Design and control of tensegrity robots for locomotion. IEEE Transaction on Robotics 22, 944–957. Pellegrino, S., 1986. Mechanics of Kinematically Indeterminate Structures. PhD Thesis, University of Cambridge, Cambridge. Rieffel, J., Valero-Cuevas, F., Lipson, H., 2009. Automated discovery and optimization of large irregular tensegrity structures. Computers and Structures 87, 368–379. Sakantamis, K., Larsen, O.P., 2004. The cocoon method: a physical modelling tool for designing tensegrity systems. International Journal of Space Structures 19, 11– 20. Tibert, A.G., Pellegrino, S., 2003. Review of form-finding methods for tensegrity structures. International Journal of Space Structures 18, 209–223. Vassart, N., Motro, R., 1999. Multiparametered formfinding method: application to tensegrity systems. International journal of Space Structures 14, 147–154. Zhang, L., Maurin, B., Motro, R., 2006. Form-finding of nonregular tensegrity systems. ASCE Journal of Structural Engineering 132, 1435–1440.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Form-finding of nonregular tensegrities using a genetic algorithm Xian Xu, Yaozhi Luo * Space Structures Research Center, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 27 May 2009 Received in revised form 15 September 2009 Available online 20 September 2009 Keywords: Form-finding Nonregular tensegrity Genetic algorithm
a b s t r a c t In this paper, the form-finding problem of nonregular tensegrities was converted into a constrained optimization problem. A genetic algorithm was used to solve this problem. Two cases of form-finding were considered. In the first case, the number of members, the rest lengths of members, the elastic moduli of members and the connectivity of members were given, and the only variables are the initial locations of nodes. In the second case, the elastic moduli of members were also treated as variables besides the initial locations of nodes. Typical examples were carried out to verify the proposed method. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction A key step in the design of tensegrity structures is the determination of their geometrical configurations and their initial prestress, known as form-finding. The topic of form-finding has received much attention since tensegrity systems were invented in the beginning of the 20th century (Motro, 1997). The pioneers of tensegrity systems, namely R.B. Fuller, K. Snelson, and D.G. Emmerich, were architects and artists. They used empirical and heuristic methods which usually included a process of trial and error to determine the form of tensegrity systems (Motro, 2003). Recently, researchers from Sheffield University have proposed a novel method for designing tensegrity systems based on the analogy between an air balloon and a tensegrity (Sakantamis and Larsen, 2004). These interactive methods are fit for designers without a strong mathematic background, and capable to handle relatively simple systems. As the applications of tensegrity systems are extending into much more scientific and engineering areas including civil structures (Geiger, 1988; Geiger et al., 1986), space structures (Furuya, 1992; Hanaor, 1993), cell mechanics (Ingber, 1993, 1997, 2003) and robotics (Paul et al., 2006), many mathematicians and engineers have paid attention to the form-finding of tensegrity systems since the 1980s. As a result, analytical methods and numerical algorithms for form-finding tensegrity systems based on mathematical theories, and mechanical principles have been developed (Juan and Tur, 2008; Tibert and Pellegrino, 2003). However, most existing form-finding methods are still limited to regular tensegrities based on polyhedrons and prisms. Though there are a few methods that claim themselves fit for form-finding of non* Corresponding author. Tel.: +86 571 8795 2349; fax: +86 571 8820 8472. E-mail address: [email protected] (Y. Luo). 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.09.003
regular tensegrities, the core idea of them is creating nonregular tensegrities by modifying the corresponding regular tensegrities (Micheletti and Williams, 2007; Vassart and Motro, 1999; Zhang et al., 2006). This idea is only suitable for a small part of nonregular tensegrities. When the form-finding problem of tensegrities is modeled as a constrained optimization problem, some solving methods that are used for constrained optimization problems become suitable for form-finding problems. An example of this is the non-linear programming method for form-finding of tensegrities proposed by Pellegrino (1986). However, the non-linear programming is an algorithm based on inexplicit enumeration, the efficiency of which is relatively low, especially for optimization problems with large solution spaces. For optimization problems with large solution spaces, stochastic search techniques such as genetic algorithms (GA) and the simulated annealing algorithm (SA), are usually applied. Recently, Paul and Valero-Cuevas (2005) proposed an evolutionary algorithm to generate irregular tensegrity structures. And based on this, Rieffel et al. (2009) developed a more efficient and general algorithm by using graphs to represent tensegrity structures. The works of Paul and Valero-Cuevas (2005) and Rieffel et al. (2009) have opened up a brand new path to solving the form-finding problem of nonregular tensegrity systems. The work presented in this paper is following their track, but using different optimization models and a much simpler binary-coded genetic algorithm, together with an interesting finding on multi-stable configurations of tensegrity systems. The paper is laid out as follows. Section 2 proposed a general constrained optimization model for form-finding of nonregular tensegrities. Two cases of form-finding were considered. In the first case, the number of members, the rest lengths of members, the elastic moduli of members and the connectivity of members were
86
X. Xu, Y. Luo / Mechanics Research Communications 37 (2010) 85–91
given, and the only variables are the initial locations of nodes. In the second case, the elastic moduli of members were also treated as variables besides the initial locations of nodes. In Section 3, an improved GA was developed to solve the optimization problem of form-finding. Typical examples were carried out to verify the proposed method in Section 4. And then Section 5 validated the feasibility of the found tensegrity structures by physical models. Finally, Section 6 discussed the advantages and limitations of the proposed method and concluded the paper.
If a more general case where the elasticities of members are also variables in a given range is considered, the corresponding optimization model becomes
2. Optimization model
where KL and KU = the lower bound and the upper bound of the elasticity of members.
Here we assume that (a) all the members are linear-elastic; (b) the number and the connectivity of members are given in advance; and (c) the members are massless and no external loads act on the structure. Hence, the potential energy caused by member strains is
1 E ¼ DLT KDL 2
ð1Þ
where E = potential energy, DL = vector of increments of member lengths, and K = matrix of linear elasticity of members. The increments of member lengths are determined by the nodal coordinates, i.e.
DL ¼ DLðXÞ;
X 2 ðXL ; XU Þ
ð2Þ
where X = vector of nodal coordinates, and XL and XU = the lower bound and the upper bound of nodal coordinates, respectively. When a structural system is in a state of stability, its potential energy should be in a minimum. The stationary condition of Eq. (1) yields the equilibrium equation:
AðXÞtðXÞ ¼ P
ð3Þ
where A = equilibrium matrix, t = vector of internal forces of members, and P = vector of equivalent nodal loads. Since there is no external load acting on the system, P is totally caused by the unbalanced internal forces of members. For a random generated X, it usually has P – 0, i.e., the stable equilibrium of the system cannot be ensured. As a result, the structure will deform and adjust to a new state of stable equilibrium:
AðX0 ÞtðX0 Þ ¼ 0
ð4Þ
0
where X = vector of nodal coordinates corresponding to the state of stable equilibrium. However, it was found that most configurations satisfying the stable equilibrium condition are usually one-dimensional or twodimensional ones which have little impact on engineering applications (Paul and Valero-Cuevas, 2005). Here we are interested in the tensegrities with three-dimensional configurations. Searching of three-dimensional tensegrities can be modeled by the following constrained optimization problem:
Max ðDÞ s:t: AðX0 ÞtðX0 Þ ¼ 0;
ð5Þ
X 2 ðXL ; XU Þ where D = space dimensions of tensegrities. Besides, in practical applications it is not only required that the tensegrity be in threedimensional space but also expected to meet some specific requirements. The optimization model corresponding to this situation is
Max ðf ðX0 ÞÞ s:t:
AðX0 ÞtðX0 Þ ¼ 0; X 2 ðXL ; XU Þ;
ð6Þ
D¼3 where f(X0 ) = is an objective function describing the practical requirements.
Max ðf ðX0 ; KÞÞ s:t: AðX0 ; KÞtðX0 ; KÞ ¼ 0; X 2 ðXL ; XU Þ;
ð7Þ
K 2 ðKL ; KU Þ; D¼3
3. Genetic algorithm In this paper, an genetic algorithm was used to solve the above optimization problem of form-finding. For the sake of simplicity, the basic and well known steps of a simple GA such as coding, selection, crossover and mutation are not discussed here in detail. Emphasis is instead placed on the new features of the GA used. 3.1. Determination of state of stable equilibrium and fitness estimation Before the fitness of an individual can be estimated, the state of stable equilibrium (i.e. X0 ) should be determined. Under any given nodal coordinates X, the corresponding internal forces of members usually cannot be equilibrated by themselves. The assembly deforms under the unbalanced internal forces. As one of the effective methods for form-finding of tensile structures, the dynamic relaxation method was adopted to determine the equilibrium state of a structural system with a randomly given set of nodal coordinates. The dynamic relaxation method consists of two iterative steps: firstly the unbalanced nodal forces are calculated and then the structural configuration is adjusted to reduce the unbalanced forces; and secondly the above procedure is repeated until the unbalanced forces are small enough according to a given numerical criterion. For detailed introduction of the dynamic relaxation method, please refer to the literature Barnes (1999). When the state of stable equilibrium is determined, the number of dimensions and other characteristics of the tensegrity can be easily determined. Then, the fitness of the corresponding individual can be estimated based on the characteristics of the tensegrity. 3.2. Niche A niche process combining a crowding check (De Jong, 1975) and a filter process is implemented in this study. For the initial generation, when fitness assignment is finished, the individuals are arranged in descending order with respect to the fitness and the first N (N 6 M) individuals are saved in a filter. Then, when a new generation is obtained and their fitness is assigned, a niche procedure as follows is conducted. Firstly, the entire new population (M individuals) is combined with the N individuals in the filter to constitute an extended population of M + N individuals. The Hamming distances between every two individuals of the extended population are computed. Secondly, they are compared with a given criterion ed. If the Hamming distance of two individuals is smaller than the criterion, the individual with a lower fitness will be identified by a penalty. In practical programming of this study it is achieved by multiplying the fitness by a penalty multiplier k, i.e.
F min ðx0i ; x0j Þ ¼ kF min ðx0i ; x0j Þ
ð8Þ
Thirdly, the individuals of the extended population are rearranged in descending order with respect to their new fitness and
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the first N individuals replace the old ones in the filter. Finally, the stopping criterion is checked to determine whether the algorithm has converged. If not, the first M individuals of the extended population are selected to constitute the parent population for next generation. Via the above process, the fitness gap between the superior individual and the inferior individual(s) within a distance ed is magnified. As a result, the inferior individuals have a higher probability of being eliminated in the next evolution process. In other words, within a space distance of ed there is only one good individual. Thus, the diversity of the population is ensured and meanwhile the even spread of individuals in the constraint space is achieved. Meanwhile, the best N individuals are recorded by the filter and flow into the parent population at every generation, which avoids the loss of good genotypes and expedites the evolution process.
4. Examples 4.1. Example 1 In this example, the target is to find as many three-dimensional tensegrities as possible under the given number of members, rest lengths of members, elastic moduli of members and connectivity of members. The only variables are the initial locations of nodes. Without loss of generality, two typical assemblies are considered here. One assembly consisting of six identical compression bars and 24 identical tension cables which connect to each other in accordance with a well known regular tensegrity called expanded octahedron tensegrity was considered, as shown in Fig. 1a. The other assembly is based on the zigzag icosahedral tensegrity which consists of 30 identical compression bars and 90 identical tension cables (Fig. 2a). The compression bars are assumed to have a rest length of unit and an elasticity of 1000. The tension cables of the first assembly and the tension cables of the second assembly are assumed to have rest lengths of 0.536 and 0.312, respectively. All the cables are assumed unit elasticity. The allowable range of nodal coordinates is [5, +5]. Each component of the vector of nodal coordinates is represented by a 15-number binary code string. As a result, the total length of a genome string = 15 3 number of nodes. The GA proposed in last section is used to carry out the optimization. Each run of the GA is conducted for 20 generations, using a population size of 20. Proportional selection is employed to probabilistically select genotypes for mutation and crossover. Pairwise one-point crossover operation and uniform mutation operation are adopted. The crossover rate and the mutation rate are empirically set to 0.6 and 0.002. For the 6-bar assembly, three stable self-equilibrium configurations are identified in the final generation, as shown by Fig. 1. The
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corresponding strain energies for the three configurations are 0.071, 0.173 and 0.241, respectively. As expected, the configuration with lowest energy is an expanded octahedron tensegrity (Fig. 1a). The configuration with the highest energy (Fig. 1c) is identical to the higher energy state reported in literature Defossez (2003). While the configuration with secondary energy (Fig. 1b) has not been reported elsewhere, to authors’ knowledge. For the 30-bar assembly, four configurations in stable self-equilibrium are found by the GA, as shown by Fig. 2. The corresponding strain energies of them are 0.090, 2.16, 3.15 and 3.22, respectively. Similar to the case of 6-bar assembly, the configuration with lowest energy is a regular tensegrity (i.e. zigzag icosahedral tensegrity) with which we are familiar. The configuration with secondary energy (Fig. 2b) has been reported (Defossez, 2003). However, the configurations with the highest two energies (Fig. 2c, d) are firstly reported, to authors’ knowledge. Since the three 6-bar configurations are corresponding to the same unprestressed assembly, as well as the four 30-bar configurations. They must be able to mutually transform between each other under certain conditions. Transformations between the configurations with different energy levels will release energy to environment or absorb energy from environment, which indicates that the tensegrity has potential applications in actuators and energy absorbers. 4.2. Example 2 In this example, we consider a more general case where elasticities of cables are also variable. In additional to the assumptions used in example 1, the variable range of the linear elasticities of cables is assumed to be [1, 10] and the rest lengths of cables are assumed to be zero. The target of this example is to find tensegrities whose spans in three-dimensions are as close as possible to each other. Hence, the root mean square of the spans in three-dimensions is used as the objective function:
f ðX0 Þ ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u3 uP 2 u ðdi dÞ ti¼1 3
ð9Þ
is the averwhere di is the span of a structure in ith dimension and d age of spans in three-dimensions. In addition, the constraint on the dimensions of the structure D = 3 was considered by converting the objective function into
gðX0 Þ ¼ D þ
1 1 þ f ðX0 Þ
ð10Þ
By doing this, it can be ensured that the structure with a higher dimension corresponds to a larger value of the objective function.
Fig. 1. Three self-equilibrium configurations of the assembly based on expanded octahedron.
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Fig. 2. Four self-equilibrium configurations of the assembly based on zigzag icosahedral tensegrity.
Meanwhile, for two structures with the same dimension, the structure that has closer spans in three-dimensions gets a larger value of the objective function than the other structure. The fitness of an individual is evaluated by directly setting it equal to the value of objective function. The GA used in this example is the same to that used in example 1 except the terminating criterion. In this example, it stops when the improvement of the best individual is lower than a critical value for 10 continuous generations or it runs for 100 generations. Formfindings on assemblies based on the expanded octahedron tensegrity (Fig. 1a), the hexagonal prism tensegrity (Fig. 4a), and a three-stage cylindrical tensegrity (Fig. 5a) were carried out, respectively. Except for the assembly based on the hexagonal prism tensegrity whose evolution stopped after 94 generations, the evolutions on the other two assemblies stopped after 100 generations. For each assembly, the tensegrities represented by the individuals of the final population are different from each other. They have very free geometrical configurations which are significantly different from the regular tensegrities well known to us. Taking the best tensegrities corresponding to the three assemblies as examples, they all have nonregular shapes and differ from their regular counterparts (Figs. 3–5). The values of the objective function for the three best tensegrities are 3.99995, 3.99994 and 3.99995, respectively. For the regular expanded octahedron tensegrity, it is obvious that its spans in three-dimensions are equal to each other when the coordinate axes are located parallel to the three orthogonal pairs of struts. Hence the value of the
objective function for the regular expanded octahedron tensegrity is 4.0. For the other two cases, it also can be deduced that the circumdiameter of the bottom polygon of a regular prim tensegrity or a regular cylindrical tensegrity can be equal to the height of the tensegrity under a certain set of force density in elements, according to the formulations given in literature Connelly and Terrell (1995). Therefore, the objective function values of the regular hexagonal prism tensegrity and the cylindrical tensegrity are also equal to 4.0. So the values of the objective function for the three nonregular tensegrities are comparable to their regular counterparts, but have free and variable shapes. In a pure viewpoint of optimization, it seems that the optimized nonregular tensegrities are not superior to the regular ones in this example. But it should be noted that the optimality of the found tensegrities heavily depends on the objective function. Here we just choose the objective function given by Eq. (10) to direct a target for the evolution of nonregular tensegrity, and the regular tensegrity becomes one of ideal solutions for the optimization problem by coincidence. If other indexes such as the cover area is taken as a maximum target, nonregular tensegrities which are now superior to the regular ones may be found.
5. Physical models Physical models of the nonregular tensegrities were made using wooden sticks and cotton threads, as shown in Figs. 6 and 7. Due to
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manual errors and viewpoint of photographs, it is unavoidable that there is a little difference between the theoretical configurations
Fig. 3. The best nonregular tensegrity corresponding to the assembly based on the expanded octahedron.
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and the physical models. But it still can be seen that the physical models retain the basic shapes of the corresponding theoretical configurations, which further verify the proposed method. It should be pointed that conceptual structural members without practical sizes were used in foregoing examples, i.e. crossing and touching of the structural members are not considered. As a result, when practical members with physical sizes are used contact between members may occur (Fig. 6c). A possible way to solve this problem is adjusting the prestress in members or the length of members to modify the configuration of the tensegrity to a state near the initial configuration but without member crossing and touching. Another point should be noted is that the axial forces of members of a nonregular tensegrity obtained are normalized ones and represent a prestress mode that is able to stabilize the structure. It is unnecessary to consider bar buckling and cable slackening at this stage of form-finding. Because we can choose appropriate bars and cables with suitable material and size to avoid the buckling and slackening of them at the stage of model making or structural design for a real tensgrity structure.
Fig. 4. Regular and nonregular tensegrities based on hexagonal prism.
Fig. 5. Regular and nonregular tensegrities based on cylindrical mast.
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Fig. 6. Physical models for the three stable configurations in Fig. 1.
Fig. 7. Physical models for the best tensegrities in example 2.
6. Conclusions and discussions In this paper, the form-finding of nonregular tensegrities was modeled as a constrained optimization problem. An GA was used to solve this problem. Examples were carried out to verify the proposed method. It was shown that tensegrities met the stable equilibrium and meanwhile optimized some practical requirements can be found by the proposed method. Because the GA is a parallel algorithm, a run of it can find more than one tensegrities that meet the stable equilibrium condition. In contrast, traditional form-finding methods are unit commitment algorithms, and at most one tensegrity can be found after each run of them. Moreover, the proposed method is more general than the traditional ones. It imposes fewer constraints on the initial assembly, and certain parameters of tensegrities can be optimized by properly choosing the objective function. As a result, tensegrities with more practical consider-
ations can be found, which advances the engineering applications of tensegrity structures. The GA used in this paper is different from the GA of Paul and Valero-Cuevas (2005) in two aspects: (a) using a much simpler binary encoding instead of a real number encoding, and (b) introducing a niche process based on strain energy to insure the diversity. Because geometrical stability is required for a tensegrity structure. An assembly with a given initial nodal position will jump into the same stable configuration when the nodal coordinates vary in a neighborhood of the given position, after a dynamic relaxation (or kinematic relaxation) process. In other words, tensegrity structures discretely exist in the space, though the search space for the initial nodal position seems continuous. Hence, we believe that a discrete representation is enough to cover the feasible solutions rather than a continuous representation. When the number of nodes increases, the proposed method will become much time
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cost. It is caused by the inherent shortage of the dynamic relaxation algorithm (kinematic relaxation algorithm) which is not effective with a large number of nodes (Motro et al., 1994). But using a much powerful computer such as a work station in stead of a PC may be able overcome this problem. This paper, as well as previously reported works on form-finding of nonregular tensegrity structures, has little control on the specific geometrical characters of the tensegrity structures obtained. They are only capable to optimize some gross and global geometrical targets such as volume and span, but have a difficult time handling more specific requirements, such as locating some nodes in given positions. Specific requirements like that mentioned above put such a heavy constraint on the problem that there may be no feasible solutions. Extending the search space by including the elastic constants of bars and connectivity of members in the variables will increase the possibility of finding feasible solution. But the efficiency of the algorithm will be affected if a larger search space is adopted. Form-finding of nonregular tensegrity structures that meet more specific requirement will be an interest subject for future work. Acknowledgements This work was supported by grants from NCET (Grant No. 060517) and National Natural Science Foundation of China (Grant No. 50638050). The authors also thanked Mr. Feng Yu, Jingmeng Xu, Zhouneng Zhong and Pengcheng Yang for their great help in making the physical models. References Barnes, M.R., 1999. Form finding and analysis of tension structures by dynamic relaxation. International Journal of Space Structures 14, 89–104. Connelly, R., Terrell, M., 1995. Globally rigid symmetric tensegrities. Structural Topology 21, 59–78. De Jong, K.A., 1975. An Analysis of the Behavior of a Class of Genetic Adaptive Systems. PhD Thesis, University of Michigan, MI. Defossez, M., 2003. Shape memory effect in tensegrity structures. Mechanics Research Communications 30, 311–316.
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