Selecting active members to drive the mechanism displacement of tensegrities

International Journal of Solids and Structures 191–192 (2020) 278–292

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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

Selecting active members to drive the mechanism displacement of tensegrities Dexi Zhu a, Hua Deng a,∗, Xiaoshun Wu b a b

Space Structures Research Centre, Department of Civil Engineering, Zhejiang University, Hangzhou 310058, China Nanchang Campus, Jiangxi University of Science and Technology, Ganzhou 341000, China

a r t i c l e

i n f o

Article history: Received 9 September 2019 Revised 22 December 2019 Accepted 23 January 2020 Available online 25 January 2020 Keywords: Tensegrity Mechanism displacement Active member Kinematic path Form finding

a b s t r a c t Based on the quasi-static assumption, the mechanism motion of tensegrities driven by the length actuation of internal active members is discussed in this paper. The self-stress and mechanism characteristics of a tensegrity are guaranteed by maintaining the zero eigenvalues of the state matrix of each configuration on a kinematic path. The sensitivity matrix between the eigenvalues of the state matrix and the joint displacements is given, and it is shown that the displacement causes only second- and higherorder changes to the zero eigenvalues. A strategy is proposed to trace the shortest path of mechanism displacement towards the target configuration of the tensegrity. The moving direction of mechanism displacement that is closest to the direction from the current configuration to the target configuration and the deviation of this direction can be calculated based on arbitrary bases of the null and row spaces of the compatibility matrix. A method for quickly constructing the basis of the row space of the compatibility matrix consisting of passive members is put forward to rapidly calculate the deviation between the current direction of mechanism displacement and the direction towards the target configuration after an active member is converted into a passive one. When the number of active members is limited, a one-by-one exclusion method is proposed to select active members to minimize the direction deviation of mechanism displacement. A four-stage quadriprism tensegrity tower and an icosahedron tensegrity are employed as illustrative examples. The proposed method is used to select active members and trace the paths of mechanism displacements, and its validity is verified by the results. © 2020 Published by Elsevier Ltd.

1. Introduction A classic tensegrity is regarded as an infinitesimal mechanism (Calladine, 1978) consisting of continuous cables and discontinuous struts (Pugh, 1976) and stiffened by prestress. The coexistence of both static and kinematic indeterminacies is the necessary condition for becoming a tensegrity. Earlier studies on tensegrities focused on explaining their structuralization (Pellegrino, 1990), i.e., how structural stability can be maintained for a prestressable mechanism (Calladine, 1978). Due to the weak stiffness, tensegrities are rarely used as load-bearing structures, such as trusses, in practical engineering. However, the theory developed to explain the unconventional structural characteristics of tensegrities has become an important basis for the analysis of flexible tensile structures such as cable nets, cable trusses and cable domes

∗

Corresponding author. E-mail address: [email protected] (H. Deng).

https://doi.org/10.1016/j.ijsolstr.2020.01.021 0020-7683/© 2020 Published by Elsevier Ltd.

(Geiger et al., 1986). As a system straddling the boundary between structures and mechanisms, the deformable tensegrity has been studied in multidisciplinary fields over the past two decades. The tensegrity was expected to be applied as aerospace structures such as the deployable antenna (Tibert and Pellegrino, 2002) and mast (Djouadi et al., 1998; Tibert and Pellegrino, 2003). For civil engineering applications, a near-full-scale deployable tensegrity footbridge has been studied both numerically and experimentally (Rhode-Barbarigos et al., 2010). In biomechanics, tensegrity is thought to be the most effective model for explaining the movement mechanism of the musculoskeletal system, in which bones act as struts and muscles are thought of as cables (Levin, 2002). This is mainly because tensegrities can not only deform by means of the length actuation of some active members, as conventional mechanisms do (Aldrich and Skelton, 2003), but also control and adjust the structural stiffness in motion by changing the prestress. Attempts to apply tensegrity to deformable robots have also emerged, such as the rolling Super Ball designed for extraterrestrial land exploration (Chen et al., 2017) and the prismatic tensegrity

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robot, which can generate crawling gait (Paul et al., 2005). Microscopically, the tensegrity model has also been used to explain the variation in structural shape and stability that occurs when cells undergo continual turnover (Stamenovic and Ingber, 2009). Whether it is a deployable tensegrity antenna or a tensegrity robot, kinematic analysis must be involved. When controls vary sufficiently slowly in time, the dynamic analysis of tensegrities can be treated as a quasi-static problem (Sultan, 2018; Mallikarachchi and Pellegrino, 2011). Thus, the deformation process can be thought of as consisting of continuous feasible configurations that are still required to maintain the self-stress and mechanism properties of tensegrity as well as the condition of being stiffened by prestress. For a deforming tensegrity, its feasibility depends on the form finding performed on the subsequent configuration when the driving variable changes. Therefore, form finding is the main focus of the latest studies on the quasi-static kinematic analysis of tensegrities. In existing methods of form finding, analytical methods are basically applicable to tensegrities with simple geometry and high symmetry (Sultan et al., 2001). A variety of numerical methods have been adopted and developed for the form finding of complicated tensegrities, such as the dynamic relaxation method (Motro, 1984), the nonlinear programming approach (Pellegrino and Calladine, 1986), the reduced coordinate method (Sultan et al., 1999), the finite element method (Pagitz and Mirats Tur, 2009) and the mixed integer linear programming approach (Kanno, 2013). Optimization algorithms are also employed, e.g., the Monte Carlo method (Li et al., 2010), the genetic algorithms (Koohestani, 2012; Xu and Luo, 2010) and the ant colony algorithm (Chen et al., 2012). Micheletti and Williams (2007) and Koohestani (2015) developed methods for the reshaping of tensegrities based on the characterization of the rank-deficiency manifold in the space of nodal coordinates. More studies on the form finding of tensegrities based on the force density method can be found in the articles of Vassart and Motro (1999), Zhang and Ohsaki (2006), Tran and Lee (2010), Koohestani (2015) and so on. Force density and structural geometry are adjusted in these methods by iterative calculations to eliminate unbalanced joint forces and make the system meet the condition of self-stress. When the structural topology is not permitted to change, all feasible configurations constitute the manifold of a tensegrity (Roth and Whiteley, 1981). If a feasible configuration deforms towards another one in the manifold, the kinematic path is usually not unique. Thus, the so-called path planning problem arises. The Rapidly-exploring Random Tree (RRT) method was employed by Xu et al. (2013) and Porta and Hernández-Juan (2016) for the path planning of the quasi-static motion of tensegrities. In addition, it is generally believed that the static stability of any equilibrium configuration on the kinematic path should be maintained to ensure that the tensegrity can move along a predetermined path. However, Sultan (2014) emphasized that the conditions of the quasi-static assumption is difficult to guarantee in practice. The dynamic equations should be employed to analyze the motion of tensegrities, and dynamic stability should be considered. A comprehensive treatment of various stability features of tensegrity from static to dynamic stability was discussed by Sultan (2013). In this paper, the motion of tensegrities is assumed to be quasi-static and only the static stability is considered. Because tensegrities can maintain structural stability due to the stiffening of prestress, any external action will generate the elongations of elements. For the deformation of tensegrities, mechanism can be utilized and "activated" by changing the lengths of some members (Sultan, 2014), which are known as "active members". The induced rigid-body displacement is referred to as the mechanism displacement in this paper. The length variation of active members will cause the deformation of a tensegrity in two ways: the resulting changes in structural internal forces will lead

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to further elastic deformation of members, and the mechanism displacement will be excited. If only the mechanism displacement is produced by the length actuation of active members, the elastic deformations of other members (passive members) are not generated (Smaili and Motro, 2007; Rhode-Barbarigos et al., 2012). To reduce the consumption of driving energy, the use of mechanism displacement is obviously the most effective way to achieve the deformation of a tensegrity. Pellegrino and Calladine (1986) stated that the direction of mechanism displacement can be expressed as an arbitrary linear combination of null space basis vectors of the compatibility matrix. Kumar and Pellegrino (20 0 0) proposed a geometrical method for tracing the kinematic path of pin-jointed bar mechanisms. Lengyel and You (2004) discussed the singularities of the compatibility paths of single-degree-of-freedom finite mechanisms consisting of rigid bars. Wang et al. (2018) employed the RRT method to perform the path planning of loaded pin-jointed bar mechanisms. If the number of active members of a tensegrity is limited, it is generally not possible to fully achieve the desired deformation through mechanism displacement. In addition, a different choice of active members will result in different moving directions of mechanism displacement as well as a different degree of closeness to the desired deformation. Therefore, the tensegrity can be driven to move as close as possible to the target direction by selecting active members appropriately, and the deformation of the tensegrity can also be controlled effectively. Although there exit many studies on the deployment of tensegrities, the topic of optimal selection of active members is rarely addressed. Some conventional methods, such as the enumeration method and the genetic algorithm, can be employed to select active members. Sultan et al. (20 0 0) used the enumeration method to select the active tendons of a tensegrity flight simulator. The genetic algorithm was employed by Yan et al. (2005) to optimize the placement of active members for truss structures. This paper will discuss the kinematics on the mechanism displacement of tensegrities driven by the length actuation of active members. The main topics include the feasibility of tensegrity configurations on the kinematic path, the moving direction of mechanism displacement, the shortest path to the target configuration and the selection of active members. The paper is organized as follows. Section 2 describes the basic characteristics of tensegrities, including prestressability (Sultan et al., 2001), and infinitesimal mechanism that can be stiffened by prestress. The sensitivity matrix between the eigenvalues of the structural state matrix and the joint displacements will be discussed in Section 3 to guarantee the feasibility of tensegrity on the kinematic path. Section 4 will discuss the solution of the mechanism displacement closest to the target configuration. A numerical strategy will be proposed in Section 5 to trace the shortest path of mechanism displacement from an initial configuration to a target configuration. A reverse selection method of gradually converting active members to passive members when the number of active components is limited is proposed in Section 6 to reduce the deviation between the direction of mechanism displacement and the direction towards the target configuration. A four-stage quadriprism tensegrity tower and an icosahedron tensegrity will be employed as illustrative examples to investigate the validity of the proposed method by tracing their kinematic paths of mechanism displacement as well as selecting active members. 2. Characteristics of tensegrities 2.1. Modelling assumptions All elements in the tensegrity are pin-jointed. Cables must remain in tension during the deformation of tensegrity. The quasistatic assumption is applied.

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2.2. Prestressability The equilibrium equation of a tensegrity with N degrees of freedom (DOFs) and b elements can be expressed as

At = p

(1)

where t is the b × 1 vector of element axial forces in which the components corresponding to cables are positive, p is the N × 1 vector of joint loads, and A is the N × b equilibrium matrix, which can be further written as



...

A= a1

...

ak

ab



Fig. 1. The shortest path of mechanism displacement from x0 to xt .

(2)

where ak (for k = 1, 2, ..., b) is the N × 1 vector contributed by element k and takes the form



x3 p−2 − x3q−2 lk

...

ak = 0

x3 p−1 − x3q−1 lk

x3 p − x3q lk

...

0

where lk is the length of element k, and (x3p − 2 , x3p − 1 , x3p ) and (x3q − 2 , x3q − 1 , x3q ) are the coordinates of its two end joints p and q, respectively. When p=0, Eq. (1) can be written as

At = 0

(4)

Tensegrities must be prestressable (Calladine, 1978; Pellegrino and Calladine, 1986), which implies that there exist non-zero solutions for t in Eq. (4). If AT A, which is called the state matrix in this paper, has s > 0 zero eigenvalues, the non-zero solutions of t must exist and can be expressed as b 

t=

αi vi

(5)

i=r+1

where r = b − s is the rank of both AT A and A; vi (i = r + 1, ...,b) are b × 1 eigenvectors corresponding to the zero eigenvalues of AT A and were named the modes of self-stress states (Pellegrino and Calladine, 1986); and αi (i = r + 1, r + 2, ..., b) are the combination coefficients, which are not all zero. It is readily known that b > r for the prestressable tensegrity, and Avi =0 (i = r + 1, ...,b). 2.3. Infinitesimal mechanism

...

x3q−1 − x3 p−1 lk

x3q −x3 p lk

...

0

T

(3)

N×1

where (d3p − 2 , d3p − 1 , d3p ) and (d3q − 2 , d3q − 1 , d3q ) are the displacements of the two end joints of element k. If the nonlinear parts 1/2BN (d)+oB (d2 ) are neglected when d is small, Eq. (7) can be simplified to

BL d = e

(11)

Note that the rank of BL is also r because BL =AT . Since the DOFs of a classic tensegrity are not less than the number of elements (Calladine, 1978), N ≥ b > r. This indicates that non-zero solutions for d must exist for

BL d = 0

(12) BTL BL

Spectral decomposition is performed for (N × N), whose rank is also r. Two matrices Ur (N × r) and Um (N × m), which include all the eigenvectors corresponding to the non-zero and zero eigenvalues of BTL BL , respectively, are then obtained, where m=N-r is the number of zero eigenvalues. Similarly, the non-zero solutions for d in Eq. (12) can be expressed as a linear combination of the eigenvectors in Um as follows

d=

The following compatibility equation must be satisfied for the tensegrity

x3q−2 − x3 p−2 lk

N 

ui βi =Um βm

(13)

i=r+1

where d is the N × 1 vector of joint displacements, e is the b × 1 vector of element elongations, and B is the b × N compatibility matrix. It should be noted that B is a nonlinear function of d and can be expressed as a Taylor series expansion as follows (Deng and Kwan, 2005)

where ui (i = r + 1, r + 2, ..., N) were named the modes of inextensional mechanism (Pellegrino and Calladine, 1986) because BL ui =0, and βm = {β r + 1 , β r + 2 , ..., β N }, where βi (i = r + 1, r + 2, ..., N) are the combination coefficients and are not all zero. Note that Ur and Um are an orthonormal basis for the row and null spaces of BL , respectively, but are not unique. If r and m are another orthonormal basis for these two spaces, the following relationships should be satisfied

B=BL +1/2BN (d )+oB (d2 )

r = Ur Tr , m = Um Tm

Bd = e

(6)

(7)

where BL =AT (Livesley, 1975) is the linear part of B, BN (d) is the nonlinear part that includes only the first-order terms of d, and oB (d2 ) represents all of the remaining nonlinear parts of B. Similar to A, BL and BN (d) can be further written as



BL =

a1



BN ( d )=

...

ak

...

aN1

...

aNk

ab ...

T

(8) aNb

T

(9)

where aNk (for k = 1, 2, ..., b) is the N × 1 vector contributed by element k and takes the form



aNk = 0

...

d3 p−2 −d3q−2 lk

d3 p−1 −d3q−1 lk

d3 p −d3q lk

...

0

...

(14)

where Tr and Tm are the rotation matrices of dimension (r × r) and (m × m), respectively, and Tr Tr T = I, Tm Tm T = I. If Um in Eq. (13) is replaced by m , the resulting d will still satisfy Eq. (12). Substituting d =Um βm into the nonlinear Eq. (6), the element elongations will still be generated, i.e., e = 0, due to the higherorder terms in Eq. (7). The existence of non-zero solutions for d in Eq. (12), i.e., m>0, is actually a necessary condition for a mechanism to allow deformation without producing any member elongation. However, tensegrities belong to the infinitesimal mechanism, which only possesses virtual mobility but no actual kine-

d3q−2 −d3 p−2 lk

d3q−1 −d3 p−1 lk

d3q −d3 p lk

...

T 0 N×1

(10)

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281

Fig. 2. A four-stage tensegrity tower.

matic mobility (Kuznetsov, 1999) and meanwhile can be stiffened by prestress (Calladine, 1978). This means that the non-zero solution for d obtained by Eq. (13) only indicates the mechanism displacement tendency of the tensegrity. The subject of the stability of prestressed mechanisms, typical of tensegrities, has been discussed in a large number of articles in the past few decades (Calladine, 1978; Kuznetsov, 1975). A "product force" criterion was also proposed by Calladine and Pellegrino (1991) to determine the stability of a prestressed mechanism as follows T T βm F Um βm >0

(15) fi =BTN (ui )t

where F= {fr + 1 , ..., fi , ..., fN } and is the product force vector corresponding to each mode of inextensional mechanism ui (i = r + 1, r + 2, ..., N ). The physical meaning of Eq. (15) is that the self-stress t existing in the structure will produce the component of unbalanced joint forces, which are opposite to Um βm , to restrain the development of mechanism displacement. Based on the energy criterion (Thompson and Hunt, 1973), Deng and Kwan (2005) proved that Eq. (15) is actually equivalent to that the prestress provides the geometry stiffness in all possible directions of mechanism displacement. Since the left-hand term of Eq. (15) is the quadratic form of βm and FT Um is a symmetric matrix, the stability of the tensegrity can be distinguished by analysing the positive definiteness of FT Um , e.g., using its minimal eigenvalue. 3. Variation of the eigenvalues of the state matrix The coexistence of the modes of self-stress state and inextensional mechanism, i.e., s>0 and m>0, is the basic characteristic of tensegrities. The values of s and m are primarily determined by

Fig. 3. The rotational relationships between (a) the top and bottom surfaces of module A or C; (b) the top and bottom surfaces of module B or D; (c) the top surfaces of module A (C) and module B (D).

the rank of the state matrix AT A (or BTL BL ). If the tensegrity deforms in a continuous manner, it is generally required that there not be any essential change in the static and kinematic properties (Tarnai, 1980) of any configuration on the equilibrium path to avoid an abrupt change in structural geometry or stability, i.e., the rank of AT A remains the same. In fact, variations in the static and kinematic properties are inevitable in some cases and may even trigger some complicated problems of structural stability, e.g., path bifurcation (Hunt, 1978; Kumar and Pellegrino, 20 0 0). However, this is not addressed in this paper. If the topology of the tensegrity remains unchanged during deformation, any eigenvalue λi (i = 1, 2, ..., b) of AT A is merely a function of the joint coordinate vector x and satisfies

(AT A − λi I )vi = 0

(16)

Left multiplying both sides of Eq. (16) by vTi and then calculating the partial derivatives of both sides with respect to any coordinate component xc (c = 1, 2, ..., N) of x gives

T

 ∂ vTi  T ∂ A ∂ λi ∂A A A − λi I vi + vTi A + AT − I vi ∂ xc ∂ xc ∂ xc ∂ xc   ∂ vi + vTi AT A − λi I =0 ∂ xc

(17)

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Fig. 4. The target configuration.

Substituting (16) into Eq. (17) results in the following

T

∂ λi ∂A ∂A = vTi A + AT v ∂ xc ∂ xc ∂ xc i

(18)

T where ∂∂Ax is the transpose of ∂∂xA . According to Eqs. (2) and c c (3), any entry A˜ in A is either equal to zero  or takes the form 2 2 (x3p − I − x3q − I )/lk , where I = 0, 1 or 2, lk = I=0 (x3 p−I − x3q−I ) ,

and p and q are interchangeable. If A˜ is not a function of xc , then ∂ A˜ ˜ ∂ xc =0. When the expression of A includes xc , the partial derivative of A˜ with respect to xc can be calculated by

⎧ 1 ⎪ ⎪ ⎨ lk −

∂ A˜ ∂ xc =

(x3 p−I −x3q−I )2

−1 +

lk

3

,

(x3 p−I −x3q−I )

2

lk

Eq. (18) that

if xc = x3 p−I ,

3 lk lk ⎪ ⎪ ⎩− (x3 p−J −x3q−J )(3x3 p−I −x3q−I ) ,

if xc = x3q−I

Fig. 5. The selected (a) 8 and (b) 16 active members in the unfolded drawing of the tensegrity tower.

(19)

if xc =x3 p−J or x3q−J

(I, J = 0, 1, 2; I = J )

Note that A is obviously differentiable for an arbitrary configuration according to Eqs. (2), (3) and (19). Thus, ( ∂∂xA )− =( ∂∂xA )+ c c holds for any nodal coordinate xc . It can be determined according ∂ λi ∂ λi to Eq. (18) that ( ∂ x )− =( ∂ x )+ , indicating that λi is differentiable c c with respect to xc . For the zero eigenvalue λi (i = r + 1, r + 2, ..., b) of AT A, the corresponding eigenvector vi satisfies Avi = 0. Thus, it follows from

∂ λi = 0(i = r + 1, r + 2, ..., b) ∂ xc

(20)

To keep the rank of the updated AT A unchanged when the tensegrity deforms a displacement d from the current configuration xj , the zero eigenvalue λi should satisfy

λi (x j +d ) = 0 (i = r + 1, r + 2, ..., b)

(21)

∂λ Since ∂ xi exists, implementing the Taylor series expansion on

the left-hand term of Eq. (21) leads to



∂ λi ∂x

T

 x=x j d + oλ (d2 ) = 0 (i = r + 1, r + 2, ..., b)

(22)

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4. Mechanism displacement 4.1. Active and passive members As mentioned in Introduction, the mechanism displacement of tensegrities can only be produced by the length actuation of internal active members. Partitioning BL and e according to active and passive members, Eq. (11) can be rewritten as

 a BL

BpL

 

d=

ea ep

(23) p

where BaL and BL are the submatrices of BL corresponding to the active and passive members, respectively, and ea and ep are the elongation vectors of the active and passive members, respectively. If d is a mechanism displacement, the resulting elongation of passive members satisfies ep = 0. According to Eq. (23), d should satisfy

BpL d = 0

(24)

In Eq. (24), the non-zero solution of d can be expressed in the p p same form as in Eq. (13), i.e., d = Um βm , where Um is the basis p matrix for the null space of BL . Accordingly, the elongations of the active members are

ea =BaL Upm βm Fig. 6. Partial process of excluding active members.

(25)

4.2. Oriented motion Assume that the tensegrity deforms a given displacement dt . p p Because all the eigenvectors of (BL )T BL form a complete basis of N-dimensional space, dt can always be expressed as the following linear combination t dt = Upr βrt +Upm βm

(26)

p Ur

where (N × r) is the matrix consisting of all the eigenvectors p p corresponding to the non-zero eigenvalues of (BL )T BL and is also p t t are the the basis matrix for the row space of BL , and βr and βm p vectors of combination coefficients for the eigenvectors in Ur and p Um , respectively, and are uniquely determined by dt . It follows p p from the orthogonality of the basis vectors in both Ur and Um that t can be obtained by βrt and βm

 T  T t βrt = Upr dt ; βm = Upm dt

(27)

According to Eqs. (26) and (27), the mechanism displacement closest to dt is



T

t dm = Upm βm =Upm Upm dt

(28)

and the deviation between dm and dt is



T

dr = dt − dm = Upr Upr dt

(29)

Obviously, dr will cause the elongation of passive members. Considering Eqs. (14), (28) and (29) can also be rewritten as Fig. 7. Components of vb for configurations x0 and xm .





dr = Upr Tr TTr (d2 )

where oλ represents the second- and higher-order terms of d. According to Eq. (20), any displacement d will never create first-order variation of the zero eigenvalue λi . The exact solution of d in Eq. (22) is, therefore, only determined by oλ (d2 ) = 0, which is difficult to solve generally. In fact, solving Eq. (22) can be regarded as a form-finding process of a tensegrity with a given primary configuration xj , and a numerical solution method must be employed.

T



T

dm = Upm (Tm TTm ) Upm dt = m pm dt



T

p



T

Upr dt = r pr dt p

(30) (31)

It follows that dm and dr can be calculated using any basis, such p p p as r and m , for the row and null spaces of BL , respectively. 5. The shortest path of mechanism displacement The tensegrity is expected to move along a path of mechanism displacement from an initial configuration x0 to a target configuration xt . It should be noted that any configuration determined by linear interpolation between x0 and xt , i.e., a point on the dashed

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Fig. 8. The (a) 8 and (b) 16 active members selected in each step on the kinematic path.

line in Fig. 1, is generally not feasible because it hardly satisfies both Eqs. (22) and (24). However, the path of mechanism displacement between x0 and xt can be traced by searching a series of feasible configurations on the path by prescribing a small step length L. For any configuration xj − 1 (j= 1, ...) starting from x0 as shown in Fig. 1, the moving direction can be determined by dt =xt − xj − 1 . p Further calculating Um for configuration xj − 1 as well as the mechanism displacement direction dm by Eq. (28), the next configuration can thus be obtained by

x j = x j−1 + dm ( j = 1, ... )

(32)

where d¯ m = (dm /dm 2 )L and ·2 denotes the L2 norm. If L is as small as possible, the shortest path of mechanism displacement between x0 and xt can therefore be traced by those configurations obtained using Eq. (32). Generally, the tensegrity should be prevented from moving away from the predetermined path due to unexpected perturbations. For example, a bifurcated path appears. For any equilibrium configuration xj on the path, maintaining its stability can ensure the tensegrity deforms definitely under the length actuation of active members. Considering the static stability of the tensegrity is usually reflected by the minimal eigenvalue λfmin of the matrix FT Um according to Eq. (15), λfmin > 0 should be satisfied for all the configurations to keep the whole kinematic path definite. Note that both Eqs. (6) and (22) include the nonlinear terms of d, and L cannot be very small in engineering computations. When

substituting d = d¯ m into Eqs. (6) and (21), ep and λi (i = r + 1, r + 2, ..., b) for the new configuration xj must be non-zero. Hence, the Newton-Raphson method can be employed to obtain the exact solution of d¯ m with the following iterative equation (q ) ¯ (q−1 ) d¯ m =dm − S+ g

(33)

(q−1 ) where d¯ m is the result of d¯ m after (q-1) iteration steps; g is the vector consisting of ep and λi (i = r + 1, r + 2, ..., b), which are non-zero; and S+ represents the Moore-Penrose pseudoinverse (Golub and Van Loan, 1996) of the gradient matrix S and can be calculated using the singular value decomposition (SVD). The expressions of S and g are

 p

S=

BL J



; g=

ep

∗



(34)

where∗ ={λr + 1 , λr + 2 , ..., λb }T and J is a Jacobian matrix ∂λ

with entries Jic = ∂ x i (i = r + 1, r + 2, ..., b; c= 1, 2, ..., N), which c (q ) can be calculated by Eq. (18). Once d¯ m is obtained by Eq. (33) in (q ) the q-th iteration step, xj can be updated by assigning d¯ m =d¯ m and substituting it into Eq. (32). Then, S and g are re-established at the updated configuration. The iterative process will continue until ep and ∗ are less than the given tolerances. In addition to there being no elongation generated in the passive members, it can be seen from Eqs. (33) and (34) that the mechanism displacement of the tensegrity should also keep all

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6. Selection of active members p

If all the elements are active members, Um will become an p N × N orthogonal matrix since Ur is void. For the given moving dit in Eq. (26) can rection dt , the vector of combination coefficients βm therefore be uniquely determined. This indicates that dt can be accomplished accurately because all the elements are extensible. However, once there exist passive members in the structure, i.e., p Ur = 0, the deviation dr between dm and dt will be generated according to Eq. (29). Generally, the range of mechanism displacement will be gradually limited with the participation of passive members, and dr will be augmented. Moreover, the direction dm obtained by Eq. (28) varies with the selected active members bep p p cause the eigenvectors of (BL )T BL , i.e., [ Upr Um ], are determined by the corresponding passive members. If the number of active members is limited, the issue of the optimal selection of active members to minimize dr will arise. According to Eq. (31), dr is the component of dt in the row space p p of BL and can be calculated by any orthonormal basis r for this space. Considering that the L2 norm of a vector represents its length in Euclidean space, dr 2 can be employed as an optimization index. Assuming that all the elements are first active members, it is easy to know that dr =0. Then, dr 2 can be calculated for each active member that is individually converted into a passive one. Therefore, the member with the smallest dr 2 is the first choice for a new passive member. A method of excluding active members can thus be built based on this idea. If n-1 passive members have been determined in the structure, the compatibility matrix (BPL )(n−1) and a basis for its row space (Pr )(n−1) can be obtained. When an active member k is further p converted into a passive member, BL is updated as follows

 p 

 p BL

(n )

BL

=



(n−1 )

(35)

aTk

According to Eq. (24), the mechanism displacement d must satp p isfy both (BL )(n−1 ) d = 0 and aTk d = 0. Since (r )(n−1 ) is the basis p for the row space of (BL )(n−1 ) , d should satisfy



pr

T

(n−1 )

d=0

(36)

If ak can be expressed as a combination of the vectors in

(pr )(n−1) , i.e., ak =(pr )(n−1) βk , where βk is the combination co-

efficient vector, then aTk d = 0 obviously holds for the solution of d p in Eq. (36). This means that (r )(n−1 ) is still a basis for the row p space of (BL )(n ) . Thus, the active member k should be converted to a passive one because dr will not change with its participation p according to Eq. (31). However, if ak = (r )(n−1 ) βk , then



p a⊥ k =ak − r

where Fig. 9. Variations in the coordinates of B7 and D7 on the kinematic paths: (a) x coordinates; (b) z coordinates.



β (n−1 ) k

= 0

βk =(pr )T(n−1) ak ,

(37)

and a⊥ is orthogonal to all vectors in k

(pr )(n−1) . Normalizing a⊥ results in k

  φk = a⊥k /a⊥k 2

(38)

Thus, the basis for the row space of

(BpL )(n)

is expanded to

[(r )(n−1) , φk ]. Using Eq. (29), it can be seen that p

the zero eigenvalues of AT A unchanged. Although the displacement only affects the high-order terms of the zero eigenvalues according to Eqs. (20) and (22), it is still necessary to eliminate the errors using the above iterative approach. p If there is no passive member in the structure, i.e., BL =0, then ∗ S = J, g =  in Eq. (34). In this case, eliminating the errors of λi (i = r + 1, r + 2, ..., b) by Eq. (33) actually adjusts the current configuration to the nearest feasible configuration. It can be regarded as the form finding of the tensegrity.





dr(n ) = (pr )(n−1) (pr )T(n−1) + φk φkT dt = dr(n−1) + φk φkT dt

(39)

where dr(n − 1) and dr(n) are the results of dr before and after member k is converted into a passive member, respectively, and p p dr(n−1) =(r )(n−1 ) (r )T(n−1 ) dt . For each active member in the structure, dr(n) can be calculated by Eq. (39) when the member is individually converted into a passive member. The active member with the smallest dr(n) 2 will finally be chosen as the new passive member. Based on the updated

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Fig. 10. The relative deviations ρ between xj on the kinematic paths and xm .

assembly of passive members, the same selection process will be restarted for the remaining active members to obtain the next optimal passive member. The process will not be terminated until the prescribed number of active members remains. If the total number of active members is limited at every step on the path of mechanism displacement, which is obtained by the tracing strategy proposed in Section 5, the active members selected using the above method will generally vary at different steps. The frequency with which each element is selected in all the steps can be counted, and those elements with higher frequency can be considered to be the active members for the entire kinematic path of the tensegrity. Compared with the conventional enumeration method and the genetic algorithm, the proposed method has the following advantages. (1) The computational efficiency is significantly higher since p p it is not necessary to repeatedly decompose the matrix (BL )T BL . (2) The priority order of the selected active members can be quantitatively evaluated. (3) The number of selection steps is definite and limited. (4) Under the driving of the selected active members, the deviation between the final configuration and the target configuration can be estimated accurately. However, the active members selected using the proposed method cannot be guaranteed to be the global optimal solution. 7. Illustrative examples 7.1. A four-stage tensegrity tower 7.1.1. Initial and target configurations A tensegrity tower consisting of four stacked quadriprism modules named A, B, C, and D from the bottom to the top is shown in Fig. 2. In the initial configuration, the bottom√and top surfaces of each module are squares with side length a=2 2. For module A or C, the top surface is rotated through an angle of θ = 1/8π counterclockwise around the z-axis with respect to the bottom surface (Fig. 3(a)). The module B or D is rotated clockwise through the same angle (Fig. 3(b)). Modules B, C and D are rotated clockwise around the z-axis by 1/4π , 0, and 1/4π , respectively, relative to module A, as shown in Fig. 3(c). Two adjacent modules are assem-

Fig. 11. The obtained final configurations corresponding to (a) 8 active members; (b) 16 active members.

D. Zhu, H. Deng and X. Wu / International Journal of Solids and Structures 191–192 (2020) 278–292

Fig. 12. Variation in the minimum eigenvalue λfmin of FT Um for the obtained feasible configurations on the kinematic path.

bled by replacing the cables in their shared surfaces with saddle cables. Diagonal cables are then added to the side surfaces of each module. In addition, the side elements are arranged in opposite rotation directions for two adjacent modules. There are 32 joints, 16 bars and 72 cables in total in the tensegrity tower. The joint numbers are also shown in Fig. 2, and each element is denoted by the numbers of its two end joints. Six DOFs, A1-x, A1-y, A1-z, A2-x, A2-z, A3-z, are constrained on the ground to prevent rigid-body motion of the tower. The height of each module is H = 8, and the overlap depth of adjacent modules is h. To be a tensegrity, a particular relationship must be satisfied between θ and h/H (Nishimura, 20 0 0). For the initial configuration shown in Fig. 2, it can be determined that h/H=0.29509 corresponds to θ = 1/8π . The numbers of the modes of self-stress state and inextensional mechanism are s = 1 and m = 3, respectively. The tensegrity tower is expected to deform from the above initial configuration to a target configuration shown in Fig. 4. The target configuration is constructed by forcing the joints in surfaces A-II, B-I, B-II, C-I, C-II, D-I and D-II to move in the negative direction of the x axis. The moving distance of joints in the same surface is d = kz2 , where k is a prescribed constant, and z is the height of the surface from the ground. Thus, the centres of all surfaces in the target configuration are located on the same parabola, as shown in Fig. 4. Furthermore, the top and bottom surfaces of all modules remain horizontal. In the following, a small and a large deformation will be analysed for the tensegrity tower by prescribing k = 1/10 0 0 and 1/100, respectively. 7.1.2. Selection of active members for the small deformation The target configuration xt is actually very close to the initial configuration x0 when k = 1/10 0 0. The tensegrity tower is therefore assumed to move towards the target configuration in a single step, i.e., dt =xt − x0 , and only cables are taken as the candidate active members. Using the method proposed in Section 6, 8 and 16 active members are selected and listed in Table 1. These active members are also highlighted in red in Fig. 5. The partial selection process is illustrated in Fig. 6, showing that 8 members from

287

Fig. 13. Variations in λmin with and without zero correction for the obtained configurations on the kinematic paths.

the remaining 16 active members are excluded as passive members one by one based on the value of dr(n) 2 . Since the number of active members is limited, ρ = xt − xf 2 /xt − x0 2 is defined according to Eq. (31) to evaluate the relative deviation between the obtained final configurations xf and xt . The results of ρ corresponding to the 8 and 16 selected active members are also listed in Table 1. It shows that ρ is reduced from 0.1452 to 0.1205 when the other 8 members are selected as active members. 8 = 1.1969e + 10 possible combinations when There are up to C72 8 elements can be selected as active members from all 72 cables in the tensegrity tower. Obviously, the computational efficiency will be challenged if the traditional enumeration method is employed. A total of 1 × 105 sets of 8 active members are randomly assigned in order to investigate the validity of the selection method developed in Section 6. The value of ρ is then calculated for each set of active members, and the results for all 1 × 105 sets are counted and listed in Table 2. It can be seen that they are all greater than that of the 8 selected active members listed in Table 1. Substituting dt into Eq. (6), the elongations of all elements in the tensegrity tower can be obtained. The 8 elements with the maximum elongations are C7D3, A5B5, A7B7, B7C7, B5C5, C5D5, C8D8 and C7D7. If these elements are designated as active members, it is intuitively believed that the mechanism displacement they drive would be closest to dt . For this set of active members, it can be calculated that ρ = 0.1934, which is however greater than the value of 0.1452 corresponding to the 8 active members listed in Table 1. It is obviously unreliable to select the active members based on the above intuition. The genetic algorithm provided in the Global Optimization Toolbox of MATLAB 2016b is also employed to select the active members. 1/ρ is adopted as the fitness function. Both the initial population size and the number of generations are set to 200. The probabilities of crossover and mutation are 0.99 and 0.1, respectively. Table 3 lists the computational time for the selection of active members and values of ρ corresponding to the method proposed in this paper, the genetic algorithm, the enumeration method with 1 × 105 samples and the method based on the maximum elonga-

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D. Zhu, H. Deng and X. Wu / International Journal of Solids and Structures 191–192 (2020) 278–292 Table 1 Two sets of selected active members and the deviations of final configurations. Number of active members

The active members selected

ρ

8 16

C1C5, A3A7, C3C7, A1A5, B3B7, A3B4, D3D7, D1D5 C1C5, A3A7, C3C7, A1A5, B3B7, A3B4, D3D7, D1D5, A1B2, B1B5, C5D5, C7D7, C4D1, C4C8, B1C4, C2D3

0.1452 0.1205

Table 2 Distribution of ρ for 1 × 105 random sets of 8 active members.

ρ

≤ 0.1452

(0.1452, 0.2]

(0.2, 0.3]

(0.3, 0.4]

(0.4, 0.5]

> 0.5

Number of sets

0

349

27,001

32,610

23,366

16,674

Table 3 Deviations ρ and computational time corresponding to the four methods. Methods

Proposed method

Genetic algorithm

Enumeration method with 1 × 105 samples

Maximum elongations

ρ

0.1452 0.512

0.1511 33.42

0.1669 85.55

0.1934 0.072

Computational time (s)

Table 4 Coordinates of some typical joints at 4 different configurations of the tensegrity tower. Joint number

Target configuration xt

Modified configuration xm

Final configuration xf with 8 active members

Final configuration xf with 16 active members

x

y

z

x

y

z

x

y

z

x

y

z

A4 A7 B4 B7 C4 C7 D4 D7

−1.414 −2.488 −2.318 −1.095 −2.686 −5.564 −4.862 −5.444

−1.414 0.765 0.0 0 0 1.848 −1.414 0.765 0.0 0 0 1.848

0.0 0 0 8.0 0 0 5.639 13.639 11.279 19.279 16.918 24.918

−1.404 −2.491 −2.323 −1.097 −2.679 −5.564 −4.865 −5.445

−1.412 0.763 −0.008 1.848 −1.416 0.766 −0.005 1.846

−0.001 8.0 0 0 5.637 13.639 11.280 19.279 16.916 24.917

−1.414 −2.769 −2.527 −1.516 −2.783 −5.150 −4.391 −5.282

−1.360 0.451 −0.674 1.842 −1.526 1.062 0.094 2.089

0.004 7.525 5.321 13.550 10.953 18.647 16.487 24.871

−1.410 −2.522 −2.558 −1.390 −2.728 −5.036 −4.687 −5.369

−1.255 1.034 −0.080 1.747 −1.629 1.286 0.219 1.741

−0.003 7.709 5.486 13.874 11.090 19.002 16.784 25.286

Table 5 Joint coordinates at 4 different configurations of the icosahedron tensegrity. Joint number

N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12

Initial configuration x0

Target configuration xt

Modified configuration xm

Final configuration xf

x

y

z

x

y

z

x

y

z

x

y

z

−1.768 −1.768 3.536 −5.303 3.536 1.768 −3.536 −1.768 5.303 −3.536 1.768 1.768

−3.062 3.062 0.000 1.021 4.082 −5.103 −4.082 5.103 −1.021 0.000 3.062 −3.062

0.000 0.000 0.000 2.887 2.887 2.887 5.774 5.774 5.774 8.660 8.660 8.660

−1.768 −1.768 0.884 −7.955 0.884 −0.884 −6.187 −1.768 5.303 −6.187 −0.884 −0.884

−3.062 3.062 0.000 1.021 4.082 −5.103 −4.082 5.103 −1.021 0.000 3.062 −3.062

0.000 0.000 0.000 2.887 2.887 2.887 5.774 5.774 5.774 8.660 8.660 8.660

−1.768 −1.768 0.880 −7.969 0.879 −0.867 −6.154 −1.845 5.334 −6.155 −0.867 −0.923

−3.062 3.099 −0.089 1.003 4.059 −5.101 −4.109 5.130 −1.000 0.016 3.053 −3.070

0.000 0.000 0.000 2.913 2.989 2.895 5.767 5.689 5.744 8.683 8.661 8.671

−1.768 −1.768 0.847 −7.964 0.978 −0.861 −6.217 −1.846 5.337 −6.219 −0.866 −0.967

−3.062 3.062 −0.038 1.005 4.073 −5.116 −4.138 5.103 −1.002 0.009 2.981 −3.142

0.000 0.000 0.000 2.943 3.097 2.966 5.770 5.773 5.746 8.727 8.834 8.761

tions caused by dt , respectively. All the calculations are performed on a desktop workstation with Intel® XeonTM E3-1226 3.3 GHz processor and 8 GB memory. As can be seen from Table 3, the proposed method has obvious advantages in terms of computational efficiency and accuracy. 7.1.3. Path tracing for the large deformation When k = 1/100, the tensegrity tower will undergo a large deformation from x0 to xt . Note that xt is infeasible because no zero eigenvalue exists for its state matrix AT A. Referring to s = 1 at x0 and taking all elements to be active members, form finding can then be performed for xt using Eqs. (32) and (33) by correcting the minimal eigenvalue λmin of AT A to zero. Thus, the tensegrity configuration nearest to xt is obtained and denoted xm . The coordinates of two typical joints in each module at xt and xm are listed in Table 4. It can be seen that their deviations are small in general. The unique modes vb of selfstress state are illustrated in Fig. 7 for configurations xm and x0 .

Fig. 7 reveals that the components of both vb can satisfy the condition that cable elements are tensile and bar elements are compressive. The large deformation path of the tensegrity tower is traced using the numerical strategy proposed in Section 5. Considering xm − x0 2 = 16.27, the step length is set at L = 1. For each step on the path of mechanism displacement, 8 and 16 optimal active members can be selected, and their corresponding feasible configurations xj are obtained by Eq. (32). The final configuration is reached if xm − xj 2 < L. However, the closest distance between the feasible configurations obtained on the path and xm is likely to be greater than L due to the limited number of active members. When xm − xj 2 can no longer decrease in successive tracking steps, this indicates that any configuration obtained in these steps should be treated as the final configuration. For this tensegrity tower, the distances between the configurations obtained in the last few steps and xm are approximately 2.1 and 2.0, corresponding to 8 and 16 active members, respectively.

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289

Fig. 15. An icosahedron tensegrity.

Fig. 16. The target configuration.

(2nd) when only correcting (a) the minimal eigenFig. 14. Variations in λmin and λmin value and (b) the second minimal eigenvalue of the state matrix at the target configuration xt to zero.

The selected active members in each step are marked in black in their corresponding grids in Fig. 8 and arranged from largest to smallest according to their counted frequencies. The 8 or 16 members with higher frequencies are finally taken as the active members for the entire kinematic path of the tensegrity. The shortest path of mechanism displacements is further traced intensively by reducing the step length L to 0.2. The coordinates of the typical joints in the final configurations are listed in Table 4. The variations in the x- and z-coordinates of joints B7 and D7 at all feasible configurations on the path are illustrated in Figs. 9(a)

and 9(b), respectively. To describe the relative distance between the obtained feasible configuration xj and xm , xt and xf in ρ are replaced by xm and xj , respectively. The variations in ρ at each step corresponding to the 8 and 16 selected active members are illustrated in Fig. 10. It can be seen that the kinematic paths driven by the 8 and 16 selected active members almost coincide in the first 60 steps, indicating that the motion of the tensegrity tower at this stage is mainly driven by the same 8 active members. On the subsequent paths, the distance from xj to xm is reduced when more active members are adopted. Obviously, it is mainly due to the contribution of the other 8 active members. It can be determined that ρ = 22.1% and 17.2% for the final configurations corresponding to the 8 and 16 selected active members, respectively, which are shown in Fig. 11. For the kinematic path driven by the 8 selected active members, the matrix FT Um in Eq. (15) is established for each configuration obtained, and its minimum eigenvalue λfmin is calculated. It can be found in Fig. 12 that the eigenvalues λfmin corresponding to all the configurations are greater than zero, indicating that the

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all obtained configurations are illustrated in Fig. 13. Since they are all greater than zero, the tower cannot maintain structural stability on the path due to the lack of prestress and kinematic indeterminacy, i.e., s = 0 and m = 2. (2) Correcting the second minimal eigenvalue to zero Adopting all elements as active members, the feasible (prestressable) configuration xm near xt is obtained by correcting the minimum eigenvalues λmin of AT A to zero using the numerical strategy put forward in Section 5. Fig. 14(a) shows the variation in (2nd ) λmin and the second minimum eigenvalue λmin of AT A for all configurations obtained in the iterative process. Fig. 14(a) reveals that λmin drops below the given tolerance of 10-12 only after 17 itera(2nd ) tion steps, and λmin changes very little. To make the tower pre-

(2nd ) stressable, one can try correcting only λmin to zero. The variation in the two eigenvalues in this case is also shown in Fig. 14(b). It can be seen that the two eigenvalues generally decrease in an os(2nd ) cillating manner as the iteration step grows. However, both λmin -12 and λmin have barely dropped below 10 after 500 iteration steps (2nd ) and cannot converge. Moreover, λmin is always smaller than λmin during the iteration. This indicates that the feasible configuration xm depends only on the effect of the structural deformation on the minimum eigenvalue λmin of the state matrix.

7.2. An icosahedron tensegrity 7.2.1. Initial and target configurations An icosahedron tensegrity with 6 bars and 24 cables is employed as another numerical example to investigate the validity of the proposed method. At the initial configuration x0 shown √ in Fig. 15, the length of all bars is 10 and that of cables is 5 6/2. The six bars are divided into three groups, and two bars in each group are parallel to each other with a distance of a = 5. The numbers of 12 joints are also shown in Fig. 15, and each element is labeled with the numbers of its two end joints. The coordinates of all joints at x0 are listed in Table 5. Joints N1, N2, N3 on the bottom surface of the icosahedron are located on the ground. Joints N1 and N2 are constrained to the ground, and joint N3 can only move horizontally before rolling. By the length actuation of active members, the tensegrity is expected to deform to a critical configuration of rolling with its center of gravity just above the edge N1N2 of the bottom support surface. The mass of bars is uniformly distributed, while that of cables is ignored. When joints N3, N4, N5, N6, N7, N10, N11, N12 move simultaneously by 2.652 in the negative direction of the x-axis, a target configuration xt (see Fig. 16) is created with the center of gravity directly above the edge N1N2. However, xt is an unprestressable configuration because λmin of AT A is 3.9 × 10-4 . The feasible configuration xm closest to xt can be obtained using Eqs. (32) and (33). The coordinates of all the joints at xt and xm are listed in Table 5.

Fig. 17. The 12 active members selected in each step on the kinematic path using (a) the proposed method and (b) the genetic algorithm.

tensegrity tower can remain stable throughout the entire kinematic path. 7.1.4. Discussion (1) The necessity of maintaining the zero eigenvalue If the minimal eigenvalue λmin of AT A is not corrected to zero, the path of mechanism displacement of this tensegrity tower can also be traced by ignoring ∗ in Eq. (34). The eigenvalues λmin of

7.2.2. Selection of active members The path of mechanism displacement from x0 to xm is also traced using the numerical strategy proposed in Section 5. The step length L is set to 0.5 considering xm − x0 2 = 7.60. 12 of the 24 cables are selected as active members in each step, and the corresponding feasible configuration xj is obtained by Eq. (32). It takes 16 steps to reach the final configuration that satisfies xm − xj 2 < L, and each feasible configuration on the path is determined to be stable using Eq. (15). As shown in Fig. 17(a), the selected active members are marked in black in their corresponding grids. The first 12 elements with higher frequencies are accepted as the active members for the entire kinematic path. The shortest path of mechanism displacement is traced again by reducing the step length L

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Fig. 18. The obtained final configuration. Table 6 Computational time corresponding to the two methods. Methods

Proposed method

Genetic algorithm

Computational time (s)

0.29

131.40

291

residuals of the zero eigenvalues of AT A are corrected, the iterative strategy is actually a type of form-finding method for tensegrities. A one-by-one exclusion method is proposed in this paper for the optimal selection of active members when their number is limited. The advantage of this method in computational efficiency depends on the proposed strategy of quickly constructing the basis for the row space of the compatibility matrix consisting of passive members. Therefore, it is convenient to obtain the deviation between the current direction of mechanism displacement and the direction towards the target configuration after an active member is converted into a passive one. This method is an implicit enumeration method, and there is no guarantee that the optimal solution will be obtained. However, the results of the illustrative example show that the selected active members have obvious optimization characteristics from the perspective of engineering. It should be noted that Eq. (15) is a loose constraint for solving the kinematic path of mechanism displacement. However, the stability of the tensegrity on the entire path can be determined by analysing the variation in the minimum eigenvalue of the quadratic matrix in Eq. (15). If the minimum eigenvalue becomes zero or negative, Eq. (15) will turn into a constraint condition for the kinematic analysis. However, this topic cannot be addressed in this paper. Declaration of Competing Interest None.

to 0.2. The obtained final configuration xf is shown in Fig. 18, and the joint coordinates are also listed in Table 5. The genetic algorithm with the same fitness function and parameter values as in the first example is also employed to select 12 active members for each step, and the results are illustrated in Fig 17(b). The 12 elements with higher frequencies are the same as those selected using the proposed method, although a few elements are slightly different in frequency. Corresponding to the two methods, the computational time for the 16 steps of active member selection are listed in Table 6. It reveals that the computational efficiency of the proposed method is significantly higher than that of the genetic algorithm. 8. Conclusions Tensegrities are infinitesimal mechanism stiffened by prestress. The mechanism displacement of a tensegrity is driven not by external action but by the length actuation of internal active members. In contrast with conventional mechanisms, the kinematics of tensegrities must also maintain the characteristics of prestressability and infinitesimal mechanisms, which are mathematically determined by the zero eigenvalues of the state matrix AT A. The zero eigenvalues of AT A are required to remain unchanged on the path of mechanism displacement to ensure structural static stability. In this paper, the sensitivity matrix between the eigenvalues of AT A and the joint displacements is given. It is found that the displacement causes only second- and higher-order change to the zero eigenvalues. A numerical method is proposed in this paper to trace the shortest path of mechanism displacement for a tensegrity moving towards a target configuration when the active members are prescribed. The mechanism displacement direction, which is closest to the direction from the current configuration to the target configuration, and its deviation can be calculated based on arbitrary bases of the null and row spaces of the compatibility matrix consisting of passive members. A numerical iterative strategy is suggested to eliminate the non-zero residuals of both the elongations of passive members and the zero eigenvalues of AT A. If all the elements are regarded as extensible active members and only the non-zero

CRediT authorship contribution statement Dexi Zhu: Conceptualization, Formal analysis, Methodology, Software, Visualization, Writing - original draft. Hua Deng: Funding acquisition, Supervision, Conceptualization, Writing - review & editing. Xiaoshun Wu: Data curation, Writing - review & editing. Acknowledgement The work was supported by the National Natural Science Foundation of China (Grant number 51878599). References Aldrich, J.B., Skelton, R.E., 2003. Control synthesis for a class of light and agile robotic tensegrity structures. In: Proceedings of the 2003 American Control Conference, Denver, CO, USA, pp. 5245–5251. doi:10.1109/ACC.2003.1242560. Calladine, C.R., 1978. Buckminster fuller’s tensegrity structures and clerk maxwell’s rules for the construction of stiff frames. Int. J. Solids Struct. 14, 161–172. doi:10. 1016/0 020-7683(78)90 052-5. Calladine, C.R., Pellegrino, S., 1991. First-order infinitesimal mechanisms. Int. J. Solids Struct 27 (4), 505–515. doi:10.1016/0020-7683(91)90137-5. Chen, L., Kim, K., Tang, E., Li, K., House, R., Zhu, E.L., Fountain, K., Agogino, A.M., Agogino, A., Sunspiral, V., Jung, E., 2017. Soft spherical tensegrity robot design using rod-centered actuation and control. ASME. J. Mechanisms Robotics. 9 (2), 025001. doi:10.1115/1.4036014. Chen, Y., Feng, J., Wu, Y., 2012. Prestress stability of pin-jointed assemblies using ant colony systems. Mech. Res. Commun. 41, 30–36. doi:10.1016/j.mechrescom.2012. 02.004. Deng, H., Kwan, A.S.K., 2005. Unified classification of stability of pin-jointed bar assemblies. Int. J. Solids Struct. 42 (15), 4393–4413. doi:10.1016/j.ijsolstr.2005.01. 009. Djouadi, S., Motro, R., Pons, J.C., Crosnier, B., 1998. Active control of tensegrity systems. J. Aerospace Eng. 11 (2), 37–44. doi:10.1061/(ASCE)0893-1321(1998)11: 2(37). Geiger, D.H., Stefaniuk, A., Chen, D., 1986. The design and construction of two cable domes for the korean olympics shells, membranes and space frames. In: Proceedings IASS Symposium, 2. Osaka, Japan, pp. 265–272. Golub, G.H., Van Loan, C.F., 1996. Matrix Computations, Third ed. Johns Hopkins University Press, Baltimore. Hunt, K.H., 1978. Kinematic Geometry of Mechanisms. Clarendon Press, Oxford. Kanno, Y., 2013. Exploring new tensegrity structures via mixed integer programming. Struct. Multidisc. Optim. 48, 95–114. doi:10.10 07/s0 0158-012-0881-6. Koohestani, K., 2012. Form-finding of tensegrity structures via genetic algorithm. Int. J. Solids Struct. 49, 739–747. doi:10.1016/j.ijsolstr.2011.11.015.

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