Modal analysis method for tensegrity structures via stiffness transformation from node space to task space

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Modal analysis method for tensegrity structures via stiffness transformation from node space to task space ⁎

Xin Li, Jingfeng He , Mantian Li, Hongzhou Jiang, Yunqi Huang School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Tensegrity structures Stiffness transformation Modal analysis Task-space stiffness

A tensegrity structure is a type of hybrid soft-rigid system, whose high compliance is prone to induce an oscillatory motion. The oscillatory characteristics can be exploited to control a tensegrity structure efficiently, and modal analysis is beneficial in guiding the exploration of the oscillatory characteristics of this type of structures. We derive node-based modes of tensegrity structures using the finite element method. By freezing the substructures as a rigid body, the node-based stiffness is converted into a task-space stiffness, during which the elasticity within the substructures is filtered out from the final modal analysis results. Compared with the nodebased modal analysis results, the transformation method proposed in this paper significantly reduces the mode dimension. Concurrently, the low-order modes almost remain unchanged when the stiffness of the substructures is much higher than that of the interconnecting cables. We select two tensegrity structures as examples to demonstrate our method.

1. Introduction Tensegrity structures are composed of compressive and tensile members, and their stability is obtained by applying a pre-stress [1–4]. The tensegrity structure is a type of hybrid soft-rigid system. Its high compliance induces an oscillatory motion and non-linearities, which are generally avoided in traditional control strategies [5,6]. However, some researchers seek to exploit the oscillatory characteristics efficiently for controlling tensegrity structures [7–12]. Modal analysis is a useful method to well comprehend these oscillatory characteristics. Bel Hadj Ali et al. analyzed the modes of tensegrity structures and shifted their natural frequencies away from excitation by modifying the self-stress level [13]. Bossens et al. used a tensegrity stage as an example to analyze the modes, and they compared the theoretical results with the experimental results [14]. For an X-frame tensegrity structure, Ashwear et al. analyzed the relationship between the lowest modes and level of the pre-stress [15]. Stiffness analysis is a critical part of the analysis of the modes of tensegrity structures. For tensegrity systems, Nagase et al. derived the stiffness expression with two parts: stiffness of rods and cables, respectively [16]. Zhang et al. represented the stiffness of tensegrity structures using both passive and active stiffness terms [17]. The stiffness matrix of tensegrity structures can also be obtained by the finite element method [13,18]. Guest introduced a new derivation for the stiffness of tensegrity

⁎

structures. However, for an elastic system, Guest proposed that the stiffness matrix needed to be modified [19,20]. All the above stiffness analyses are based on the node stiffness expression. For a complex tensegrity structure, there will be lots of nodes on it and the dimension of the node-based stiffness matrix is high. The node-based stiffness matrix is inconvenient to use for a few complex tensegrity structures. Thus, the objective of this paper is to propose a method of stiffness dimension-reduction for modal analysis of tensegrity structures. When we research on the stiffness and mode of a complex tensegrity structure, we don’t always care about the stiffness of all nodes and high-order modes of the structure. In particular, some parts of the structure are more rigid than that of the other parts. These rigid parts have little effect on the low-order modes of the whole structure. For example, the musculoskeletal systems of animals or humans are generally regarded as hierarchical tensegrity structures in which the bones serving as the substructures are more rigid than other parts [21–25]. Although there may be numerous connection nodes on bones, we are generally concerned with their task-space stiffness rather than their stiffness in the node space when the bones are being viewed as rigid bodies. In this paper, the task-space stiffness of tensegrity structures is proposed. We present a method of transforming the node-based stiffness into taskspace for some substructures with a high rigidity and remaining other parts unchanged. In this process, the dimension of the stiffness matrix is reduced. To demonstrate our method, we use tensegrity spines as

Corresponding author. E-mail address: [email protected] (J. He).

https://doi.org/10.1016/j.engstruct.2019.109881 Received 3 September 2019; Received in revised form 31 October 2019; Accepted 1 November 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Xin Li, et al., Engineering Structures, https://doi.org/10.1016/j.engstruct.2019.109881

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examples. In Section 2, we describe the structure of tensegrity spines. In Section 3, we derive the stiffness and mass matrices of tensegrity structures by the finite element method. In addition, in this section, different stiffness calculation methods of tensegrity structures are compared. In Section 4, we propose a stiffness transformation method converting a node-based stiffness expression into a task-space one. In Section 5, the modes of two different tensegrity spines are analyzed, and the mode shapes are illustrated. In Section 6, the characteristics of the stiffness transformation method and impact of the different element stiffness on the structure modes are discussed. 2. Structural description For a d-dimensional tensegrity structure with s cables as tensile members, r rods act as compressive members, and a total of n nodes, m (m = s + r ) members, and a connectivity matrix can be used to describe the topology of the tensegrity structure. Let there be p substructures in the tensegrity structure. Generally, the value of d is 3. If a member k connects nodes i and j (i < j ) , then the ith and jth elements of the kth row of C(∈m × n) are set as 1 and −1, respectively [17]. The expression is as follows:

⎧ 1 for p = i C(k, p) = − 1 for p = j ⎨ ⎩ 0 otherwise

Fig. 2. Relationship of the coordinate transformation.

⎡1 0 0⎤ where I0 = ⎢ 0 0 0 ⎥, L, E , and A are the length, Young’s modulus, and ⎣0 0 0⎦ cross-sectional area of the members. The stiffness matrix is expressed in a local coordinate system, {xL , yL , zL} , where xL is along the member axis. Tensegrity structures are generally composed of multiple rods and cables. Each rod and cable are viewed as truss elements to compute their stiffness. The local stiffness matrix of each member can be expressed by Eq. (2). The global stiffness matrix can be obtained by a coordinate transformation. The transformation relationship between the local coordinates, {xL , yL , zL} , and global coordinates, {x , y, z } , are illustrated in Fig. 2. The transformation matrix, T (i) (∈6 × 6) , of the ith member is denoted as follows [26]:

(1)

In this study, we performed a modal analysis for two types of tensegrity spines with interconnecting tetrahedron substructures. In Fig. 1(a), a truss tetrahedron acts as a substructure, each substructure is composed of six interconnected rods. In Fig. 1(b), a tensegrity tetrahedron is a substructure. Each substructure contains four rods and fourteen cables. If the tetrahedron substructures in Fig. 1(a) are rigid and stiffness of cables unchanged, then it is equivalent to the one in Fig. 1(c). Each substructure is a rigid body composed of four rods.

T T T (i) = ⎡ 11 12 ⎤ T ⎣ 21 T22 ⎦

3. Node-based stiffness and mass matrices

where T12 = T21 = 0 .

3.1. Node-based stiffness matrix

⎡l ⎢ T11 = T22 = ⎢ m ⎢ ⎢h ⎣

We can obtain the stiffness and mass matrices of a truss element [18] by the finite element method. The local passive axial stiffness matrix, k(i) (∈6 × 6) , of the ith member is expressed as follows:

k(i) =

EA ⎡ I0 − I0 ⎤ L ⎣− I0 I0 ⎦

(3)

l = cosα =

(2)

Fig. 1. Tensegrity spine. 2

−lm

−h

l2 + h2

l2

+

h2

−mh l2 + h2

⎤ ⎥ 0 ⎥ ⎥ l ⎥ 2 2 l +h ⎦

l2 + h2

(4)

xj − xi lij

(5)

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m = cosβ =

h = cosγ =

yj − yi lij

where the mass matrix is denoted in a local coordinate system, {xL , yL , zL} . The total mass matrix, M(∈3n × 3n) , of the structure is denoted as follows:

(6)

zj − z i lij

(7)

M=

The global passive stiffness matrix(the stiffness matrix for an element referred to as the global axes) k(pi) (∈6 × 6) of the ith member is expressed as follows [26]:

k(pi)

T (i) k(i) (T (i) )T

=

(8)

∑ L¯ (i) k(pi) (L¯ (i) )T

In this part, we froze substructures with a high rigidity as rigid bodies, following which applied the transformation method between the node-space stiffness and task-space stiffness. We use an example (Fig. 1(a)) to introduce the concept of freezing substructures. For the structure shown in Fig. 1(a), we regarded six interconnecting rigid rods as one rigid substructure with four nodes. Each rigid substructure is composed of more than one rod(n > 2p ). Task space is used to represent the position and orientation of the robotic end effector. In this paper, we extend “task space” from the robotic area to describe the position and orientation of substructures in tensegrity structures. Each substructure is regarded as an end effector. The position and orientation of all substructures can be represented in task space.

(9)

L¯ (i) (∈3n × 6)

is the Boolean localization matrix defined by the where connectivity matrix, and the sum extends over all the elements [18,15]. The global active stiffness matrix, k (ai) (∈6 × 6) , of the ith member is expressed as follows [13]:

T I −I ⎤ ⎡ L ⎣− I I ⎦

k (ai) =

(10) 4.1. Stiffness matrix of task space

⎡1 0 0⎤ where I = ⎢ 0 1 0 ⎥ is an identity matrix and T is the axial load. ⎣0 0 1⎦ The total active stiffness matrix, K a (∈3n × 3n) , of the tensegrity structures can be obtained by assembling the element matrices, which can be expressed as follows: Ka =

∑ L¯ (i) k a(i) (L¯ (i) )T

The relationship between the velocity of these nodes on the substructure and the task-space velocity of the substructure as a rigid body is represented as follows: i Ṅ g = t ̇g + ωg × Rig aig = t ̇g − Rig aig × ωg

(11)

We obtained the stiffness expression of the tensegrity spines using the finite element method, following which our result was compared with those of Nagase [16] and Zhang [17]. All these results agree with each other. However, these results do not consider the variation in the stiffness with the member length. For an elastic system, in fact, the stiffness of an element generally varies with its length. This effect will introduce some differences in the final stiffness matrix expression. From the derivation of the stiffness of pre-stressed frameworks by Guest in reference [20], an extra stiffness term, Km (∈3n × 3n) , is required to be added to the stiffness matrix expression, which is given by the above three methods.

Km = −AQAT

transform the local coordinates to the global coordinates. Then the velocity transformation relationship in these two spaces is represented as

ω ⎡ 1⎤ t1̇ ⎥ 1 ⎢ ̇ ⎡ N1 ⎤ ⎢⋮⎥ ⎢ ⋮ ⎥ ⎢ ωg ⎥ ⎢ i⎥ ⎢ ⎥ ⎢ Ṅ g ⎥ = JZ ⎢ t ̇ ⎥ g ⎢ ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢⋮⎥ n ⎢ Ṅ p ⎥ ⎢ ωp ⎥ ⎣ ⎦ ⎢ ṫ ⎥ ⎣ p⎦

(12)

where A(∈3n × m) is the equilibrium matrix, which describes the equilibrium relationship between the internal forces in the members, t , and applied loads at the nodes, p and At = p [19]. Q(∈m × m) is the force density matrix, which can be obtained by form-finding. Because Km is related to the pre-stress, we added it to the active stiffness matrix. Furthermore, the total active stiffness matrix, K am (∈3n × 3n) , for the entire structure is represented as

The node-based stiffness matrix, Kn

(∈3n × 3n) ,

T

Ṅ = ⎡ Ṅ 11 ⋯ Ṅ ig ⋯ Ṅ np ⎤ (∈3n × 1) ⎣ ⎦

⎡ q̇1 ⎤ ⎢⋮⎥ ⎢ ⎥ T q̇ = ⎢ q̇ g ⎥ = [ ω1 t1̇ ⋯ ωg t ġ ⋯ ωp tṗ ] (∈6p × 1) ⎢⋮⎥ ⎢ q̇ ⎥ ⎣ p⎦

is denoted as (14)

(20)

The velocity transformation matrix between the node and task spaces is denoted as follows:

The global mass matrix, m(i) (∈6 × 6) , of a truss element is represented as follows [18]:

ρAL 2I I ⎤ ⎡ 6 ⎣ I 2I ⎦

(19)

where q̇ denotes the global matrix of the velocity in the task space.

3.3. Node-based mass matrix

m(i) =

(18)

where Ṅ represents the global matrix of the node velocity.

(13)

K am = K a + Km

(17)

i where Ṅ g (∈3 × 1) is the ith node velocity on the gth substructure, t ̇g (∈3 × 1) is the translation velocity of the gth substructure in the task space, aig (∈3 × 1) is the ith local node coordinate of the gth substructure, ωg (∈3 × 1) is the angular velocity of the gth substructure in the task space, and Rig (∈3 × 3) is the ith rotation matrix of the gth substructure to

3.2. Comparison of various stiffness calculation methods

Kn = Kp + K am

(16)

4. Stiffness transformation method

The total passive stiffness matrix, Kp (∈3n × 3n) , of the tensegrity structures can be obtained by assembling the element matrices, which can be expressed as follows:

Kp =

∑ L¯ (i) m(i) (L¯ (i) )T

(15)

∂N = JZ (∈3n × 6p) ∂q

(21)

F·∂q = Fn·∂N

(22)

According to the principle of virtual work defined in Eq. (22), we 3

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can derive the transformation between the node resultant forces and task-space resultant forces as

F = JTZ Fn

(23)

The definition of the task-space stiffness is presented in Eqs. (24)–(26).

KT =

∂ (JTZ ) ∂ (JTZ Fn ) ∂ (Fn ) ∂F Fn + JTZ = = ∂q ∂q ∂q ∂q

∂Fn ∂F ∂N = n ∂q ∂N ∂q KT =

∂JT ∂ (JTZ Fn ) ∂F ∂F = Z Fn + JTZ n JZ = ∂q ∂N ∂q ∂q

(24)

(25)

(26)

(∈6p × 6p)

represents the total stiffness matrix in the task where KT space. For a tensegrity structure in equilibrium, the value of the resultant forces of the nodes, Fn , is zero. We can obtain the task-space stiffness matrix that is transformed from the node-based stiffness as follows:

KT =

∂ (JTZ Fn ) ∂F ∂F = JTZ n JZ = JTZ KnJZ = ∂q ∂N ∂q

Fig. 3. Velocities of the nodes and rods on the gth substructure.

(27)

From Eq. (27), we can derive the transformation relationship between the active nodes stiffness and active task-space stiffness as follows:

K ã = JTZ K amJZ

i+1

⎡ Ṅ g i+1 i Vgj = lTni Ṅ g − lTni Ṅ g = [ lTni − lTni ] ⎢ i Ṅ ⎢ ⎣ g

(28)

(29)

i i ⎡ skew(−R g a g ) I ⎤ ⎡ ωg ⎤ Vgj = [ lTni − lTni ] ⎢ ⎥ ṫ ⎥ g skew(−Rig+ 1aig+ 1) I ⎢ ⎦⎣ ⎦ ⎣

The task-space stiffness matrix of a tensegrity structure is expressed as

KT = K ã + Kp̃

i i ⎡ skew(−R g a g ) I ⎤ JT(j) = [ lTni − lTni ] ⎢ ⎥=0 skew(−Rig+ 1aig+ 1) I ⎦ ⎣

(36)

The transformation matrix, JT(j) , is a part of JZ . Thus, the elasticity of the elements in a substructure is filtered out in the process of the stiffness transforming from node space to task space. The final taskspace stiffness matrix is dominated by stiffness of external elements.

4.2. Filter element stiffness impact of substructures Here, we present an example (Fig. 3)) to show that the elasticity of the members of the substructure that is frozen as a rigid body will not be included in the final stiffness expression. As shown in the Fig. 3, rods and cables are denoted by different lines. Transformation vector from global coordinate, {x , y, z } , to local coordinate, {xL , yL , zL} , is represented by tg . According to Eq. (18), we can derive the velocities of the ith and (i + 1) th nodes on both ends of the corresponding rod on the gth substructure as

ωg i+1 Ṅ g = [skew(−Rig+ 1aig+ 1) I ] ⎡ ̇ ⎤ ⎢ tg ⎥ ⎣ ⎦

(35)

The transformation matrix of the jth rod on the gth rigid body can be denoted as JT(j) (∈3 × 6) , which is a zero matrix.

(30)

There are multiple nodes on a complex substructure which is composed of more than one rod (n > 2p ). Thus, the dimension of the taskspace stiffness matrix, KT (∈6p × 6p) , is lower than the dimension of the node-based stiffness matrix, Kn (∈3n × 3n) .

ωg i Ṅ g = [skew(−Rig aig ) I ] ⎡ ̇ ⎤ ⎢ tg ⎥ ⎣ ⎦

(34)

where lni denotes the identity vector along the rod direction. The transformation matrix, which is expressed in Eq. (35), between the taskspace velocity and rod velocity on the gth rigid body can be derived by combining Eqs. (33) and (34).

Similar to the active stiffness, the transformation relationship between the passive node-based stiffness and passive task-space stiffness can be derived as follows:

Kp̃ = JTZ KpJZ

⎤ ⎥ ⎥ ⎦

4.3. Mass matrix of task space We can derive Eq. (37) based on the kinetic energy principle.

1 ̇T 1 ̃ ̇ N MṄ = q̇ T Mq 2 2

(37)

Relating Eq. (37) to Eq. (21), the task-space mass matrix, M(̃ ∈6p × 6p) , can be derived as

(31)

M̃ = JTZ MJZ

(32)

With this approach, we can transform the high-dimensional nodebased mass matrix, M(̃ ∈6p × 6p) , into the low-dimensional task-space mass matrix, M(∈3n × 3n) (n > 2p ).

(38)

We stack Eqs. (31) and (32). 5. Modal analysis

i+1 ⎡ Ṅ g ⎤

i i ⎡ skew(−R g a g ) I ⎤ ⎡ ωg ⎤ ⎢ i ⎥=⎢ ⎥ t ̇g ⎥ skew(−Rig+ 1aig+ 1) I ⎢ Ṅ ⎢ ⎦⎣ ⎦ ⎣ g ⎥ ⎦ ⎣

After obtaining the stiffness and mass matrices, in this part, we analyze the modes of tensegrity structures in the node space and task space, respectively.

(33)

Velocity of the jth rod on the gth substructure can be represented as 4

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50

With the tangent stiffness matrix, Kn , and mass matrix, M, of the node space assembled, the small free undamped vibrations of the tensegrity structure around the evaluated equilibrium state can be obtained from the generalized eigenproblem [15]:

40

ωi2 Mϕi − Knϕi = 0

Frequency(Hz)

5.1. Modes in nodes space

(39)

where ωi is a natural frequency and ϕi is the corresponding eigenvector in the node space.

task space modal node space modal

30 20 10 0

5.2. Modes in task space

1

12000

ωi2̃ M̃ ϕi ̃

(40)

11000

where ωĩ is the natural frequency and ϕi ̃ is the corresponding eigenvector in the task space.

10000

Frequency(Hz)

− KT ϕi ̃ = 0

λi ϕi = JTZ ϕi ̃

30

35

40

45

50

54

9000 8000 7000

5000 4300 55

60

65

Table 1 Stiffness and density of different elements.

Rods External cables Rods

Internal cables External cables

Stiffness(

EA )(N/m) L

80

85

90

95

100

105 108

frequencies of node space and all task-space natural frequencies are shown in Fig. 4. The residual natural frequencies of node space are shown in Fig. 5. These natural frequencies are very high. As shown in Table 1, the stiffness of the rods is sufficiently high and that of the cables is much lower than that of the rods. From Figs. 4 and 5, we can see that when the stiffness of the rods is sufficiently high, all natural frequencies in the task space almost coincide with the first 54 natural frequencies in the node space. Moreover, the node-based stiffness matrix contains the elasticity of the substructures with a high rigidity. By contrast, the elasticity of the substructures is excluded from the taskspace stiffness matrix. The natural frequencies of the tensegrity structure are ranked, and there are six zero natural frequencies, which represent the rigid-body modes. Besides, we list the first five low-order nonzero natural frequencies of the node and task spaces in Table 2, and they are identical within a finite digit number. For comparison, the modes of the structures with tensegrity tetrahedrons (Fig. 1(b)) as their substructures are also analyzed. As shown in Table 1, the stiffness of the rods and internal cables of the substructures is set higher than that of the external cables interconnecting adjacent substructures. The stiffness of the rods is sufficiently high, but the stiffness of the internal cables is not as much as that of the rods. For this structure, there are 216 natural frequencies in the node space and 54 natural frequencies in the task space. From Fig. 6 we can see that the stiffness of the elements of the internal tensegrity substructures is high,

We used nine-level tensegrity spines, as shown in Fig. 1(a) and (b), as examples to analyze the modes of tensegrity structures by Eqs. (39) and (40) in the task and node spaces. In this paper, wood or bamboo is selected as material of rods, similar materials have been used to build tensegrity structures [27]. Young’s modulus and density of bamboo are 20GPa and 0.8 g/cm3 respectively [28]. Polyethylene is usually selected as the material of cables. Young’s modulus and density of this material are 0.63GPa and 0.95 g/cm3 respectively [28]. Due to the lengths and cross-section areas of rods and cables are different in one structure, the stiffness of these rods and cables is not identical. For the same value of EA, the longest rod of the substructure has the minimum stiffness. Stiffness ranges of the rods and cables are listed in the Table 1. In the first case, we analyzed the modes of the structures with truss elements, which are shown in Fig. 1(a). The numbers of nodes and rigid bodies of this structure are 36 and 9 respectively. The modal numbers of node space and task space are 108 (3n ) and 54 (6p ), respectively. The results show that there are 108 and 54 natural frequencies in the node and task spaces, respectively. The first 54 node-based natural

EA(N)

75

Fig. 5. Natural frequencies in node space.

5.4. Numerical results of modal analysis

Elements

70

Mode number

where ϕi denotes the ith eigenvector in the node space, ϕi ̃ represents the ith eigenvector in the task space, and λi is a non-zero value.

Substructures

25

node space modal

(41)

Substructures

20

6000

According to the definition of the transformation matrix in Eq. (21), the eigenvector transformation relation in different spaces can be derived as

Case2

15

Fig. 4. Comparison of the natural frequencies.

5.3. Mode shape transformation

Case1

10

Mode number

We can obtain the undamped natural frequencies and corresponding eigenvectors using the coupling stiffness matrix KT with mass matrix, M̃ , in the task space.

Cases

5

ρ (g/cm3)

1.5708e6 14.2997

0.9778e7~1.2533e7 176.5121~ 210.2904

3.5343e6 1.9792e3

3.7255e7~3.9270e7 2.0617e 4 ~7.4594e 4

0.95

14.2997

205.0366~ 297.9113

0.95

Table 2 Natural frequencies of the structure with truss tetrahedra as the substructures.

0.8 0.95

0.8

5

Mode

Frequency(Node space) (Hz)

Frequency(Task space) (Hz)

Error(% )

Mode1 Mode2 Mode3 Mode4 Mode5

2.808 2.808 5.416 6.453 6.453

2.808 2.808 5.416 6.453 6.453

0.000% 0.000% 0.000% 0.000% 0.000%

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70

Frequency(Hz)

by these elements in the node space. Further, these stiffness elements are filtered during the transformation. Thus, there is a slight difference in the natural frequencies between the node and task spaces, which is illustrated in Table 3. This difference decreases with increasing internal cables stiffness. To substantiate this statement, we will explore the influence of different internal cables stiffness on different space modes. For the structures with tensegrity substructures (Fig. 1(b)), there are seven cases for comparative tests. As mentioned above, stiffness of rods and cables in one structure is different, K e and Ki represent the stiffness ranges of external and internal cables respectively. As shown in Table 4, we set K e unchanged and gradually increase Ki (case 1 to case 7). Relative error percentage of the first five natural frequencies between the node and task spaces is calculated and depicted in Fig. 7. It can be seen that the difference of the natural frequencies between the node and task spaces decreases with increasing internal cables stiffness.

task space modal node space modal

60 50 40 30 20 10 0

1

5

10

15

20

25

30

35

40

45

50

54

Mode number Fig. 6. Comparison of the natural frequencies. Table 3 Natural frequencies of the structure with tensegrity tetrahedra as the substructures. Mode

Frequency(Node space) (Hz)

Frequency(Task space) (Hz)

Error(% )

Mode1 Mode2 Mode3 Mode4 Mode5

4.412 4.432 8.812 8.958 9.668

4.428 4.446 8.861 9.000 9.757

0.363% 0.316% 0.556% 0.469% 0.921%

5.5. Mode shape Natural frequency and corresponding eigenvectors of the tensegrity spines can be obtained by modal analysis. For the same mode, the corresponding eigenvector of the node space is the same as the vector that is transformed from the task space by Eq. (41). Based on the modal analysis of two tensegrity structures(Fig. 1(a), (b)), we plot the mode shapes in Figs. 8 and 9. As shown in Fig. 8, there are three mode shapes of the structure with truss substructures. The first one (Fig. 8(a)) is a bending mode corresponding to the lowest natural frequency. This is a “C”-like shape. The torsional mode shape corresponding to the third natural frequency is shown in the Fig. 8(b). As shown in the Fig. 8(c), there is a “S”-like bending mode shape corresponding to the fifth natural frequency. For the structure with tensegrity substructures, we plot modal shapes in Fig. 9. There are also “C”-like bending, torsional and “S”-like bending mode shapes in the figure. Comparing these two figures (Figs. 8 and 9), we find that mode shapes of these two different structures are similar. If we want to excite the tensegrity spine by vibration to achieve a specific modal shape, we can use truss or tensegrity structures as substructures of the tensegrity spine. These modal shapes may be able to help the tensegrity spines to achieve locomotion in the future.

Table 4 Stiffness ranges of external and internal cables. Cases

K e (N/m)

Ki (N/m)

Case 1

205.0366~ 297.9113 205.0366~ 297.9113 205.0366~ 297.9113

1.4896e3~5.3894e3

5

3∗(1.4896e3~5.3894e3 ) 5∗(1.4896e3~5.3894e3 )

15

205.0366~ 297.9113 205.0366~ 297.9113

10∗(1.4896e3~5.3894e3 ) 15∗(1.4896e3~5.3894e3 )

50

205.0366~ 297.9113 205.0366~ 297.9113

20∗(1.4896e3~5.3894e3 ) 50∗(1.4896e3~5.3894e3 )

100

Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

Relative error percentage

11% 9%

min(Ki )/max(K e )

25 75 250

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

6. Conclusion

7%

In this paper, the task-space stiffness of tensegrity structures is proposed. We present a method of stiffness dimension-reduction for modal analysis of tensegrity structures via stiffness transformation from node space to task space. In the process of the transformation, the impact of the stiffness elements on each corresponding substructure is filtered out from the final stiffness expression. This causes the stiffness of the cables interconnecting the substructures to dominate the final expression. When the stiffness of the elements within each substructure is sufficiently high, there is almost no impact on the low modes in the node space. Thus, there is almost no difference in the low natural frequencies between the node and task spaces, which is illustrated in Table 2. This method can be viewed as a modal filter that freezes substructures as rigid bodies, excluding the minor flexibility of the elements in the substructures and retaining the entire flexibility of the cables interconnecting the substructures. We compare the results of the modal analysis and illustrate the mode shapes of the tensegrity structures to prove the validity of the modal analysis method, which is proposed in this paper.

5% 3% 1% Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7

Fig. 7. The influence of internal cables stiffness on different space mode.

and the first 54 eigenvalues in the task space almost coincide with the eigenvalues in the node space. The remaining 162 natural frequencies of the node space are relatively higher, similar to the case shown in Fig. 5, and we do not repeat the illustration here. We list the first five low-order nonzero natural frequencies of the node and task spaces in Table 3. As shown in Table 1, when the stiffness of elements on the substructures is much higher than that of the cables interconnecting multiple substructures, but is not sufficiently high, such as for the internal cables of the tensegrity tetrahedron substructures (Fig. 1(b)), then the low-order natural frequencies are slightly affected

Declaration of Competing Interest The authors declared that there is no conflict of interest. 6

Engineering Structures xxx (xxxx) xxxx

X. Li, et al.

Fig. 8. Mode shapes of structure with truss tetrahedrons.

Fig. 9. Mode shapes of structure with tensegrity tetrahedrons.

Acknowledgments

[11] Bliss TK, Iwasaki T, Bart-Smith H. Resonance entrainment of tensegrity structures via cpg control. Automatica 2012;48(11):2791–800. [12] Khazanov M, Jocque J, Rieffel J. Evolution of locomotion on a physical tensegrity robot. Artificial life conference proceedings, vol. 14. MIT Press; 2014. p. 232–8. [13] Ali NBH, Smith I. Dynamic behavior and vibration control of a tensegrity structure. Int J Solids Struct 2010;47(9):1285–96. [14] Bossens F, De Callafon R, Skelton R. Modal analysis of a tensegrity structure–an experimental study. 2004. [15] Ashwear N, Eriksson A. Natural frequencies describe the pre-stress in tensegrity structures. Comput Struct 2014;138:162–71. [16] Nagase K, Skelton R. Minimal mass tensegrity structures. J Int Assoc Shell Spatial Struct 2014;55(1):37–48. [17] Zhang J, Ohsaki M. Adaptive force density method for form-finding problem of tensegrity structures. Int J Solids Struct 2006;43(18–19):5658–73. [18] Paultre P. Dynamics of structures. John Wiley & Sons; 2013. [19] Guest SD. The stiffness of tensegrity structures. IMA J Appl Mathe 2010;76(1):57–66. [20] Guest S. The stiffness of prestressed frameworks: a unifying approach. Int J Solids Struct 2006;43(3–4):842–54. [21] Scarr G. Biotensegrity. United Kingdom: Handspring Publishing; 2014. [22] Levin S. A suspensory system for the sacrum in pelvic mechanics: biotensegrity. Movement Stabil Lumbopelvic Pain 2007;2:229–37. [23] Scarr G. Helical tensegrity as a structural mechanism in human anatomy. Int J Osteopathic Med 2011;14(1):24–32. [24] Scarr G. A consideration of the elbow as a tensegrity structure. Int J Osteopathic Med 2012;15(2):53–65. [25] Levin SM. The importance of soft tissues for structural support of the body. SpinePhiladelphia-Hanley Belfus- 1995;9:357. [26] Logan DL. A first course in the finite element method. Cengage Learning; 2011. [27] Lessard S, Castro D, Asper W, Chopra SD, Baltaxe-Admony LB, Teodorescu M, et al. A bioinspired tensegrity manipulator with multi-dof, structurally compliant joints. 2016 IEEE/ RSJ International conference on intelligent robots and systems (IROS). IEEE; 2016. p. 5515–20. [28] Department CUE. Materials data book. Mater Courses 2003;:1–41.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Skelton RE, de Oliveira MC. Tensegrity systems vol. 1. Springer; 2009. [2] van de Wijdeven J, de Jager B. Shape change of tensegrity structures: design and control. Proceedings of the 2005, American Control Conference, 2005. IEEE; 2005. p. 2522–7. [3] Branam NJ, Arcaro V, Adeli H. A unified approach for analysis of cable and tensegrity structures using memoryless quasi-newton minimization of total strain energy. Eng Struct 2019;179:332–40. [4] Sychterz AC, Smith IF. Using dynamic measurements to detect and locate ruptured cables on a tensegrity structure. Eng Struct 2018;173:631–42. [5] Agogino AK, SunSpiral V, Atkinson D. Super Ball Bot-structures for planetary landing and exploration. 2018. [6] Iscen A, Caluwaerts K, Bruce J, Agogino A, SunSpiral V, Tumer K. Learning tensegrity locomotion using open-loop control signals and coevolutionary algorithms. Artif Life 2015;21(2):119–40. [7] Rieffel J, Mouret JB. Adaptive and resilient soft tensegrity robots. Soft Robot 2018;5(3):318–29. [8] Lakatos D, Görner M, Petit F, Dietrich A, Albu-Schäffer A. A modally adaptive control for multi-contact cyclic motions in compliantly actuated robotic systems. 2013 IEEE/RSJ International conference on intelligent robots and systems. IEEE; 2013. p. 5388–95. [9] Mirletz BT, Bhandal P, Adams RD, Agogino AK, Quinn RD, SunSpiral V. Goal-directed cpg-based control for tensegrity spines with many degrees of freedom traversing irregular terrain. Soft Robot 2015;2(4):165–76. [10] Lakatos D, Garofalo G, Petit F, Ott C, Albu-Schäffer A. Modal limit cycle control for variable stiffness actuated robots. 2013 IEEE international conference on robotics and automation. IEE; 2013. p. 4934–41.

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