Muhammad Basit Chandio, Ani Luo, Yaohui Li, Sanaullah Khushak and Asif Raza

(College of Mechanical & Electrical Engineering, Harbin Engineering University, Harbin 150001, China)

Abstract: Tensegrity structures have identical members in an orientation that have correlated dynamics under external force. To study this interdependent dynamics in different members in compression and expansion processes, it is vital to analyze the dynamics of the whole structure. In this study, six bar tensegrity structure was studied under compression and expansion, and interdependent movement of different members of the structure in both processes was obtained. First, the relationship between external force and members force densities was analytically developed based on the assumption that each bar moves with the same distance when an external force is applied on the six bar tensegrity ball structure along one plane that either compresses or expands the structure. Then, two individual simulations were carried out to analyze the movement of each bar in compression and expansion under the effect of external force, and elongation in all strings was studied in both processes. Finally, comparative dynamic study of different members in compression and expansion of the structure with the effect of external force was performed, which were categorized according to dynamic symmetry.

Keywords: tensegrity; compression and expansion; interdependent dynamics; dynamic symmetry

1 Introduction

Tensegrity structures are composed of flexible strings and rigid bars[1-3]that deliver any external force or torque to internal member force in the form of torsional or compressional forces[4-5]. Strings in tensegrity structures withstand the torsional internal force that always directs along the strings axis, while bars experience a compressional internal force that always directs along the bars axis[6-7]. Tensegrity structures possess a property which assures that the shape of the tensegrity can be changed with little changes in the potential energy of the structure[8-9]. Since six bar tensegrity ball structure is symmetric in motion in hexagonal packing, nodes and members can be divided into categories by the same dynamics[10-11]. With the properties of being deployable and tunable, tensegrity structures can easily change shapes under the provision of some external force or torque, therefore they are considered the most suitable to be utilized in tough environmental conditions and for space exploring missions[8,12-13]. Six bar tensegrity ball robots can be packed, separated, and launched easily without any hazard to the payload, and thus have been applied experimentally for the exploration of other planets by NASA[14-16]. Zhang et al.[17]conducted drop tests of three different modular tensegrity lattice systems of a six bar tensegrity ball robot to analyze the impact of resistance in landing. They found that lattice is not enough as a structural solution that strut material should also be considered and the landing of the robot on two vertical bars is more effective in protecting the payload. The investigation of symmetric motion in different members of the structure in compression and expansion is essential for packing, landing, rolling, and moving the six bar ball tensegrity structure for exploring planetary missions. However, there is no study that has demonstrated the dynamic symmetry in different members of six bar tensegrity structure and the effect of external force on structure in compression and expansion processes.

In this paper, the dynamic symmetry of compression and expansion processes was analyzed in six bar tensegrity ball structure with the application of external force which changes the positions of members and nodes and deforms the structure. A comparative analysis of compression and expansion processes was performed to assure the processes following the motion laws and the deformation of the structure. The paper aims to investigate different members (either flexible or rigid) that obey same laws of motion and experience same movement with the study of deformation velocity, displacement, momentum, and deformation in compression and expansion. Direct and indirect effects of external force on different members of six bar tensegrity structure were analyzed. The relationship between external force and force densities of different members in compression and expansion was also formulated in this study on an assumption basis.

2 Defining the Tensegrity Structure

Six bar tensegrity ball structure is a highly symmetrical structure containing 12 rigid compressive bars and 24 flexible strings in tension[18](Fig.1). Each bar length isLband displayed by thick lines, and each string length isLsshown by thin lines. All parallel bars have the same distanceLd.

Fig.1 Six bar tensegrity ball structure

The six bar tensegrity ball structure has 12 nodes and a nodeni, which can be represented in Cartesian coordinates as

Node matrix can be written by arranging all nodes along columns as

N=[n1n2...n12]3×12

All members of the structure are connected by two nodes and shape the structure as a self-balanced structure. Every member can be described by two of end nodes that it is linked with. Here, members are represented by node vectors. There are six bars in the structure (Fig.2(a)), and barbi(ibar vector) can be described by a node vector that starts fromnjand ends atnkas follows:

where,i∈(1,…,6) andk,j∈(1,…,12).

Bar matrix can be described by arranging bars along columns as

The structure has 24 flexible strings (Fig.2(b)), and stringsi(istring vector) can be described by a node vector that starts fromnpand ends atnqas follows:

where,i∈(1,…,24) andp,q∈(1,…,12).

String matrix can be defined by arranging all strings along columns as

Fig.2 Strings and bars in six bar tensegrity ball structure

3 Relationship between External Force and Force Densities

An analytical approach was adopted to find out the relationship between external force and members force densities in compression and expansion of the structure.

3.1 Relationship between External Force and Force Densities in Compression

Fig.3 Compressing the structure by external force along one plane

(1)

Fig.4(a) shows the structure in a static position when there is no external force, and Fig.4(b) presents the compressed structure when an external forceWacts along the inward direction of the structure. To study the effect of the external force on the structure, node 1 (n1) was considered in calculation. Therefore, all forces acting on the node are

Here note that equal compression in all directions has been assumed, so force densityγ′in all strings is the same for this analysis and bar force density is expressed byλ′as

Fig.4 External force compressing the structure

The equation in matrix form defines three axes of coordinates as

Then, forces along thex-axis are

0+λ′(0.5Lb-(-0.5Lb))+γ′(0+0+

Forces along they-axis are

Therefore, the following equation is obtained:

W=-4γ′δd

(2)

Since there is no bar force along they-axis, Eq.(2) of external force is obtained which compresses the structure in terms of string force density.

Forces along thez-axis are

0.5Lb-0.5Lb-4×0)=0

There is no effect of force along thez-axis direction, so forces are added along thex-axis and they-axis directions as follows:

∵Lb=2Ld′+4δd

4γ′δd=0

Therefore,

(3)

It represents the relationship between bar force density and string force density in compression.

3.2 Relationship between External Force and Force Densities in Expansion

Fig.5 External force expanding the structure

Fig.6(a) shows the structure in a static position when there is no external force, and Fig.6(b) shows the changes in the structure under the application of external forceWthat expands the structure. To study the effect of this external force on the structure in an outward direction, node 1 (n1) was analyzed in the calculation, and all forces acting on node are

Here note that equal expansion has been assumed, so force densityγ′in all strings is the same for our analysis and bar force density can be expressed byλ′as

This equation in matrix form that defines three axes of coordinates can be written as

Fig.6 External force expanding the structure

Put all the values of the coordinates of nodes into

So forces along thex-axis are

Forces along they-axis are

Therefore,

W=4γ′δd

(4)

There is no bar force along they-axis, so Eq.(4) of external force is applied in terms of string force density.

Forces alongz-axis are

Since there is no effect of force along thez-axis direction, all forces are added to thex-axis and they-axis directions as follows:

4γ′δd=0

Thus the following equation is obtained:

(5)

It demonstrates the relation between string force density and bar force density in expansion.

4 Simulation Model

Two individual models were established for analyzing the structure in compression and expansion processes, and the parameters are listed in Table 1.

Table 1 Parameters of the simulation model

4.1 Simulation for Compressing the Structure

A simulation work was performed in ADAMS for analyzing the motion of all bars and the elongation in all strings when same external forceWacts on all four nodes of two parallel bars lying on one plane and compresses the structure (Fig.3).

All strings have been replaced here by spiral springs and all bars have been displayed by solid lines (Fig.7(a)). External forceWwas applied to act on barsb1andb2onxoyplane alongy-axis direction, which squeezed the structure by relocating all nodes and decreased the distance between parallel barsb1andb2lying onxoyplane,b3andb4onyozplane, andb5andb6onzoxplane (Fig.7(b)).

Fig.7 Simulation model for compression analysis

4.1.1Barsmomentanalysis

All bars in the structure moved inwards, squeezing the structure and decreasing the distance between two parallel bars. All parallel bars moved with the same distance in the opposite direction to each other and decreased the distance betweenLd. When an external forceWwas applied on two parallel barsb1andb2along they-axis, it was observed that different movement in bars lay on different planes while same and opposite movement in two parallel bars lay on the same plane. Parallel barsb1andb2lying onxoyplane experienced more movement and more decrease in the distance because of the direct effect of force, compared with those of parallel barsb3andb4onyozandb5andb6onzoxplanes. Since parallel barsb3andb4received more indirect effect from the external force in compression, they experienced more movement and decrease in distance compared with parallel barsb5andb6(Fig.8).

4.1.2Elongationinstringlength

When an external force is applied on the structure in inward direction, it squeezes the whole structure and causes movement in all bars, resulting in the elongation of all strings in length. The 24 strings can be divided into three groups of 8 strings as per experiencing the same elongation in their length. All these 8 strings in each group which connect with bars that lie on two different planes had same elongation and deformation velocity (Fig.9).

Fig.9 Elongation in length of strings in compression

Fig.2(b) and Fig.7(a) show the details of all strings. Stringss1,s2,s5,s6,s9,s10,s13,s14, (categorized in Group I) that connect with barsb1andb2, andb3andb4, had the same elongation in length and experienced deformation with same deformation velocity. Stringss3,s4,s7,s8,s11,s12,s15,s16(categorized in Group II)that connect with barsb5andb6, andb1andb2had the same elongation in length and deformed with same deformation velocity. All other 8 stringss17,s18,s19,s20,s21,s22,s23,s24(categorized in Group III) connecting with barsb3andb4andb5andb6had the same elongation in length and received deformation with same deformation velocity.

The strings in Group I experienced more elongation than all other strings, and those in Group III had more elongation than those in Group II in compressing the structure.

4.2 Simulation for Expanding the Structure

A similar simulation work was performed in ADAMS for analyzing the motion of all bars and the elongation in all strings when same external forceWacts on all four nodes of two parallel bars lying on one plane and expands the structure (Fig.5).

All strings have been replaced here by spiral springs and all bars have been displayed by solid lines (Fig.10(a)). External forceWwas applied on barsb1andb2onxoyplane along they-axis direction, which expanded the structure by relocating all nodes and increased the distance between parallel barsb1andb2lying onxoyplane,b3andb4onyozplane, andb5andb6onzoxplane (Fig.10(b)).

Fig.10 Simulation model for expansion analysis

4.2.1Barsmomentanalysis

All bars in the structure moved outwards, expanding the structure and increasing the distance between two parallel bars, when an external forceWwas applied along one plane on the structure. All two parallel bars in one set of two bars moved with the same distance in the opposite direction to each other, leading to the increase in distance between these bars. When an external force was applied on two parallel barsb1andb2along they-axis direction, it was found that different movement in bars lay on different planes whereas same and opposite movement in all two parallel bars lay on the same plane. Parallel barsb1andb2lying onxoyplane experienced more movement and more increase in distance, due to the direct effect of force compared with the other two sets of parallel bars (i.e.,b3andb4lying onyozplane andb5andb6onzoxplane). Parallel barsb5andb6received more indirect effect of the external force in expansion, so they experienced more movement and increase in distance than parallel barsb3andb4(Fig.11).

Fig.11 Movement analysis in all six bars in expansion

4.2.2Elongationinstring

When an external force is applied to the structure in outward direction, it expands the whole structure and causes the movement in all bars, leading to the elongation of all strings in length. The 24 strings can be divided into three groups of 8 strings as per experiencing the same elongation in their length. All these 8 strings in each group which connect with bars lying on two different planes had same elongation and deformation velocity (Fig.12).

Fig.12 Elongation in length of strings in expansion

For details of all strings refer to Fig.2(b) and Fig.10(b). All 8 strings1,s2,s5,s6,s9,s10,s13,s14(categorized in Group I) that connect with barsb1andb2andb3andb4had the same elongation in length and deformed with the same deformation velocity. Stringss3,s4,s7,s8,s11,s12,s15,s16(categorized in Group II) that connect with barsb5andb6andb1andb2had same elongation in length and experienced deformation by same velocity. While all other 8 stringss17,s18,s19,s20,s21,s22,s23,s24(categorized in Group III) that connect with barsb3andb4andb5andb6had the same elongation in length and deformed with the same deformation velocity.

Strings in Group II experienced more elongation than all others, and those in Group III had more elongation than those in Group I in the expansion of the structure.

4.3 Comparison of Structure in Compression and Expansion

In this section, all the graphs mentioned above were zoomed in to compare with the results in compression and expansion processes of the structure regarding to dynamic symmetry.

4.3.1Comparisoninbarsmovement

Since compression and expansion are opposite processes, the results obtained are also opposite as displayed in Table 2.

Two parallel barsb1andb2lying on the same plane had the same but opposite movement in compressing and expanding the structure (Table 2), which was the same for all other parallel bars that lay on the same plane. Furthermore, other two non-parallel barsb3andb5lying on different planes had the same movement but opposite in direction in compression and expansion. Same situation occurred for non-parallel barsb4andb6lying on different planes (Fig.13).

Table 2 Movement in all bars in compression and expansion (mm)

Fig.13 Movement analysis in all six bars in compression and expansion processes

4.3.2Comparisonofelongationinstrings

Since expansion is the inverse process of compression, when same external force was applied on a structure with same parameters in compression and expansion, noteworthy symmetrical and reversed results in the elongation of different strings were obtained. Strings (Group I) that experienced maximum elongation in compression had a minimum elongation in the expansion of the structure, whereas strings (Group II) that experienced minimum elongation in compression had a maximum elongation in the expansion, and those (Group III) experienced medium elongation in compression had the same elongation in expansion (Table 3).

Stringss1,s2,s5,s6,s9,s10,s13,s14(Group I) in compression and stringss3,s4,s7,s8,s11,s12,s15,s16(Group II) in expansion had equal and maximum elongation. Strings (Group I) in expansion and strings (Group II) in compression had same and minimum elongation. Stringss17,s18,s19,s20,s21,s22,s23,s24(Group III) had same and medium elongation in compression and expansion (Fig.14).

Table 3 Elongation in strings during compression and expansion (mm)

Fig.14 Elongation in strings

5 Conclusions

In this paper, direct and indirect effects of external force on six bar tensegrity ball structure were studied in compression and expansion processes. The dynamic symmetry of the structure in both processes was analyzed, in which similar and reversed results were obtained. Unequal compression and expansion of the structure were observed when an external force was applied along one plane on two parallel bars by simulation. The parallel bars under direct effect of external force had maximum movement in compression and expansion compared with the other two sets which experienced indirect effect of force. Therefore, the assumption for the analytical calculations to develop the relationship between member force densities and the applied external force can be modified by the obtained correct simulation results. All strings can be divided into three groups as per the same dynamics, elongation, and deformation velocity.