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Deformation of a Tensegrity Structure Using Tendon Actuators
Yusuke Hagiwara*, Mitsushige Oda**
*Department of Mechanical and Aerospace Engineering, Tokyo Institute of Technology, Japan e-mail: [email protected]
**Aerospace Research and Development Diractorate, JAXA, Japan e-mail: [email protected]
Abstract The tensegrity structure is made of some rigid bars
and flexible wires, and can maintain its shape when the members of the structure are balanced. The tensegrity structure can be deformed by changing the lengths of its members. However, if each wire has one actuator, the number of actuators increase and the transforming mechanism becomes more complicated. In this paper, we suggest connecting wires to each other and reducing the number of the actuators used for transformation, and the leaning motion is studied. Configurations of the tensegrity structure and a stringing pattern of connected wires are shown. Then, a form-finding analysis is conducted to obtain bounds of valid shape of the tensegrity structure. A method for obtaining the deforming path of the tensegrity structure is suggested. Finally, a transforming experiment is conducted, and the leaning motion is demonstrated.
1 Introduction
The tensegrity structure was proposed by K.Snelson and named by R.B.Fuller in 1948 [1]. It is made of several rigid bars and flexible wires. The rigid bars are often called ‘struts’ and the flexible wires ‘tendons’. Figure 1 shows an example of the tensegrity structure. A tensegrity structure cannot always maintain its own shape because the tendons cannot have axial compression force. When axial member forces are balanced, the tensegrity structure is in equilibrium and can maintain its shape. Struts always have axial compression force and tendons always have tensional force. A tensegrity structure can be quite large while still very light-weight. Therefore, the tensegeity structure is used in large dome-shaped structures [2], [3] on the ground. Additionally, because the tensegrity structure can be transformed by extending or shortening its members, the tensegrity structures are studied as locomotion robots [4], [5].
In space, the tensegrity structure is being studied for application to several types of space structures and instruments, such as space telescopes [6], deployable masts [7], and planar reflectors [8], because the tensegirity structure can assume various shapes
depending on the configurations of the members. Then, constructing a Solar Power Satellite (SPS) using the tensegrity structure is being studied [9] as example of the application of a massive space exploration mission. The tensegrity structure is taken advantage of the characteristic of a deformable structure, and can be used as the deployable space structure.
Figure 1. A tensegrity structure
Various studies to deploy tensegrity have been
conducted. G. Tibert and S. Pellegrino demonstrated deploying the mechanism of a tensegrity mast using tape-spring rolling (TSR) hinges [7]. The mast is deployed by opening the hinges. However this mechanism cannot be used repeatedly or perform a retracting motion. H. Furuya used variable length struts and fixed their length with latch mechanisms [10]. However, this is also a one-shot mechanism and retraction was not considered. H. Furuya and H. Nakahara et al. applied inflatable structures to the struts [11]. However, the structure cannot maintain its stiffness while changing its shape. A Variable Geometry Truss (VGT) is similar to the deployable tensegrity structure because it can also be transformed by changing its members’ lengths. In general, a VGT consists of variable length members and fixed length members, and all of which are rigid. Each variable length member has an actuator, therefore, the number of actuators increases and the transformation mechanism becomes complicated as the number of members increases. K. Hanamura and Y. Tada suggested replacing variable length members with flexible wires [12]. A VGT with wired actuators is lighter than other VGTs and can be reduced to machinery parts.
i-SAIRAS 2010August 29-September 1, 2010, Sapporo, Japan
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However, the number of actuators has not been considered. The deploying mechanism should be simple and consist of as few actuators and mechanical parts as possible for ease of maintenance. Compared with other deployable space structures, for example, the inflatable structure, the storable tubular extendable member (STEM), and the coilable mast, the tensegrity structure has a high degree of freedom for deformation and can be activated repeatedly. Therefore, the tensegrirty structure may be applied not only to stable space structures but also to smart structures and space manipulators.
The objective of this paper is to demonstrate transformation of the experimental model of a tensegrity structure. The next section shows the configuration of a tensegrity structure whose tendons are connected each other. In the third section, numerical form-finding analyses are conducted and it is shown that the tensegrity structure whose tendons are connected can be realized not only axially symmetric shape but also dissymmetric shape. In the fourth section, a derivation method of the transformation path of the tensegrity structure is studied. In the fifth section, the transforming experiment is conducted. The transformation is conducted by open-loop control. Therefore, the transforming path must be obtained in advance.
2 Configuration of a tensegrity structure
2.1 Configuration of a tensegrity mast In this paper, we deal with a mast-shaped tensegrity
structure called a ‘tensegrity mast’. A tensegrity mast is composed of several tensegrity units shown in the left side of Figure 2. One tensegrity unit is composed of three struts and nine tendons. A triangle consisting of the lower edges of the struts of the tensegrity unit is called the “bottom surface”, and one consisting of the upper edges of the struts is called the “top surface”. The tensegrity units are arranged linearly and connected each other at the end of the struts with pin-joints. There are two types of tensegrity unit. One is twisted in a clockwise fashion, and the other is twisted in a counterclockwise fashion. The two types are connected alternately. Therefore, all odd-numbered tensegrity units and all even-numbered tensegrity units are each same shape. In this paper, we consider the two-stage tensegrity mast. ‘Two-stage’ means that the tensegrity mast is composed of two tensegrity units. If all tendons are not connected each other, the two-stage tensegrity mast has six struts and fifteen tendons.
Fugure 2. A tensegrity unit (left) and a tensegrity
mast (right)
2.2 Definition of parameters To express the shape of the tensegrity structure,
various parameters are needed. Figure 3 shows the parameters for determining the shape of the i-th tensegrity unit. The edge of the strut is called node . The length of the strut is . The lower edges of the struts define the bottom surface. Position and attitude of the bottom surface are expressed by the Cartesian coordinate system Oi-1-Xi-1Yi-1Zi-1 whose origin position is placed at the center of the circumscribed circle defined by the lower edges of the struts. Then the Cartesian coordinate system Oi-XiYiZi, which determines position and attitude of the top surface, is defined similar to Oi-1-Xi-1Yi-1Zi-1. The positions of the nodes, which attach to the surface defined by Oi-XiYiZi, is defined by and
, as shown in the left side of Figure 4. is always equal to zero for fixing the attitude of the coordinate system. Additionally, the direction of the strut is defined by and , as shown in the right side of Figure 4. Figure 5 shows the parameters for providing the N-stage tensegrity mast. The Cartesian coordinate system O0-X0Y0Z0, which is placed on the lowest surface, is defined as the reference frame. The position of surface i is determined by , which is defined by the position of the origin of Oi-XiYiZi from the reference frame O0-X0Y0Z0.
Figure 3. Parameters for providing the tensegrity unit
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Figure 4. Parameters to define a strut’s nodal position
and direction
Figure 5. Parameters to define the N-stage tensegrity
mast
2.3 Stringing Pattern of Connected Tendons If each tendons of the tensegrity structure have one
actuator, the number of actuator increases with the growth of the number of stages of that. Table 1 shows relation between the number of stages of the tensegrity mast and the number of tendons. When one stage is added, the number of tendons increases by nine. The increase of the number of actuators makes the deploying mechanism of the tensegrity mast more complicate. Therefore, the number of actuators should be as few as possible.
To reduce actuators in the transformation, we suggest that the tendons of the tensegrity structure are connected each other. Figure 6 is a developed figure of the two-stage tensegrity mast. The gray rectangles are struts, and the lines are tendons. Blue, red, and green lines are connected tendons. Lines of the same color and same pattern are connected and used as actuators. Orange lines are separated tendons. Each of them is not connected and their lengths are constant. In this paper, it is assumed that all separated tendons are same length. In Figure 6, green and orange lines are called ‘horizontal tendons’ and are denoted by H. Blue lines and red lines are called
‘diagonal tendons’ and denoted by D. If the tendons are connected as shown in Figure 6, the number of tendons of the two-stage tensegrity mast is reduced from twenty-one to thirteen, and the number of connected tendons is seven. The structure of the 2N-stage tensegrity mast has 7+N connected tendons.
Table 1. Relation between the number of stages and tendons of the tensegrity mast
Number of Stages 1 2 3 4 5 Number of Tendons 12 21 30 39 48
Figure 6. Configuration of the connected tendons
3 Form-finding analysis
The tensegrity structure does not always maintain its shape because tendons cannot have axial compression forces and the equilibrium of axial member forces must be satisfied. ‘Form-finding’ means to find the shape which is satisfied the equilibrium and can be maintained. In this section, numerical form-finding analyses are conducted and valid shapes of the tensegrity mast are obtained.
3.1 Conditions for the validity of the tensegrity structure Let us consider a tensegrity structure composed of M members and N nodes. When there is a strut or a tendon whose edges are provided by node and node , a direction vector of the member, is shown as
(1)
and are positional vectors of node and node , and is the length of the member. When the
equilibrium of the internal and external forces is satisfied, relation between the internal forces and the external forces can be written as the following equation-
(2)
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is an axial member force vector and is an external nodal force vector. is called an equilibrium matrix, and which is an submatrix of , is defined as follows: i) The ith member is provided by node j and node k
(j<k).
ii) The ith member is provided by node j and node k
(j>k).
iii) Node j is not an edge of the ith member.
If there are no external forces that act at each node,
is substituted to (2).
(3) We can obtain solutions of t which is from (3) if A is satisfied rank . To maintain the shape of the tensegrity structure, each member must have appropriate axial force. In other words, when struts have positive forces, tendons must have negative forces. Therefore, the equilibrium of the axial members’ forces of the tensegrity structure is judged by the signs of elements of t, which are obtained from (3). Then, let us consider connecting the tendons each other to reduce the number of actuators for deformation. Connected tendons must have the same value of tensional force. Therefore, equilibrium matrix is arranged and rewritten as . When some tendons of the two-stage tensegrity mast are connected each other, as shown in Figure 6, equilibrium matrix is arranged as . Equilibrium matrix of the two-stage tensegrity structure is shown as follows-
Additionally, the radius of struts must be considered to be prevented interference of the struts. When the radius of all struts is , distance between two struts must be satisfied as follows:
(4)
3.2 Numerical analyses of form-finding To figure out the limit of valid parameters which
defining the shape of the two-stage tensegrity mast, numerical form-finding analyses were conducted. ‘Valid’ means that the conditions for maintaining the shape of the tensegrity structure are satisfied. A certain shape of the tensegrity structure is identified as the valid shape, if (3) and (4) are satisfied, and if all tendons have axial tensional forces. In these analyses, the position of the center of surface two were obtained. The position of the center of surface two , which is written by , , and , is defined as follows:
(5)
indicates transposition of the matrix. Values of other parameters are shown in Table 2. was varied from 0.0 to 2.0, in 0.05. was from 0.0 deg, in 1.0 deg. was from 0 to 350 deg, in 10 deg. Then, the results of the analysis are evaluated by counting the set of valid ’s parameters, which is , , and .
Table 2. Constant parameters for form-finding analyses
α α α 1 0 120 240
β 0.3 0 or 0.05 165
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Results of the analyses are shown in Figure 7 and Figure 8. They are plotted the valid positions of the center of surface two. Figure 7 is the result , and Figure 8 is about . The two-stage tensegrity mast which has connected tendons shown in Figure 6 can be realized not only axial symmetric ( , and ) shapes but also asymmetric ( ) shapes. In addition, the range of the valid when is 0, is much larger than that of when is 0.05. This is because interference of struts.
Figure 7. Set of the position of the center of surface 2
( )
Figure 8. Set of the position of the center of surface 2
( )
4 Numerical analysis for the deforming path
4.1 Relation between changing ratio of nodes and tendons
When the tensegrity structure is deformed, tendons are extended or shortened to control its shape. Therefore, we must obtain changing paths of tendons’ lengths for the transformation of tensegrity structures. We can obtain changing velocities of members from moving velocities of the nodes using the following equation.
(6)
Then, is the vector of the nodes’ moving velocity, and is the vector of extending or shortening velocity of the
members.
4.2 An algorithm to obtain the deformation path
A flowchart of an algorithm to obtain the deformation path of the tensegrity structure is shown in Figure 9. At first, shape parameters, such as the height of the tensegrity structure or the position of the center of the top surface, are provided. It is very difficult for users to deal with nodal position and nodal moving velocity correctly because there are many parameters concerning the nodes. Therefore, shape parameters should be provided to reduce the burden on the user. In this paper, the position of the center of top surface H is applied to the shape parameter. Then, all nodal positions and nodal moving velocities are obtained from the shape parameters. If every nodal position is obtained, we can obtain all members’ length and equilibrium matrix A. Then, stability analysis is conducted and axial members’ forces are calculated numerically using (3). If each force obtained from (3) is correct, the algorithm is continued and each member’s deforming velocity is obtained from (4). The shape parameter is updated if the shape parameter has not reached the target value.
Figure 9. Flowchart for obtaining the deformation
path
4.3 Obtaining deformation path of leaning motion
Numerical analysis is conducted using the algorithm shown in Figure 9 as the preparation of a deformation experiment. Leaning motion is dealt with in this analysis. Table 3 shows the initial and target parameters concerning this analysis. Parameters shown in Table 3 are assumed the experimental model of the two stage tensegrity structure. Initial shape parameter is
, and target shape parameter
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is . , , , and shown in Table 3, remain constant during the transformation. Direction of leaning motion is defined as
deg. Then, velocity of the transformation is . Figure 10 shows the initial
shape and the target shape of the numerical analysis model of the two-stage tensegrity structure. In Figure 10, blue lines are struts and red lines are tendons. Figure 11 shows axial forces of the tendons when the strut ( s’ axial force is -1, and Figure 12 shows deforming velocities of the connected tendons. All tendons’ forces are always positive, while struts’ forces are negative. Table 3. Parameters for obtaining deformation path
530 0.4 0 120 240
[deg/s] [deg]
166 1 80
Figure 10. Initial shape (left) and target shape (right)
of the leaning motion
Figure 11. Result of the numerical analysis on
members’ axial force
Figure 12. Result of the numerical analysis on
tendons’ deforming velocity
5 Transformation experiment
5.1 Experimental model Figure 13 shows the experimental model of the
two-stage tensegrity structure. The lower struts were connected to the ground by hinge joints. Therefore, nodes connected to the ground could not be moved. Other edges were connected with joints shown in Figure 14. The joints have two holes to string the tendons. Tendon was strung through the left hole of the joint and was strung through the right. At the nodes, which comprise the top surface, and at the ground, red spherical markers, shown in Figure 13, were attached. The position of the center of the top surface was measured by obtaining these markers’ position using a pair of cameras, as shown in Fig. 15. The three-dimensional position of the markers could be obtained by the parallax between the two cameras. Figure 16 shows a reel mechanism to extend or shorten the tendons. The reel has a brushless DC motor and a rotary encoder, and its current value and rotation speed can be measured.
Figure 13. Experimental model of the two-stage
tensegrity structure
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Figure 14. Joint connecting struts and hole for
stringing
Figure 15. Cameras for measuring the top surfaces’
nodal position
Figure 16. The reel mechanism
5.2 Experiment of bending motion We conducted an experiment of leaning motion at
which the velocity control method was applied to all motors. Figure 17 and 18 shows variations of the positions of the node , , and during transformation. Green line is the result of . Blue line is that of . And Red line is that of . Circles attached the edge of lines in Figure 17 and 18 are initial positions of the nodes. Colored allows indicate the directions of nodes’ deformation. Results of numerical analysis for deformation path shown in the former section are given as profile of the motion of the experiment. Figure 19 shows the experimental model’s pictures during transforming. The leaning motion is demonstrated qualitatively.
Figure 17. Variation of the positions of , , and
(birds-eye view)
Figure 18. Variation of the positions of , , and
(top view)
6 Conclusions
To transform a tensegrity structure using fewer actuators and to realize a simpler transforming mechanism, we suggested connecting tendons and stringing the pattern of connected tendons, as in the two-stage tensegrity mast shown. From the form-finding analysis, it is shown that the tensegrity structure can be realized not only axial symmetric shape but also asymmetric shape. The transforming path is obtained from all nodal positions and nodal moving velocities. Then, leaning motion of the tensegrity structure is demonstrated. References [1] C. Sultan, “Tensegrity Structures Research
Evolution”, Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 2294-2299
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[2] K. Kawaguchi and Z.-Y. Lu, “Construction of three-strut tension systems”, Proceedings of the 5th International Conference on Space Structures, 2002, pp. 1-10
[3] A. Hanaor, “Aspects of Design of Double Layer Tensegrity Domes”, International Journal of Space Structures, vol. 7, 1992, pp. 101-113
[4] C. Paul, F. J. Valero-Cuevas, and H. Lipson, “Design and Control of Tensegrity Robots for Locomotion”, IEEE Transactions on Robotics, vol. 22, no. 5, 2006, pp. 944-957
[5] M. Shibata, F. Saijyo, and S. Hirai, “Crawling by Body Deformation of Tensegrity Structure Robots”, Proceedings of the 2009 IEEE international conference on Robotics and Automation, 2009, pp. 4375-4380
[6] C. Sultan, M. Corless, and R. E. Skelton, “Peak to Peak Control of an Adaptive Tensegrity Space Telescope”, Proceedings of SPIE Symposium on Smart Structures and Materials, vol. 3667, 1999, pp. 190-201
[7] G. Tibert and S. Pellegrino, “Deployable Tensegrity Masts”, The 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2003, AIAA 2003-1978
[8] G. Tibert and S. Pellegrino, “Deployable Tensegrity Reflectors for Small Satellites”, AIAA Journal of Spacecraft and Rockets, vol. 39, 2002, pp. 701-709
[9] S. Murata, D. Jodoi, H. Furuya, Y. Terada, and H. Takadama, “Inflatable Tensegrity Module for a Large-Scale Space Structure and its Construction Scenario”, The 56th International Astronautical Congress, 2005, IAC-05-D1.1.01
[10] H. Furuya, “Tensegrity Structures in Space Application”, International Journal of Space Structures, vol. 7, no. 2, 1992, pp. 143-151
[11] H. Furuya, H. Nakahara, S. Murata, D. Jodoi, Y. Terada, and K. Takadama, “Concept of Inflatable Tensegrity for Large Space Structures”, Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2006, pp. 1322-1330
[12] K. Hanahara and Y. Tada, “Statically Indeterminate Adaptive Truss with Wire Member Actuators (Topology Design and Kinematics)”, Transactions of the Japan Society of Mechanical Engineers, vol. 66, 2000, pp. 539-544
Figure 19. The leaning motion
