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Deployment of a Class 2 Tensegrity Boom



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Deployment of a Class 2 Tensegrity Boom Jean Paul Pinauda Soren Solaria and Robert E Skeltona a University of California San Diego 9500 Gilman Drive La Jolla California 92093 U S A Abstract Tensegrity structures are special truss structures composed of bars in compression and cables in tension Most tensegrity structures under investigation to date have been of Class 1 where bars do not touch In this article however we demonstrate the hardware implementation of a 2 stage symmetric Class 2 tensegrity structure where bars do connect to each other at a pivot The open loop control law for tendon lengths to accomplish the desired geometric reconfiguration are computed analytically The velocity of the structure s height is chosen and reconfiguration is accomplished in a quasi static manner ignoring dynamic effects The main goal of this research was to design build and test the capabilities of the Class 2 structure for deployment concepts and to further explore the possibilities of multiple stage structures using the same design and components Keywords Tensegrity structures deployment reconfiguration stable unit 1 INTRODUCTION Tensegrity structures are built of bars and tendons attached to the ends of the bars 1 The bars can resist compressive force and the strings cannot Most bar string configurations which one might conceive are not in equilibrium and if actually constructed will collapse to a different shape Only bar string configurations in a stable equilibrium will be called tensegrity structures 2 5 If well designed the application of forces to a tensegrity structure will deform it into a slightly different shape in a way which supports the applied forces Tensegrity structures are very special cases of trusses where members are assigned special functions Some members are always in tension and others are always in compression We will adopt the words tendons for the tensile members and bars for compressive members 6 A tensegrity structure s bars cannot be attached to each other through joints that impart torques The end of a bar can be attached to tendons or ball jointed to other bars Tensegrity structures are natural candidates to be actively controlled structures since the control system can be embedded in the structure directly for example tendons can act as actuators and or sensors 7 9 Shape control of the tensegrity structure can be accomplished by moving along its equilibrium manifold The tensegrity unit shown in Fig 1 is the simplest three dimensional tensegrity unit which comprises three bars held together in space by strings so as to form a tensegrity unit A tensegrity unit comprising of three bars will be called a 3 bar tensegrity A 3 bar tensegrity is constructed by using three bars in each stage which are twisted either in clockwise or in anti clockwise direction The top strings connecting the top of each bar support the next stage in which the bars are twisted in a direction opposite to the bars in the previous stage In this way any number of stages can be constructed which will have an alternating clockwise and anti clockwise rotation of the bars in each successive stage This is the type of structure in Snelson s Needle Tower Contact info For animations movies of figures contact authors J P Pinaud Email jpinaud mae ucsd edu S Solari Email ssolari ucsd edu Figure 1 Simplest tensegrity unit three bar unit In contrast to the Class 1 structure 10 11 a Class 2 structure can also be constructed with the same 3 bar tensegrity unit Assembly of two units with clockwise and anti clockwise sense can be stacked directly on each other resulting in bars connecting shown in Fig 2a The following section addresses the analysis and construction of a prototype Class 2 tensegrity structure that verifies the deployment methodology used 2 TWO STAGE CLASS 2 TENSEGRITY STRUCTURE Design of a two stage Class 2 tensegrity structure begins with the design of the base The allowable twist angle of a two stage Class 2 structure with fixed base nodes is 2 n where n is the number of bars in each stage n 3 in this paper The addition of a Reinforcing tendon to be described in the next section increases the admissable twist angle range to 6 2 12 14 In addition from these papers by Masic we conclude the forces in the three bar tensegrity unit are realisable and can be scaled with respect to each other We now turn our attention to geometry of the structure and writing the equations that describe the reconfiguration of the structure These equations are vital in deriving the motor command signals that actuate the structure Choosing the coordinate axes as shown in Fig 2 we write the coordinates of each node as follows First Stage Nodal Number 1 2 3 4 5 6 Second Stage 7 8 9 Fixed Node yes no yes no yes no Coordinates rbase 0 0 r cos r sin h rbase cos rbase sin 0 r cos 2 r sin 2 h rbase cos 2 2 rbase sin 2 2 0 r cos 2 3 r sin 2 3 h top platform top platform top platform rbase 0 2h rbase cos rbase sin 2h rbase cos 2 2 rbase sin 2 2 2h Table 1 Table of nodal coordinates shown in Fig 2 where the chosen parameters to describe the geometry of the structure are r h and From symmetry the angle is defined 2 3 a VRML Simulation b Top View Figure 2 Views of 2 stage tensegrity indicating nodal positions and coordinate axes The saddle tendon length can be computed by calculating the distance between nodes 4 and 2 as S 12 2 2 r cos r cos 2 r sin r sin 2 r 3 3 3 1 Similarly the vertical tendon length can be computed by calculating the distance between nodes 2 and 1 as 1 V r cos rbase 2 r2 sin2 h2 2 2 and the reinforcing tendon length can be computed from the distance between nodes 2 and 3 as R r cos rbase cos 2 2 2 r sin rbase sin 2 h2 3 3 12 3 The parameter r can be shown to depend on h and therefore only two independent parameters are needed to describe the structure completely Figure 3 can be used to derive the dependence as follows Apply the law of cosines to 4ABC 2 2 2 r2 2rrbase cos lbar h2 rbase 4 3 and solving for r via the quadratic formula yields r rbase cos 1 2 2 2 2 2 rbase cos2 lbar h2 rbase 2 3 3 5 a Top View b Side View of a single bar Figure 3 Geometric relations used to derive the dependence of r on height h Equations 1 and 2 can now be written as functions of h and although will be chosen to be in the admissible range 6 2 and h will be determined by choice of the function h t In practice the bars of the structure will intersect at 3 so the admissible range of interest will be either 6 3 or 3 2 Since the


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