44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 7-10 April 2003 Norfolk, VA

Transcription

1 AIAA -45 Closed-Loop Deploable Structures W.W. Gan and S. Pellegrino Uniersit of Cambridge, Cambridge, CB P, UK 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dnamics, and Materials Conference 7- April Norfolk, VA For permission to cop or republish, contact the American Institute of Aeronautics and Astronautics 8 Aleander Bell Drie, Suite 5, Reston, VA 9 444

2 Closed-Loop Deploable Structures W.W. Gan and S. Pellegrino Uniersit of Cambridge, Cambridge, CB P, UK This paper is concerned with a noel kind of deploable structures that form closed loops that fold into a bundle of bars using simple hinges. These structures, first enisaged b Dr J.M. Hedgepeth, hae potential applications for ultra-lightweight solar arras, solar sails, and radar structures. Asstematic stud of the kinematics of closed-loop structures is carried out, and a scheme for simulating the deploment of both smmetric and nonsmmetric linkages is presented. Introduction and Background Man deploable structures form single or multiple, often interlaced, closed loops. Some of these structures form space frames, and hae alread had a ariet of applications,, but this paper is concerned with a different tpe of structure, a kind of mechanical linkage that forms a segmented hoop that folds into a bundle of bars. The first application of this tpe of structure was enisaged, to our knowledge, in a stud co-authored b J.M. Hedgepeth in 97. Figure shows an eample of the main solution proposed, which was later adopted in the edge beam of the Hoop-Column Antenna, see Figure. In these structures the rim consists of an een number (four or more) of equal-length hinged segments and each joint inoles two hinges and an intermediate block. Howeer, Appendi A of Ref. [] considered an alternatie hinging method, which apparentl had been suggested b A. Buseman during a progress reiew meeting at NASA Langle. 4 This alternatie solution is illustrated in Figure and is the object of the present stud. An adantage oer the preious scheme is that it requires half the number of hinges and can proide better controlled deploment. Although new as a deploable structure, the four-sided ersion of this linkage is an eample of the classical 4-bar linkage discoered in 9 b the Cambridge mathematician G.T. Bennett. 5 The si-sided ersion of this linkage was also known, as it had been discoered b M. Goldberg in It is shown as model H on the right-hand side of Figure 4, together with man other models made b Goldberg. Structures of this kind hae potential applications for solar arras as enisaged in Ref. [] and also solar sails. The could also be used to deplo and support fleible actie surface for SAR structures. Here the configuration that is most frequentl required is rectangular, and preliminar studies of closed-loop Research Student, Department of Engineering, Trumpington Street. Professor of Structural Engineering, Department of Engineering, Trumpington Street. Associate Fellow AIAA. pellegrino@eng.cam.ac.uk Copright c b S. Pellegrino. Published b the American Institute ofaeronautics and Astronautics, Inc. with permission. Fig. Two-hinge method for deploing rim (from Ref. []). structures whose full-deploed configuration is rectangular hae been carried out. 7 For eample, Figure 5 shows a si-bar structure with self-locking hinges that forms a rectangular frame. In this paper we present a sstematic stud of the kinematics of closed-loop structures. The proposed approach is suitable for both smmetric linkages such as those enisaged in Ref. [] and for much less smmetric structures such as that of Figure 5. Modelling of Closed-Loop Linkages A general formulation for linkages connected b reolute joints is naturall set up b considering a series of sets of cartesian coordinate aes, as eplained net. Gien a fied coordinate sstem,,,,, the general position and orientation of a local coordinate sstem, P,,,, can be described b a translation of9 American Institute of Aeronautics and Astronautics Paper -45

3 Fig. 5 Deploable 6-bar structure with self-locking hinges. Fig. Deploment of Hoop-Column Antenna (from Ref. []). from to P followed b a rotation to P,,,. The two quantities that describe this transformation are a ector ( ) and a matri R ( ). Following Denait and Hartenberg 8 it is conenient to write them as a single 4 4 matri T = R () a α Here R is defined according to the standard - conention for the Euler angles φ, ω, ψ that transform 9 the global coordinates sstem into,, cos φ sin φ R = sin φ cos φ cos ω sin ω sinω cos ω cos ψ sin ψ sin ψ cos ψ () α Note that R transforms the local coordinates,, into the global coordinates,,. R P Fig. ne-hinge method for deploing rim (from Ref. []). Fig. 4 Models made b Goldberg (from Ref. [6]). General transformation of a coordinate ss- Fig. 6 tem. Net, consider a further transformation of this set of aes, from P,,, to P,,, as shown in Figure 7. This transformation is represented b T = R () of9 American Institute of Aeronautics and Astronautics Paper -45

4 R P P ', R R P ' P Fig. 7 Transformation from P,,, to P,,,. where = P P and its components are defined in the local coordinate sstem P,,, and hence transform from P,,, to P,,,. The relatie rotation R transforms from the sstem,, to,, ; note that this goes in the opposite direction to. Consider the compound transformation T,, i.e. T followed b T. The translation ector is followed b R, because the second translation needs to be transformed to the first coordinate sstem, hence ', = + R (4) The rotation matri is obtained from Hence, = R = R = R R (5) Therefore, clearl T, = R R + R (6) This can also be obtained b post-multipling T b T : T T = R R = R R + R (7) R, Fig. 8 Compound transformation T,, i.e. T followed b T. A more general compound transformation, consisting of n transformations can be epressed as T,n = T T... T n = R... R n n (8) and is useful in describing the configurations of a linkage of n rods connected b reolute joints. Closed Loop Linkages We will describe a general method of analsis for closed loop linkages consisting of n rods connected b reolute joints. We will describe first the case n = 4, which corresponds to a Bennett linkage. Figure 9 shows a square framework that folds into a compact bundle of rods. This model is made from wooden bars connected together b door hinges. During folding, the model preseres two planes of smmetr, and one -fold smmetr ais. Single Rod with Joints Consider a rod with two non-parallel reolute joints attached to the ends. Its geometr can be represented b the transformation matri T, where describes the direction and length of the member and R describes the relatie orientation of the two joints. If the rod has unit length (a +d = ), we hae a = = d The relatie orientation of the two reolute joints is obtained b performing a series of elementar rotations φ, ω and ψ, respectiel, about the, ξ and ζ 6 of9 American Institute of Aeronautics and Astronautics Paper -45

5 a d P plane φ ζ = η P ξ ζ η' η ζ' P ξ' = ξ ω η' plane P ξ' = ζ' Model of four-member frame (Bennett link- Fig. 9 age). ψ aes, as shown in Figure. Here φ = ω = +9 + tan (/ 8) = ψ = 5 The corresponding transformation matri is T = Two Rods Connected b a Joint Consider the two rods P and P P, as shown in Figure, connected at point A where P = P. The second member P P is arranged in such a wa that is collinear with when the hinge rotation θ A is ero. We can write the following epression for the trans- Fig. Formulation of transformation matri for element of Bennett linkage. formation matri T, = T T θa T cos θ A sin θ A = R sin θ A cos θ A R Note that the hinge rotation appears onl in the ma- P 4of9 American Institute of Aeronautics and Astronautics Paper -45

6 P θ A A P Finall, consider a radial section of the structure in its deploed configuration, Figure, through one of the hinges. This section is also an isosceles triangle, and the angles at the base are α = arctan(sin α tan α ) () The hinges are mounted alternatel on the inner and outer side faces of the isosceles triangles. Hence, we can obtain the following epressions for the angles φ, ω, ψ and the lengths a, d P φ = arctan(tan α sin α ) ω = π + arctan + ψ = π φ P = A a = cos α d = sinα where P = A = cos α tan α sin α cos α = sinα tan α + cos α cos α P Adjusted to fit = sinα The corresponding epressions for the second rod are φ = φ Fig. Two rods of a Bennett linkage. tri in the middle; T and T are constant. Closed Loop of Four Rods B an obious etension of the aboe two-rod eample we can formulate a closed-loop, four-rod linkage as the product of four constant matrices and four angledependent matrices: T T θa T T θb T T θc T T θd = I (9) Here, the identit matri on the right-hand-side imposes the condition that the loop should remain closed in all configurations, and hence (i) the first and last points are the same, and (ii) the aes of rotation of the first and last reolute joints coincide. We will call this equation the loop closure equation. Loop of n Rods Consider a closed loop of the tpe shown in Figure, consisting of n rods with identical cross-section, their cross-section being isosceles triangles. For these rods to fit together as a bundle in the folded configuration, the ape angle of the isosceles triangle must be We also define α = π n () α = α () ω = ω ψ = ψ a = a d = d Using these epressions, we can set up the loop closure equation for this linkage. It is analogous to Equation 9, but now includes n constant matrices and n angle-dependent matrices. The hinge angle for which the linkage is full deploed is θ A = arccos[ cos α (sin( π π N )+sin α tan α ) ] Analtical Solution of Loop Closure Equation For the Bennett linkage smmetr can be assumed, hence θ A = θ C and θ B = θ D. Thus, Equation 9 can be simplified to 6 cos θ sin θ A 6 6 sin θ A cos θ A 6 cos θ B sin θ B 6 6 sin θ B cos θ B = I 5of9 American Institute of Aeronautics and Astronautics Paper -45

7 and soling this equation smbolicall ields θ B = ( tan cos θ A sinθ A cos θ A + sin θ A 4 cos θ A + ) /( cos θ A sinθ A cos θ A + ) sinθ A cos θ A + () A plot of θ A against θ B is shown in Figure. θ (deg) B θ (deg) Α Substituting into, e.g. the transformation matri T A, ields cos(θ A + θ A ) sin(θ A + θ A ) T A = sin(θ A + θ A ) cos(θ A + θ A ) sin θ A cos θ A T A + cos θ A sin θ A θ A = T A + U A θ A (6) where U A is the deriatie of T A calculated at A. Substituting Equation 6 and similar into (4) ields T (T A +U A θ A ) T (T B +U B θ B ) = I (7) Epanding Equation 7 and rearranging, we hae T T A T T B T T C T T F +(T U A T T B T T C T T D ) θ A Fig. θ A s θ B for Bennett linkage. +(T T A T U B T T C T T D ) θ B +(T T A T T B T U C T T D ) θ C +(T T A T T B T T C T U D ) θ D = I (8) Numerical Solution of Closure Equation Here we present a general solution method, based on a predictor-corrector scheme suitable for implementation in a standard Newton-Raphson iteration, for the loop closure equation. For definiteness, we refer to a Bennett linkage. We assume that the linkage is gien in its initial configuration, e.g. the hinge angle s θ A, θ B, θ C and θ D are equal to ero and all four rods fit together as a bundle. Predictor Step From the closure equation T T A T T B T T C T T D = I (4) Consider small geometr changes in each hinge with ero strain or deformation in the rods. For the linkage to remain intact it needs to satisf the equation: T T A T T B T T C T T D = I (5) where T A, T B, T C and T D correspond to the general configuration defined b θ A, θ B, θ C and θ D. Consider the Talor epansion of the hinge angles. B ignoring higher-order terms we hae sin(θ A + θ A ) sin θ A + cos θ A ( θ A ) cos(θ A + θ A ) cos θ A sin θ A ( θ A ) From Equation 4, the first term of Equation 8 is equal to the identit matri. Hence, the sum of the remaining terms must be ero. Defining A = T U A T T B T T C T T D, etc. this equation can be rearranged into A θ A + B θ B + C θ C + D θ D = [] (9) where [] is a 4 4 null matri. It is interesting to note the form of these matrices a, a, a,4 a, a, a,4 a, a, a,4 θ A b, b, b,4 + b, b, b,4 b, b, b,4 θ B +... = []() At first glance, the matri equation () would appear to be equialent to 9 scalar equations in 4 unknowns, not 6, since the 7 equations corresponding to the diagonal and the last row are alwas satisfied. The fact that the sub-matri in the top-left corner is skew-smmetric corresponds to the fact that an rotation can alwas be defined b onl three elementar rotations, hence 6 of the 9 coefficients of this sub-matri are dependent on the other. 6of9 American Institute of Aeronautics and Astronautics Paper -45

8 Hence, this ields a 6 4 sstem of equations a, b, c, d, a, b, c, d, θ A a, b, c, d, θ B a,4 b,4 c,4 d,4 θ C = () a,4 b,4 c,4 d,4 θ D a,4 b,4 c,4 d,4 The rank of the matri in Equation equals, which implies that the equation has a single infinit of solutions. This also indicates that the linkage has an internal mechanism. B soling Equation we can find sets of θ s which describe infinitesimal, ero-strain motions of the linkage. Corrector Step A finite, et small motion of the linkage where each hinge angle is changed b an amount proportional to the infinitesimal mechanism computed aboe is likel to induce small errors. Let C be a configuration obtained from the predictor step, i.e. b imposing a small, and et finite change of angles to the initial configuration. We wish to compute a configuration C, near C, where all errors hae been remoed. Hence, we need to compute C, the configuration change from C to C. In configuration C we will hae T T A T T B T T C T T D = I () Although we do not know C, we do know C. In this configuration the closure equation is not satisfied, hence we can compute an error matri E from T T Ā T T B T T C T T D = I + E () Epanding the hinge angles in configuration C in term of C, wehae θ A = θ Ā + θ A, θ B = θ B + θ B,... Now, writing the closure equation in configuration C, substituting a Talor epansion for each hinge angle, and then manipulating the equations as aboe we obtain T T Ā T T BT T CT T D +(T U Ā T T BT T CT T D) θ A +(T T Ā T U BT T CT T D) θ B +(T T Ā T T BT U CT T D) θ C +(T T Ā T T BT T CT U D) θ D = I (4) From Equation, the first term in Equation 4 is equal to I + E. Hence substituting Equation into Equation 4, we obtain +(T U Ā T T BT T CT T D) θ A +(T T Ā T U BT T CT T D) θ B +(T T Ā T T BT U CT T D) θ C +(T T Ā T T BT T CT U D) θ D = E (5) which can be written in form P θ A + Q θ B + R θ C + S θ D = E (6) where P = T U Ā T T BT T CT T D,... etc. The error matri E on the right hand side has the structure e,4 F e,4 e,4 We decompose F into its smmetric and skewsmmetric components, i.e. (F + F T )/, (F F T )/, and consider onl the skew-smmetric part, whose coefficients are denoted b ē,, ē, and ē,. This transformed equation can be treated in the same wa as Equation, thus rearranging it into 6 scalar equations p, q, r, s, ē, p, q, r, s, θ A ē, p, q, r, s, p,4 q,4 r,4 s,4 p,4 q,4 r,4 s,4 p,4 q,4 r,4 s,4 θ B θ C θ D = ē, e,4 e,4 e,4 (7) The least squares solution of Equation 7 is used to determine the minimal correcting angles θ due to the errors in configuration C θ = r i= w i u T i i,i e (8) where F = UVW T is the singular alue decomposition of the 6 4 coefficient matri in Equation 7 and r is the number of non-ero singular alues. Also, w i is the i th column of matri W ; u i is the i th column of U; i,i is the i, i term in matri V ; e is the ector on the right hand side of Equation 7. The predictor-corrector stages of the algorithm presented aboe are repeated until the hinges hit their phsical stops, at which point the linkage is full deploed. Results Using the aboe numerical solution method for the closure equation, we hae analsed the deploment behaiour of the 4-rod linkage (Bennett linkage) but this time without assuming smmetric behaiour. Figure shows a plot of the hinge angles ariation during deploment; note that θ B = θ D at all stages, hence confirming that the structure remains smmetric at all stages. Also note that the behaiour predicted b the numerical scheme is identical to the analtical results shown in Figure. Figure 4 shows si snapshots from the deploment sequence. The same solution method, applied to the 6-rod linkage also predicts that the structure remains smmetric at all stages, see Figure 5 and Figure 6. 7of9 American Institute of Aeronautics and Astronautics Paper -45

9 θ, θ, θ (deg) B C D θ C θ B, θ D θ B, θ C, θ D, θ E, θ F (deg) θ C, θ E θ B, θ D, θ F θ (deg) A Fig. Variation of hinge angles during deploment of 4-rod linkage θ (deg) A Fig. 5 Variation of hinge angles during deploment of 6-rod linkage θ Α = θ Α = θ Α = θ Α = θ Α = 96 θ Α = θ Α = 9 θ Α = θ Α = 9 θ Α = θ Α = 8 θ Α =. Fig. 4 Deploment sequence of 4-rod linkage. Fig. 6 Deploment sequence of 6-rod linkage. 8of9 American Institute of Aeronautics and Astronautics Paper -45

10 Discussion The formulation presented in this paper proides considerable insight into the kinematic behaiour of closed-loop linkages, from which issues important for design, such as sensitiit to imperfections, can be analsed. It also becomes possible to design assemblies with special properties, e.g. frames with some particular dimensions. References Hoberman, C. (99). Reersibl epandable structures. USA Patent No Pellegrino, S. and ou,. (99). Foldable ring structures. In Space Structures 4. Proceedings of Fourth International Conference on Space Structures, Guildford, 6- September 99, (Edited b G.A.R. Parke and C.M. Howard), pp Thomas Telford, London. Crawford, R. F., Hedgepeth, J. M. and Preiswerk, P. R. (97). Spoked wheels to deplo large surfaces in space: weight estimates for solar arras. NASA-CR Mikulas, M. (998) Personal Communication. 5 Bennett, G.T. (9) A new mechanism, Engineering 76, Goldberg, M. (94). New Fie-Bar and Si-Bar Linkages in Three Dimensions. Transactions of the ASME, Pellegrino, S., Green, C., Guest, S.D. and Watt, A. () SAR adanced deploable structure, Technical Report CUED/D-STRUCT/TR9, Department of Engineering, Uniersit of Cambridge. 8 Denait, J. and Hartenberg, R.S. (955) A kinematic notation for lower-pair mechanisms based on matrices, Journal of Applied Mechanics, June 955, Goldstein, H. (98). Classical Mechanics, Second Edition. Addison-Wesle, Reading, MA. 9of9 American Institute of Aeronautics and Astronautics Paper -45