Snappability and singularity-distance of pin-jointed body-bar frameworks
SSnappability and singularity-distanceof pin-jointed body-bar frameworks
Georg Nawratil
Institute of Discrete Mathematics and Geometry & Center for Geometry and Computational DesignTU WienWiedner Hauptstrasse 8-10/104, Vienna 1040, AustriaEmail: [email protected]
ABSTRACT
It is well-known that there exist rigid frameworks whose physical models can snap between different realizationsdue to non-destructive elastic deformations of material. We present a method to measure these snapping capabilitybased on the total elastic strain energy density of the framework by using the physical concept of Green-Lagrangestrain. As this so-called snappability only depends on the intrinsic framework geometry, it enables a fair comparisonof pin-jointed body-bar frameworks, thus it can serve engineers as a criterion within the design process either toavoid snapping phenomena (e.g. truss structures) or to utilize them (e.g. multistable materials).Moreover, it turns out that the value obtained from this intrinsic pseudometric also gives the distance to theclosest shaky configuration in the case of isostatic frameworks. Therefore it is also of use for the kinematicscommunity, which is highly interested in the computation of these singularity-distances for diverse mechanicaldevices. In more detail we study this problem for parallel manipulators of Stewart-Gough type.
In this paper we study frameworks composed of bars and bodies linked by pin-joints, which are rotational joints in theplanar case and spherical joints in the spatial case. Note that all joints are assumed to be without clearance. A body is eithera polyhedron or a polygonal panel . For both of these cases it is assumed that the body does not possess any unnecessaryvertices; i.e. every of its vertices is pin-jointed. An additional assumption is that the inner graph of each body is globallyrigid, where an inner graph is defined as follows: Definition 1.
Connect all vertices of the polyhedron (polygonal panel) by edges, which are either located on the boundaryof the polyhedron (polygon) or in its interior. The resulting graph is called inner graph.
Note that our studied class of geometric structures known as pin-jointed body-bar frameworks also contains hinge-jointed frameworks, as a hinge between two bodies can be replaced by two pin-joints.By defining the combinatorial structure of the framework as well as the lengths of the bars and the shapes of the bodies,respectively, the intrinsic geometry of the framework is fixed. In general the assignment of the intrinsic metric does notuniquely determine the embedding of the framework into the Euclidean space, thus such a framework can have differentincongruent realizations.A realization is called a snapping realization if it is close enough to another incongruent realization such that the physicalmodel can snap into this neighboring realization due to non-destructive elastic deformations of material. Shakiness can beseen as the limiting case where two realizations of a framework coincide; e.g. [50, 47].The open problem in this context is the meaning of closeness , which is tackled in this article. In more detail, we present amethod to measure the snapping capability (shortly called snappability ) of a realization. The provided distance is of interestfor practical applications, because it can be used in the early design phase of a framework to avoid snapping phenomena (e.g.engineering of truss structures) or to utilize them (e.g. multistable materials and structures). The latter approach has receivedmuch attention in the last few years within a wide field of applications; ranging from origami structures (e.g. [31, 10, 30])over mechanical metamaterials (e.g. [18, 44, 58]) to metastructures (e.g. [25]). A polygonal panel can be seen as a body with coplanar vertices according to [26]. a r X i v : . [ c s . C G ] J a n ut the snappability also provides a distance to the next shaky configuration in the case of isostatic frameworks. There-fore this singularity-distance can also be of use for mechanism engineers in the context of singularity-free path planing ofrobotic devices. In two recent conference articles [37,38] the author already started to investigate this topic. In [37] a first attempt towardsthe computation of the snappability of bar-joint frameworks was done based on the definition of Cauchy/Engineering strain.In [38] the author extended the approach to frameworks composed of bars and triangular panels. For this it was necessary toswitch to the concept of Green-Lagrange strain, as the elastic strain energy of triangular panels using Cauchy/Engineeringstrain is not invariant under rotations. In the articles [37, 38] the author restricted to planar examples; namely the trivial caseof a triangular framework and the more sophisticated example of a pinned 3-legged planar parallel manipulator. In the paperat hand, we render the approach of [38] more precisely and generalize it to polygonal panels and polyhedra, respectively,and study some spatial structures, which already appear in existing literature on this topic reviewed next.By the well-known technique of deaveraging (e.g. [47], [43, page 1604] and [21]) snapping frameworks can be con-structed in any dimension R d . Moreover, for snapping bipartite frameworks in R d an explicit result in terms of confocalhyperquadrics is known (cf. [47, page 112] under consideration of [46]). Most results are known for the dimension d = buckling polyhedral surfaces and Siamese dipyramids are introduced. Snapping structuresare also related to so-called model flexors (cf. [34]) as in some cases the model flexibility can be reasoned by the snappingthrough different realizations. Examples for this phenomenon are the so-called four-horn [57] or the already mentioned Siamese dipyramids . The latter are studied in more detail in [13], especially how relative variations on the edge lengthsproduce significant relative variations in the spatial shape. The authors of [13] also suggested estimates to quantify theseintrinsic and extrinsic variations. Recently, a more general approach for estimating these kinds of quantities for arbitrarybar-joint frameworks was presented in [19], where inter alia also the Siamese dipyramid was studied as an example.Beside the above reviewed mathematical studies on snapping structures, there are also the following application drivenapproaches. The snapping behavior of structures is studied by(1) numerical simulations based on (a) finite element methods [44, 18, 58] or (b) force method approaches like [16] or ageneralized displacement control method [10, 31, 30],(2) theoretical approaches based on the variation of the total potential energy [58, 25, 27, 8].In contrast, our approach only relies on the total strain energy of the structure (i.e. no a priori assumptions on externalloads have to be made) paving the way for the definition of the snappability, which only depends on the intrinsic frameworkgeometry. Finally it should be noted, that a short review on approaches from the robotic community towards the computationof singular-distances is given in the section dealing with Stewart-Gough (SG) manipulators, which brings us straight to theoutline of the paper.After introducing notations and summarizing fundamentals of rigidity theory in Section 1.2, we present the underlyingphysical model of deformation in Section 2, which is used for building up the pseudometric on the space of intrinsic frame-work geometries in Section 3. Based on some theoretical considerations, we discuss the computation of the snappability andthe singularity-distance in Sections 4 and 5, respectively, and demonstrate the presented methods in two examples. Afterwardwe adopt our theoretical results for the singularity-distance computation of SG platforms in Section 6, which is also closedby a practical example. Finally we conclude the paper in Section 7. Moreover, in Appendix A the Siamese dipyramid andthe four-horn are studied, and the obtained results are compared with existing literature.
A pin-jointed body-bar framework G ( K ) consists of a knot set K = (cid:110) V , . . . , V r , B d ( n ) , . . . , B d q q ( n q ) (cid:111) and an 2-edgecolored (green, red) graph G on K . A knot B d i i ( n i ) represents a body, where d i ∈ { , } gives the additional informationif the body is a polyhedron ( ⇔ d i =
3) or a polygonal panel ( ⇔ d i = d = . . . = d p = d p + = . . . = d q = ≤ p ≤ q . The number n i gives the number of vertices of the body B i . A knot V i corresponds to rotational/spherical joint linking bars. A green edge connecting two knots corresponds to a bar. A red edgeis only allowed to connect two bodies and represents a pin-joint.Due to the assumed global rigidity of the inner graph of a body we can replace each body by a globally rigid bar-jointsubframework according to [26, page 437]. Note that the combinatorial characterization of global rigidity is only known for R (cf. [22]), but still open for R [26, page 450]. Mathematically these structures do not posses a continuous flexibility but due to free bendings without visible distortions of materials their physicalmodels flex. emark 1.
The completeness of an inner graph is a sufficient condition for global rigidity (cf. [2]). This implies that thebody B d i i ( n i ) has to be convex as all n i ( n i − ) / edges are in the interior of the polyhedron (polygon) or on its boundary.Note that the globally rigid bar-joint subframework of a polygonal panel B i ( n i ) is not infinitesimal rigid in R for n i > because every vertex can be infinitesimally flexed out of the plane spanned by the remaining n i − vertices. (cid:5) By replacing the bodies by globally rigid bar-joint subframeworks resulting from the inner graphs, we end up with a bar-joint framework G ∗ ( K ∗ ) which is equivalent to the given pin-jointed body-bar framework G ( K ) . This bar-joint framework G ∗ ( K ∗ ) can be used for defining the intrinsic geometry of the framework G ( K ) in a mathematical rigorous way. For doingthis, we introduce the following notation.By denoting the vertices of the body B d i i ( n i ) by V s i + , . . . , V s i + n i with s i = r + ∑ i − j = n j we get the set K ∗ = { V , . . . , V w } with w = r + ∑ qj = n j . Moreover, we denote the edge connecting V i to V j by e i j with i < j . Now we can fix the intrinsic metricof the framework G ∗ ( K ∗ ) (and therefore also of G ( K ) ) by assigning a length L i j ∈ R > to each edge e i j . Moreover, wecollect all these lengths in the b -dimensional vector L = ( . . . , L i j , . . . ) T of the space R b of intrinsic framework metrics, where b gives the number of edges of the graph G ∗ . Finally we collect the indices i j of edges e i j of G ∗ ( K ∗ ) which correspond togreen edges of G ( K ) in the set G .We denote a realization of the framework G ( K ) and G ∗ ( K ∗ ) by G ( V ) and G ∗ ( V ) , respectively, where the configurationof vertices V = ( v , . . . , v w ) ∈ R wd is composed by the coordinate vectors v i = ( x i , y i , z i ) T for d = v i = ( x i , y i ) T for d =
2, respectively, of V i for i = , . . . , w .Now we consider a realization G ∗ ( V ) of the equivalent bar-joint framework G ∗ ( K ∗ ) . In the rigidity community (e.g. [6])each edge e i j is assigned with a stress (coefficient) ω i j ∈ R . For every knot V i we can associate a so-called equilibriumcondition ∑ i < j ∈ N i ω i j ( v i − v j ) + ∑ i > j ∈ N i ω ji ( v i − v j ) = o (1)where o denotes the d -dimensional zero-vector and N i the knot neighborhood of V i ; i.e. the index set of knots ∈ K ∗ connectedwith V i by bars. If for all w knots this condition is fulfilled, then the b -dimensional stress-vector ω = ( . . . , ω i j , . . . ) T is refereedas self-stress (or equilibrium stress ). Algebraic approach to rigidity theory.
The relation that two elements of the knot set are edge-connected can also beexpressed algebraically. They are either quadratic constraints resulting from the squared distances of vertices (implied by agreen edge) or they are linear conditions, which are steaming from the identification of vertices (implied by a red edge) orthe elimination of isometries . In total this results in a system of n algebraic equations c = , . . . , c n = m unknowns ,which constitute an algebraic variety A .If A ( c , . . . , c n ) is positive-dimensional then the framework is flexible; otherwise rigid. The framework is called mini-mally rigid (isostatic) if the removal of any algebraic constraint (resulting from an edge) will make the framework flexible.In this case m = n has to hold. Rigid frameworks, which are not isostatic, are called overbraced or overconstrained ( n > m ).Note that there is also a combinatorial characterization of isostaticity for dimension 2 according to the work of Laman [29],but for dimension 3 this is still an open problem.If A ( c , . . . , c n ) is zero-dimensional, then each real solution corresponds to a realization G ( V i ) of the framework for i = , . . . , k . If there is exactly one real solution, then the framework is called globally rigid. But one can also considerthe complex solutions of the set of realization equations c , . . . , c n resulting in complex knot configurations V i with i = k + , . . . , k + f and f ∈ N ∗ as they always appear in pairs. According to [9] they imply complex realizations G ( V i ) .We can compute in a realization the tangent-hyperplane to each of the hypersurfaces c i = R m for i = , . . . , n . Notethat this is always possible as all hypersurfaces are either hyperplanes or regular hyperquadrics. The normal vectors of thesetangent-hyperplanes constitute the columns of a m × n matrix R G ( V ) , which is also known as rigidity matrix of the realization G ( V ) . If its rank is m then the realization is infinitesimal rigid otherwise it is infinitesimal flexible; i.e. the hyperplaneshave a positive-dimensional affine subspace in common. Therefore the intersection multiplicity of the n hypersurfaces is atleast two in a shaky realization. As a consequence shakiness (of order one ) can also be seen as the limiting case where tworealizations of a framework coincide [45, 47, 50].Clearly, by using the rank condition rk ( R G ( V ) ) < m one can also characterize all shaky realizations G ( V ) algebraicallyby the affine variety A ( J ) – which is referred as shakiness variety – where J denotes the ideal generated by all minors of R G ( V ) of order m × m . Let us assume that the polynomials g , . . . , g γ form the Gröbner basis of the ideal J . Note that forminimally rigid framework γ = g : det ( R G ( V ) ) =
0. Another approachtowards this so-called pure condition in terms of brackets is given in [48]. This are 6 linear constraints for d = d = Note that for bar-joint frameworks this number equals wd , where w is the number of vertices. Each additional coinciding realization raises the order of the infinitesimal flexibility by one [50].
Physical model of deformation
The snappability index presented in this paper is based on the physical model of deformation relying on the concept ofGreen-Langrange strain, which is reduced to its geometric core by eliminating the influence of material properties. In orderto do so, we make the following assumption.
Assumption 1.
All bars and bodies of the framework are made of the same homogeneous isotropic material, which isnon-auxetic; i.e. the Poisson ratio ν ∈ [ , / ] , and has a positive Young modulus E > . Due to the fact that the elastic deformation during the process of snapping are expected to be small, we can applyHooke’s law. As a consequence, the relation between applied stresses and resulting strains is a linear one, which can begiven for the spatial case by ε x ε y ε z γ xy γ xz γ yz (cid:124) (cid:123)(cid:122) (cid:125) e = E − ν − ν − ν − ν − ν − ν ( + ν ) ( + ν )
00 0 0 0 0 2 ( + ν ) (cid:124) (cid:123)(cid:122) (cid:125) = : D ( ν ) δ x δ y δ z τ xy τ xz τ yz (2)where δ i denotes the normal stress in i -direction and ε i its corresponding normal strain with i ∈ { x , y , z } . Moreover, τ i j denotes the shear stress in the i j -plane and γ i j its corresponding shear strain with i (cid:54) = i and i , j ∈ { x , y , z } .In the case of planar stress ( xy -plane) the shear stresses τ xz and τ yz are zero as well as the normal stress δ z . Then Eq. (2)simplifies to: ε x ε y γ xy (cid:124) (cid:123)(cid:122) (cid:125) e = E − ν − ν ( + ν ) (cid:124) (cid:123)(cid:122) (cid:125) = : D ( ν ) δ x δ y τ xy . (3)In the case of a bar (in x -direction) the relation reduces to ε x = D δ x with D : = E .For the later done computation of the elastic strain energies we need the inverse relations, which map the strains to thestresses. This can be obtained by inverting the matrices D i for i = , , D does not depend on Poisson’s ratio ν and itsinverse read as D − = E .As D is regular for all possible Poisson ratios ν ∈ [ , / ] , we can always compute D − ( ν ) = E − ν ν − ν ν − ν − ν
00 0 ( + ν ) for 0 ≤ ν ≤ . (4)For the spatial case, D − ( ν ) is only not defined if ν equals the upper boarder of , thus we get: D − ( ν ) : = E ν − ν + ν − − ν ν + ν − − ν ν + ν − − ν ν + ν − ν − ν + ν − − ν ν + ν − − ν ν + ν − − ν ν + ν − ν − ν + ν − ( + ν ) ( + ν )
00 0 0 0 0 ( + ν ) for 0 ≤ ν < . (5)or ν = we compute the Moore-Penrose pseudo inverse of D which yields: D − ( ) : = E − − −
29 49 − − −
29 49
00 0 0 0 0 . (6)As this matrix has rank five, there exists a 1-dimensional set of strains ( α , α , α , , , ) with α ∈ R yielding zero stresses. The study of the deformation of a polyhedron is based on the deformation of tetrahedra, which also play a central rolein the stress analysis within the finite element method (e.g. see [32, Chapter 6]). The strain computation for 4-simplicesaccording to Green-Lagrange is outlined next (e.g. see [40, Section 2.4.2]).Let V a , V b , V c , V d denote the vertices of the tetrahedron in the given undeformed configuration and V (cid:48) a , V (cid:48) b , V (cid:48) c , V (cid:48) d in thedeformed one. Then there exists a uniquely defined 3 × A which has the property A ( (cid:98) v b − (cid:98) v a ) = (cid:98) v (cid:48) b − (cid:98) v (cid:48) a , A ( (cid:98) v c − (cid:98) v a ) = (cid:98) v (cid:48) c − (cid:98) v (cid:48) a , A ( (cid:98) v d − (cid:98) v a ) = (cid:98) v (cid:48) d − (cid:98) v (cid:48) a , (7)where (cid:98) v i (resp. (cid:98) v (cid:48) i ) is a 3-dimensional vector of V i (resp. V (cid:48) i ) for i ∈ { a , b , c , d } with respect to a Cartesian frame F (resp. F (cid:48) ) attached to the undeformed (resp. deformed) tetrahedron. The Cartesian frame F can always be chosen in a way that itsorigin equals V a , the vertex V b is located on its positive x -axis and V c is located in the xy -plane with a positive y coordinate;i.e. (cid:98) v a = ( , , ) T , (cid:98) v b = ( x b , , ) T , (cid:98) v c = ( x c , y c , ) T , (cid:98) v d = ( x d , y d , z d ) T , (8)with x b > y c >
0. Similar considerations can be done for the Cartesian frame F (cid:48) with respect to the tetrahedron V (cid:48) a , V (cid:48) b , V (cid:48) c , V (cid:48) d ending up with exactly the same coordinatisation as above but only primed. Then the normal strains and theshear strains can be computed as ε x γ xy γ xz γ xy ε y γ yz γ xz γ yz ε z = (cid:0) A T A − I (cid:1) . (9)Reassembling these quantities in the vector e (cf. Eq. (2)) the elastic strain energy of the deformation can be calculated as U abcd = Vol abcd e T D − ( ν ) e (10)where Vol abcd denotes the volume of the undeformed tetrahedron and D − ( ν ) the stress/strain matrix (constitutive matrix)from Eq. (5) and Eq. (6), respectively.The same procedure can be done for the computation of the elastic strain energy U abc of a triangular panel with vertices V a , V b and V c , which is outlined in detail in [38]. As final formula we obtain in this case: U abc = Vol abc e T D − ( ν ) e (11)where D − ( ν ) denotes the stress/strain matrix from Eq. (4) and Vol abc the volume of the undeformed panel, which can becomputed as the product of the triangle’s area Area abc and the panel height h abc . For a bar with end-points V a and V b we endup with the following simple expression: U ab = E Vol ab L ab (cid:16) L (cid:48) ab − L ab (cid:17) (12)where Vol ab denotes the volume of the undeformed bar, which can be computed as the product of the length L ab of theundeformed bar and its cross-sectional area Area ab . The deformed bar length is given by L (cid:48) ab . A pseudometric on the space of intrinsic framework metrics
In this section we set up a pseudometric on the space of intrinsic framework metrics, which is based on the totalstrain energy density of the framework, because in this way the distance is invariant under scaling (change of unit length).Moreover, it allows to compare pin-jointed body-bar frameworks differing in the number of knots, the combinatorial structureand intrinsic metric.
We assume that the intrinsic metric of the framework G ( K ) is given by the edge-length vector L = ( . . . , L i j , . . . ) T ∈ R b of the equivalent framework G ∗ ( K ∗ ) . In the same way the intrinsic metric of the deformed framework is determined by L (cid:48) =( . . . , L (cid:48) i j , . . . ) T ∈ R b . As the strain energy of a polyhedron (polygonal panel) depends on its tetrahedralization (triangulation )we compute the strain energy over all tetrahedra of the polyhedron (triangles of the polygonal panel). To do so, we definethe index set C i containing all index 4-tuple abcd (3-tuple abc ) with a < b < c < d (with a < b < c ) of non-degenerated tetrahedra (triangles) within a polyhedron B i ( n i ) (polygonal panel B i ( n i ) ). Using this notation we can formulate the strainenergy density within the next lemma. Lemma 1.
The strain energy density of a pin-jointed body-bar framework given byu ( L (cid:48) ) : = ∑ ab ∈ G U ab ( L (cid:48) ) + ∑ pi = Vol ( B i ) (cid:20) ∑ abc ∈ C i U abc ( L (cid:48) ) ∑ abc ∈ C i Vol abc (cid:21) + ∑ qj = p + Vol ( B j ) (cid:20) ∑ abcd ∈ C j U abcd ( L (cid:48) ) ∑ abcd ∈ C j Vol abcd (cid:21) ∑ ab ∈ G Vol ab + ∑ pi = Vol ( B i ) + ∑ qj = p + Vol ( B j ) (13) is defined by the intrinsic metric L of the undeformed framework, the cross-sectional areas Area ab of its bars, the panelheights h abc and the material constants E and ν . The argument of the density function is given by the intrinsic metric L (cid:48) of the deformed framework. It is a fourth order polynomial with respect to the variables L (cid:48) i j which only appear with evenpowers.Proof. We prove this lemma by investigating each summand in the numerator of Eq. (13) for the stated properties. As theenergy functions differ for bars, triangular panels and tetrahedra, we have to split up the proof into these three cases: • Bar:
For bars this result follows directly from Eq. (12). • Triangular panel:
We choose a planar Cartesian frame F in a way that the coordinates of the triangle V a , V b , V c read as (cid:98) v a = ( , ) T , (cid:98) v b = ( x b , ) T and (cid:98) v c = ( x c , y c ) T with x b = L ab , x c = L ab + L ac − L bc L ab , y c = √ ( L ab + L ac + L bc )( L ab − L ac + L bc )( L ab + L ac − L bc )( L ac + L bc − L ab ) L ab (14)where the coordinate y c can have positive or negative sign for planar frameworks depending on the orientation of thetriangle V a , V b , V c . For spatial frameworks one can always assume a positive sign. Similar considerations can be done forthe planar Cartesian frame F (cid:48) with respect to the triangle V (cid:48) a , V (cid:48) b , V (cid:48) c ending up with exactly the same coordinatisation asabove but only primed. Inserting these coordinates of the six vectors (cid:98) v a , (cid:98) v b , (cid:98) v c , (cid:98) v (cid:48) a , (cid:98) v (cid:48) b , (cid:98) v (cid:48) c into Eq. (11) shows the result fortriangular panels by taking into account that the area Area abc of the triangle can be computed by Heron’s formula. Notethat the obtained expression is independent of the signs of the y -coordinates of (cid:98) v c and (cid:98) v (cid:48) c . • Tetrahedron:
We choose the same Cartesian frame F as in Section 2.2 which implies the coordinatisation of the tetrahe-dron V a , V b , V c , V d given in Eq. (8) with x b , x c and y c from Eq. (14) and x d = L ab + L ad − L bd L ab , y d = L ab L ac + L ab L ad + L ab L bc + L ab L bd − L ab L cd − L ac L ad + L ac L bd + L ad L bc − L bc L bd − L ab L ab √ ( L ab + L ac + L bc )( L ab − L ac + L bc )( L ab + L ac − L bc )( L ac + L bc − L ab ) z d = ( − L ab L cd − L ab L ac L bc + L ab L ac L bd + L ab L ac L cd + L ab L ad L bc − L ab L ad L bd + L ab L ad L cd + L ab L bc L cd + L ab L bd L cd − L ab L cd − L ac L bd + L ac L ad L bc + L ac L ad L bd − L ac L ad L cd + L ac L bc L bd − L ac L bd + L ac L bd L cd − L ad L bc − L ad L bc + L ad L bc L bd + L ad L bc L cd − L bc L bd L cd ) / / √ ( L ab + L ac + L bc )( L ab − L ac + L bc )( L ab + L ac − L bc )( L ac + L bc − L ab ) (15)where the coordinate z d can have positive or negative sign depending on the orientation of the tetrahedron V a , V b , V c , V d .We get the same coordinatisation for the tetrahedron V (cid:48) a , V (cid:48) b , V (cid:48) c , V (cid:48) d as above but only primed. Inserting these eight vectors Decomposition into a set of disjoint tetrahedra without adding new vertices. Decomposition into a set of disjoint triangles without adding new vertices. The tetrahedron (triangle) does not degenerate into a plane (line). Note that C i = ( n i − )( n i − )( n i − ) n i /
24 ( C i = ( n i − )( n i − ) n i /
6) holds ifthe polyhedron (polygonal panel) is strictly convex. v a , (cid:98) v b , (cid:98) v c , (cid:98) v d , (cid:98) v (cid:48) a , (cid:98) v (cid:48) b , (cid:98) v (cid:48) c , (cid:98) v (cid:48) d into Eq. (10) under consideration that Vol abcd can be computed by the Cayley-Menger determi-nant shows the stated result. Note that the obtained expression is independent of the sign of the z -coordinate of (cid:98) v d and (cid:98) v (cid:48) d .Moreover, it should be mentioned that the obtained polynomial is homogenous of degree 4 in L (cid:48) i j for ν = / (cid:3) Remark 2.
Concerning Lemma 1 the following should be noted: (cid:63)
The expressions given in the square brackets of Eq. (13) can be seen as the mean densities of the polygonal panels andpolyhedra, respectively. (cid:63)
Note that the height h abc of each triangular panel belonging to B i ( n i ) equals the height h i of B i ( n i ) . (cid:63) Due to Lemma 1 the formula for u ( L (cid:48) ) can be written in matrix formulation as u ( Q (cid:48) ) = Q (cid:48) T MQ (cid:48) where M is a symmetric ( b + ) -matrix and Q (cid:48) : = ( , . . . , Q (cid:48) i j , . . . ) T is composed of the b squared edge lengths Q (cid:48) i j : = L (cid:48) i j and the number . (cid:5) The pseudometric on the space R b of intrinsic framework metrics is defined within the next lemma: Lemma 2.
The following functiond : R b × R b → R ≥ with ( L (cid:48) , L (cid:48)(cid:48) ) (cid:55)→ d ( L (cid:48) , L (cid:48)(cid:48) ) : = | u ( L (cid:48) ) − u ( L (cid:48)(cid:48) ) | E (16) is a pseudometric on the b-dimensional space of intrinsic framework metrics given by L (cid:48) and L (cid:48)(cid:48) , respectively. Moreover, thepseudometric does not depend on the choice of E.Proof. One has to check the axioms for a pseudometric ( ) d ( L (cid:48) , L (cid:48)(cid:48) ) ≥ , ( ) d ( L (cid:48) , L (cid:48) ) = , ( ) d ( L (cid:48) , L (cid:48)(cid:48) ) = d ( L (cid:48)(cid:48) , L (cid:48) ) , ( ) d ( L (cid:48) , L (cid:48)(cid:48)(cid:48) ) ≤ d ( L (cid:48) , L (cid:48)(cid:48) ) + d ( L (cid:48)(cid:48) , L (cid:48)(cid:48)(cid:48) ) , (17)which is a trivial task and remains to the reader.Due to Assumption 1, Young’s modulus E factors out of the density u ( L (cid:48) ) . Therefore it factors out of the numerator ofthe distance function and cancels with the numerator. (cid:3) From Lemma 1 it is clear that the pseudodistance of Eq. (16) does not only depend on the intrinsic metric L of theundeformed framework but also on the cross-sectional areas of the bars and the heights of the panels, which are needed forthe computation of their volumes. In the following section we fix these parameters by relating them to the intrinsic geometryof the undeformed framework. As mentioned in Section 1.2 each pin-jointed body-bar framework G ( K ) can be replaced by an equivalent bar-jointframework G ∗ ( K ∗ ) . In order to ensure a fair comparability of both frameworks, we came up with the following assumption. Assumption 2.
The frameworks G ( K ) and G ∗ ( K ∗ ) have the same volume; i.e. they are built from the same amount ofmaterial. Moreover, we assume that all bars have the same cross-sectional area noted by Area (cid:31) . We start with the volumes of the polyhedra Vol (cid:16) B j ( n j ) (cid:17) , which are already determined by L , and compute Area (cid:31) asArea (cid:31) : = ∑ qj = p + Vol (cid:16) B j ( n j ) (cid:17) ∑ qj = p + ∑ ab ∈ I j W ab L ab (18)where I j is the index set of all pairs of vertices belonging to an edge of the inner graph of the polyhedron B j ( n j ) . Moreover,the weight factor W ab is one over the number of bodies hinged along the corresponding bar .Now having Area (cid:31) one can also compute the height h i of the polygonal panel B i ( n i ) over the bar-joint subframeworkequivalent to B i ( n i ) as: h i : = Area (cid:31) ∑ ab ∈ I i W ab L ab Area (cid:0) B i ( n i ) (cid:1) (19) If the bar does not hinge bodies, then the weight factor is one. here I i is the index set of all pairs of vertices belonging to an edge of the inner graph of the polygonal panel B i ( n i ) . Inthe case that the framework does not contain any polyhedra, then we can compute h i in the same way but depending on theunknown Area (cid:31) . In this case it can easily be seen that Area (cid:31) factors out in the numerator as well as in the denominatorof Eq. (13). Therefore Eq. (16) does not depend on Area (cid:31) . This also holds if the given framework is already a bar-jointframework. Under consideration of Section 3.2.1 the pseudometric d only depends on Poisson’s ratio ν beside the intrinsic metric L of the undeformed framework. From the geometric point of view the most satisfying choice is ν = / isochoric deformation. This does not pose any problemsfor a bar or triangular panel, as one can always define the cross-sectional area Area (cid:48) ab of the deformed bar by Vol ab / L (cid:48) ab and aheight h (cid:48) abc of the deformed panel by Vol abc / Area (cid:48) abc , respectively, where Area (cid:48) abc is the area of the deformed triangle V (cid:48) a , V (cid:48) b , V (cid:48) c . But for each tetrahedron we get the additional condition that Vol abcd = Vol (cid:48) abcd holds, which fits very well into our theoryfor the following reason: At the end of Section 2.1 we have seen that the strain vector ( α , α , α , , , ) with α ∈ R yieldszero stresses in a tetrahedron. Plugging the entries of this strain vector into Eq. (9) shows that it results from an equiformmotion (Euclidean motion plus a scaling). But an equiform motion keeping the volume fixed has to be an Euclidean motion( ⇔ α = abcd = Vol (cid:48) abcd resolves also the problem arising fromthe singularity of the stress/strain matrix given in Eq. (6). Therefore one is only allowed to compute the pseudodistance ofEq. (16) for ν = / abcd = Vol (cid:48) abcd holds for all abcd ∈ C j for j = p + , . . . , q .Clearly, theoretically one can also use another Poisson ratio 0 ≤ ν < / Remark 3.
If one does not want to use the Poisson ratio ν = / , then one is confronted to make a choice within theinterval [
0; 1 / [ . One can circumvent the arbitrariness in the choice by determining ν within a constrained optimization( ≤ ν < / ) in such a way that the distance of Eq. (16) is minimal. The disadvantage of this approach is that the triangularinequality of Eq. (17) cannot longer be guaranteed thus the pseudometric degenerates to a so-called premetric. (cid:5) Finally, it should be noted that the results of the next sections are general ones; i.e. the assumptions done in Section3.2.1 and 3.2.2 are not necessary for their validity.
If we want to compute the distance between L and L (cid:48) then the pseudometric d ( L , L (cid:48) ) simplifies to u ( L (cid:48) ) / E as this func-tion is positive-definite, which is clear from the underlying physical interpretation but one can also prove this mathematicallyby decomposing it into a sum of squares (see e.g. [41]).As we can replace L (cid:48) i j in u ( L (cid:48) ) by (cid:107) v (cid:48) i − v (cid:48) j (cid:107) the function u can be computed in dependence of V (cid:48) ; i.e. u ( V (cid:48) ) . Theorem 1.
For ≤ ν < / the critical points of the total elastic strain energy density u ( V (cid:48) ) of a pin-jointed body-barframework correspond to realizations G ∗ ( V (cid:48) ) of the equivalent bar-joint framework that are either undeformed or deformedwith a non-zero self-stress. This also holds for ν = / under the side conditions of constant tetrahedral volumes.Proof. The system of equations characterizing the critical points of u ( V (cid:48) ) reads as follows: ∇ i u ( V (cid:48) ) = o with ∇ i u ( V (cid:48) ) = (cid:16) ∂ u ∂ x (cid:48) i , ∂ u ∂ y (cid:48) i (cid:17) for d = ∇ i u ( V (cid:48) ) = (cid:16) ∂ u ∂ x (cid:48) i , ∂ u ∂ y (cid:48) i , ∂ u ∂ z (cid:48) i (cid:17) for d = i = , . . . , w , where ( x (cid:48) i , y (cid:48) i ) T and ( x (cid:48) i , y (cid:48) i , z (cid:48) i ) T is the coordinate vector of v (cid:48) i for the planar and spatial case, respectively.Due to the sum rule for derivatives we only have to investigate ∇ i of the following three functions: U ab ( v (cid:48) a , v (cid:48) b ) of Eq. (12), U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) given in Eq. (11) and U abcd ( v (cid:48) a , . . . , v (cid:48) d ) of Eq. (10), respectively.1. Due to ∇ a U ab ( v (cid:48) a , v (cid:48) b ) = Area ab ( L (cid:48) ab − L ab ) L ab ( v (cid:48) a − v (cid:48) b ) Theorem 1 is valid for frameworks, which only consist of bars, as ∇ a u ( V (cid:48) ) can be written in the form of Eq. (1) with ω ab = Area ab ( L (cid:48) ab − L ab ) L ab .. If polygonal panels are involved we consider a representative triangular panel with vertices V a , V b , V c and compute ∇ a U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) , ∇ b U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) and ∇ c U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) . Straight forward symbolic computations (e.g. using Maple)show that the following system of equations ω ab ( v (cid:48) a − v (cid:48) b ) + ω ac ( v (cid:48) a − v (cid:48) c ) − ∇ a U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) = o ω ab ( v (cid:48) b − v (cid:48) a ) + ω bc ( v (cid:48) b − v (cid:48) c ) − ∇ b U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) = o ω ac ( v (cid:48) c − v (cid:48) a ) + ω bc ( v (cid:48) c − v (cid:48) b ) − ∇ c U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) = o (21)which is overdetermined , has a unique solution for ω ab , ω ac and ω bc if V (cid:48) a , V (cid:48) b , V (cid:48) c generate a triangle. If these pointsare collinear we even get a positive dimensional solution set. Hence, one can replace ∇ a U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) by a linearcombination ω ab ( v (cid:48) a − v (cid:48) b ) + ω ac ( v (cid:48) a − v (cid:48) c ) where the coefficients ω ab and ω ac are compatible with the other equations of(21).3. If bodies are involved we consider a representative tetrahedron with vertices V a , V b , V c , V d and compute ∇ a U abcd ( v (cid:48) a , . . . , v (cid:48) d ) , ∇ b U abcd ( v (cid:48) a , . . . , v (cid:48) d ) , ∇ c U abcd ( v (cid:48) a , . . . , v (cid:48) d ) and ∇ d U abcd ( v (cid:48) a , . . . , v (cid:48) d ) . Now we are faced with the system of equations ω ab ( v (cid:48) a − v (cid:48) b ) + ω ac ( v (cid:48) a − v (cid:48) c ) + ω ad ( v (cid:48) a − v (cid:48) d ) − ∇ a U abcd ( v (cid:48) a , . . . , v (cid:48) d ) = o ω ab ( v (cid:48) b − v (cid:48) a ) + ω bc ( v (cid:48) b − v (cid:48) c ) + ω bd ( v (cid:48) b − v (cid:48) d ) − ∇ b U abcd ( v (cid:48) a , . . . , v (cid:48) d ) = o ω ac ( v (cid:48) c − v (cid:48) a ) + ω bc ( v (cid:48) c − v (cid:48) b ) + ω cd ( v (cid:48) c − v (cid:48) d ) − ∇ c U abcd ( v (cid:48) a , . . . , v (cid:48) d ) = o ω ad ( v (cid:48) d − v (cid:48) a ) + ω bd ( v (cid:48) d − v (cid:48) b ) + ω cd ( v (cid:48) d − v (cid:48) c ) − ∇ d U abcd ( v (cid:48) a , . . . , v (cid:48) d ) = o (22)which is again overdetermined (12 equations in six unknowns ω ab , ω ac , ω ad , ω bc , ω bd and ω cd ). Again direct computationsshow that there exists a unique solution if V (cid:48) a , V (cid:48) b , V (cid:48) c , V (cid:48) d span a 3-space; otherwise even a positive dimensional solution setexists. Thus one can substitute ∇ a U abcd ( v (cid:48) a , . . . , v (cid:48) d ) by a linear combination ω ab ( v (cid:48) a − v (cid:48) b ) + ω ac ( v (cid:48) a − v (cid:48) c ) + ω ad ( v (cid:48) a − v (cid:48) d ) where the coefficients ω ab , ω ac and ω ad are compatible with the other equations of (22).Summing up the results of the three items shows that ∇ a u ( V (cid:48) ) can be written in the form of Eq. (1) which proves the theoremfor ν < / ν = / F ( V (cid:48) , λ ) = u ( V (cid:48) ) − λ f − . . . − λ ϕ f ϕ with λ : = ( λ , . . . , λ ϕ ) , (23)where f , . . . , f ϕ are the isochoric constraints of the form Vol (cid:48) abcd − Vol abcd = abcd ∈ C j for all j = p + , . . . , q ;i.e ϕ = ∑ qj = p + C j . Due to the Cayley-Menger determinant we get the squared volume V (cid:48) abcd of the tetrahedron spannedby V (cid:48) a , . . . , V (cid:48) d , as a polynomial in the squared distances of these vertices. Now one can replace in Eq. (22) the function U abcd ( v (cid:48) a , . . . , v (cid:48) d ) by V (cid:48) abcd ( v (cid:48) a , . . . , v (cid:48) d ) and do the analogous computation ending up with the same conclusion. Therefore ∇ a F ( V (cid:48) , λ ) is again of the form of Eq. (1) which proves the theorem for ν = / (cid:3) Remark 4.
One can also ask for the critical points of the elastic strain energy density u ( L (cid:48) ) ; i.e. we have to consider thepartial derivatives with respect to the edge lengths L (cid:48) i j . It can easily be seen that there is only one valid critical point namely L (cid:48) = L as all other solutions of the resulting system imply at least one edge of zero length. These invalid solutions can beavoided by considering u ( Q (cid:48) ) of Remark 2 and its partial derivatives with respect to Q (cid:48) i j ending up in a linear system. (cid:5) For the formulation of the next theorem we also need the notation of stability. A realization G ( V (cid:48) ) is called stable if itcorresponds to a local minimum of the total elastic strain energy of the framework, which is also a minimum of the strainenergy density u ( V (cid:48) ) . Theorem 2.
If a pin-jointed body-bar framework snaps out of a stable realization G ( V ) by applying the minimum strainenergy needed to it, then the corresponding deformation passes a realization G ( V (cid:48) ) at the maximum state of deformation,where the equivalent bar-joint framework G ∗ ( V (cid:48) ) has a non-zero self-stress.Proof. We think of u as a graph function over the space R wd of knot configurations. In order to get out of the valley of thelocal minimum ( V , u ( V )) , which corresponds to the given stable realization G ( V ) , with a minimum of energy needed, onehas to pass a saddle point ( V (cid:48) , u ( V (cid:48) )) of the graph, which corresponds to a realization G ( V (cid:48) ) . As a local extrema as well assaddle points of the graph function are given by the critical points of u we can use Theorem 1, which implies that these pointscorrespond with self-stressed realizations of the equivalent bar-joint framework. As u ( V (cid:48) ) > G ∗ ( V (cid:48) ) is deformed which has to imply a non-zero self-stress; i.e. the stress-vector ω differs from the b -dimensional zero vector. (cid:3) As a triangle is planar, we get in total 6 equations in three unknowns from Eq. (21). orollary 1.
If the equivalent bar-joint framework of Theorem 2 is minimally rigid, then “non-zero self-stress” can bereplaced by “shakiness”.Proof.
If the equivalent bar-joint framework is minimally rigid then the existence of a non-zero self-stress implies a rankdefect of the square rigidity matrix (cf. end of Section 1.2), which results in an infinitesimal flexibility. (cid:3)
But also without the assumption of minimal rigidity used in Corollary 1, one can give the following connection betweenshakiness and snapping.
Theorem 3.
One can replace “non-zero self-stress” by “shakiness” in Theorem 2 if there exists a deformation such thatthe path from G ∗ ( V ) to G ∗ ( V (cid:48) ) is identical to the path of G ∗ ( V (cid:48)(cid:48) ) to G ∗ ( V (cid:48) ) in the space of intrinsic framework metrics R b ,where G ∗ ( V ) and G ∗ ( V (cid:48)(cid:48) ) are not related by a direct isometry.Proof. Let us assume that there exists a path ( V t , u ( V t )) on the graph with parameter t ∈ [ , ] such that for t = ( V , u ( V )) and for t = ( V (cid:48) , u ( V (cid:48) )) . This deformation implies a path L t in the space R b of intrinsic metrics with L t (cid:12)(cid:12) t = = L (cid:48) and L t (cid:12)(cid:12) t = = L .But vice versa the path L t corresponds to several 1-parametric deformations in R d , where one of these deformations ( V t , u ( V t )) has to lead towards ( V (cid:48)(cid:48) , u ( V (cid:48)(cid:48) )) according to our assumption. Moreover, tracking the realizations of the path L t with t ∈ [ , ] shows that in G ∗ ( V (cid:48) ) two realizations coincide, which implies that G ∗ ( V (cid:48) ) is shaky. (cid:3) Remark 5.
Note that G ( V (cid:48)(cid:48) ) can also be a complex realization. In this case the deformation ( V t , u ( V t )) towards ( V (cid:48)(cid:48) , u ( V (cid:48)(cid:48) )) get stuck on the boarder of reality. Therefore the snap ends up in a realization of the equivalent bar-joint framework, whichis shaky as a real solution of an algebraic set of equations can only change over into a complex one through a double root. (cid:5) If the two realizations G ∗ ( V ) and G ∗ ( V (cid:48)(cid:48) ) of Theorem 3 are thought infinitesimal close to G ∗ ( V (cid:48) ) then we get thefollowing characterization of shakiness: Corollary 2. G ∗ ( V (cid:48) ) is shaky, if three exist two instantaneous snapping deformations ( (cid:54) = infinitesimal isometric deforma-tions) out of G ∗ ( V (cid:48) ) represented by two non-zero vectors in R wd pointing into distinct directions, whose corresponding twovectors of instantaneous changes of the intrinsic metric in R b are identical. Remark 6.
Note that within the set of pairs of vectors fulfilling Corollary 2, there exists at least one pair of oppositelydirected vectors in R wd , which both correspond to the zero-vector in R b . (cid:5) Based on Theorem 2 we can evaluate the snappability of the pin-jointed body-bar framework G ( K ) in the undeformedrealization G ( V ) by the value s ( V ) : = d ( L , L (cid:48) ) = u ( L (cid:48) ) / E , which we call local snappability . As in general a framework hasseveral undeformed realizations G ( V ) , . . . , G ( V k ) we can define a global snappability by s ( L ) : = min { s ( V ) , . . . , s ( V k ) } . We compute the critical points of the Lagrange function of Eq. (23) by using the homotopy continuation method (e.g.Bertini; cf. [3]) as other approaches (e.g. Gröbner base, resultant based elimination) are not promising due to the number ofunknowns and degree of equations. By choosing a suitable reference frame we can reduce the number of unknowns by 6 for d = d =
2, respectively, which also eliminates isometries (cf. footnote 3).
Remark 7.
The computation of the critical points, which depends heavily on the number of unknowns, can be a timeconsuming task in the first run of the homotopy. But if one changes the inner metric of the framework within the designprocess the critical points of the resulting new system of equations can be computed from the already known critical pointsof the initial system more efficiently by means of parameter homotopy [3]. (cid:5)
First of all we can restrict to the obtained real critical points as only these correspond to realizations. This resulting set R of realizations is split into a set E , whose elements correspond to local extrema of u ( V (cid:48) ) , and its absolute complement S = R \ E of so-called saddle realizations . This separation can be done by the second partial derivative test ; i.e. all eigenvaluesof the Hessian matrix of the function u ( V (cid:48) ) are either positive ( ⇔ local minimum) or negative ( ⇔ local maximum). Let usdenote the set of stable realizations by M ⊂ E , which correspond to local minima.One way for computing the local snappability is to start at saddle points and apply gradient descent algorithms to findneighboring local minima. This was done in the following example of a full quad. Example 1.
We consider a full quad with vertices A , B , C , D. Its intrinsic metric L = ( AB , AC , AD , BC , BD , CD ) is given by:AB = , CD = √ , AC = BD = √ , AD = BC = √ . (24) ( V ) G ( V ) G ( V ) G ( V ) G ( V ) G ( V ) G ( V ) G ( V ) G ( V ) G ( V ) G ( V ) G ( V )
11 221 12 213 1322
Fig. 1. Directed graph relating the stable realizations (undeformed green and deformed yellow) and saddle relations (red), where the orien-tation points towards local minima. The number i beside an arrow refers to the gradient flow in direction of the i -th smallest main curvature.If this number is missing, then there is only one negative main curvature direction. The two blue/violet dotted arrows indicate that these twoflows end up in a realization obtained from G ( V / ) by reflecting it at the y -axis ( ⇒ x < which contradicts our assumption). The edgeand arrow between G ( V ) and G ( V ) are dotted as under this gradient flow all points move along the x -axis, which imply that some edgelengths become zero during the deformation. But taking small perturbations into account these zeros can be avoided. As this arrow appearsbetween two saddle realizations, it means that G ( V ) can also be deformed into G ( V ) and G ( V ) , respectively. The second black dottedarrow pointing upwards indicate that this flow ends up in the mirrored realization of G ( V ) ( ⇒ x < which contradicts our assumption).Finally it should be noted that the transition between the undeformed realizations G ( V ) and G ( V ) can not only be done by snaps over theshaky realizations G ( V ) and G ( V ) , respectively, but also by two subsequent snaps from G ( V ) over G ( V / ) to G ( V / ) and furtherover G ( V / ) to G ( V ) , without passing a shaky realization.Fig. 2. Local minima: The two undeformed realizations G ( V ) and G ( V ) and the two deformed stable realizations G ( V ) and G ( V ) (from left to right). The point A is colored red, B green, C blue and D yellow, respectively. This color-coding is also used in the Figs. 3–5. For the computation we coordinatise the vertices as follows:A = ( − x , ) , B = ( x , ) , C = ( x , y ) , D = ( x , y ) . (25) In this way the bar from A to B is attached to the x-axis of the reference frame. Due to the fact that a bar cannot have zerolength, we restrict ourselves to realizations of M and S where no points coincide. Moreover, we can assume without loss ofgenerality that x > holds, as a continuous deformation between a realization with x < and a realization with x > hasto pass x = ( ⇔ A = B). A in-depth analysis of the remaining critical points results in Fig. 1, which shows a directed graphrelating the local minima and saddle points, where the orientation points towards local minima. The stable realizations areillustrated in Fig. 2 and the saddle realizations in Figs. 3 and 4, respectively. Representative snaps (transitions) betweenstable realizations are illustrated in Fig. 5.Even though this framework is globally rigid we get two undeformed realizations G ( V ) and G ( V ) , which are mirror-symmetric with respect to the x-axis (Fig. 2). Moreover, it is possible to snap from G ( V ) into G ( V ) (cf. Figs. 1 and 5).Thus the property of being globally rigid cannot save the framework from the snapping phenomenon.The values for u ( V i ) / E as well as x , x , y , x , y of the stable/saddle realizations are given in Table 1. From thesevalues and the graph given in Fig. 1 one sees that the two local snappabilities are equal; thus s ( L ) = s ( V ) = s ( V ) = . holds. In practice one starts with the saddle realization G ( V i ) with the lowest value for u ( V i ) / E and compute gradient flowstowards the stable realizations, hoping that one ends up in the undeformed realization under consideration. If one does notfind such a descent path, then one repeats the procedure for the saddle realization with the next higher value for u ( V i ) / E . ig. 3. Saddle points: These four realizations G ( V ) , . . . , G ( V ) (from left to right) correspond to saddle points. Three points look to becollinear but they are not (cf. values given in Table 1).Fig. 4. Saddle points: These four realizations G ( V ) , . . . , G ( V ) (from left to right) correspond to saddle points. Moreover, the realizations G ( V ) and G ( V ) are shaky realizations.Fig. 5. The gradient flows starting at some saddle realizations in direction of negative main curvature. In one direction the paths of thepoints are plotted solid and in the opposite direction dotted. Upper left: Saddle realization G ( V ) and the corresponding paths of the pointstowards the realizations G ( V ) and G ( V ) . Lower left: Following the gradient flow starting at the saddle realization G ( V ) in direction ofthe second-smallest main curvature shows that both paths end up in the realization G ( V ) . Therefore this realization can snap into itself overthe saddle realization G ( V ) . Upper (center) right: Saddle realization G ( V ) and the gradient flow in direction of the smallest (second-smallest) main curvature. This shows the snapping between G ( V ) and G ( V ) ( G ( V ) and G ( V ) ) over the shaky realization G ( V ) . Bycombining the different gradient flows G ( V ) can also snap into G ( V ) over G ( V ) , but one needs more deformation energy compared tothe snap over the saddle realization G ( V ) illustrated in the upper left corner (cf. Table 1). Lower right: Saddle realization G ( V ) and thegradient flow in direction of the third-smallest main curvature. This shows an alternative snapping between G ( V ) and G ( V ) , which alsoneeds more deformation energy as the one over G ( V ) illustrated in the upper right corner (cf. Table 1). Finally note that the three snapsillustrated in the right column also demonstrate Theorem 3. The problem of this approach is that one has no guarantee to detected all local minima which can be reached from a saddlepoint by a descent path. With guarantee one can only give a lower bound for the global snappability by the following value:
Lemma 3.
Let us assume that G ( V − ) ∈ S implies the minimal value for d ( L , L − ) for all elements of S . As a resulto ( L ) : = d ( L , L − ) ≤ s ( L ) has to hold. (cid:5) Therefore we want to present a more sophisticated approach, which allows us to check directly if a saddle realization anda stable realization can be deformed continuously into each other, whereby the deformation energy density has to decreasemonotonically. The minor drawback of this method is that it only works for pin-jointed body-bar framework, which areisostatic. x y x y u ( V i ) / E V V V V V V V V V V V V Table 1. Coordinates of stable/saddle realizations and their corresponding u ( V i ) / E value. We assume that the given pin-jointed body-bar framework G ( K ) is isostatic . By replacing the bodies by the cor-responding globally rigid subframeworks, the equivalent body-bar framework G ∗ ( K ∗ ) is also isostatic, if all polyhedra(polygonal panels) are tetrahedral (triangular). But in the general case the subframeworks are overbraced.The edges of the overbraced subframeworks are involved in the computation of the density function given in Lemma1. One can get rid of this property by allowing only affine (homogeneous) deformations of the bodies. In this way everypolyhedron (polygonal panel) can be represented by a tetrahedron V a , V b , V c , V d (triangle V a , V b , V c ) and the remaining verticesof the deformed polyhedron (polygonal panel) can then be obtained by the affine transformation determined by V i (cid:55)→ V (cid:48) i for i ∈ { a , b , c , d } (resp. i ∈ { a , b , c } ).As a consequence we can consider a minimal set of lengths L i j which contains the lengths of the bars ∈ G plus foreach polyhedron (polygonal panel) we get six (three) additional lengths. We collect these lengths within the vectors (cid:101) L : =( . . . , L i j , . . . ) T ∈ R a and (cid:101) Q : = ( , . . . , Q i j , . . . ) T ∈ R a + with a ≤ b .Using this notation the density function can be rewritten as follows: u ( (cid:101) L (cid:48) ) : = ∑ ab ∈ G U ab ( (cid:101) L (cid:48) ) + ∑ pi = Vol ( B i ) (cid:104) U abc ( (cid:101) L (cid:48) ) Vol abc (cid:105) + ∑ qj = p + Vol ( B j ) (cid:104) U abcd ( (cid:101) L (cid:48) ) Vol abcd (cid:105) ∑ ab ∈ G Vol ab + ∑ pi = Vol ( B i ) + ∑ qj = p + Vol ( B j ) (26)for an arbitrary abc ∈ C i and abcd ∈ C j , respectively. As we can replace L (cid:48) i j in u ( (cid:101) L (cid:48) ) by (cid:107) v (cid:48) i − v (cid:48) j (cid:107) the function u can becomputed in dependence of (cid:101) V (cid:48) ∈ R ˜ wd with ˜ w = r + p + ( q − p ) , where (cid:101) V (cid:48) contains the vectors of the vertices V , . . . , V r as well as three vertices of each polygonal panel and four vertices of each polyhedron, respectively. With respect to theresulting density function u ( (cid:101) V (cid:48) ) we compute the set M of stable realizations and the set S of saddle realizations. Moreover,the following theorem holds true: Theorem 4.
If an isostatic pin-jointed body-bar framework snaps out of a stable realization G ( V ) by applying the minimumstrain energy needed to it, then the framework passes a realization G ( V (cid:48) ) at the maximum state of deformation (under theassumption that each body is deformed affinely), which is either shaky or contains at least a body of reduced dimension.Proof. For the analysis of the equations ∇ a u ( (cid:101) V (cid:48) ) = V a ∈ B i is a vertex of a tetrahedron (triangle) representing a polyhedron (polygonal panel).From Theorem 1 it is already known that ∇ a U abcd ( v (cid:48) a , v (cid:48) b , v (cid:48) c , v (cid:48) d ) (resp. ∇ a U abc ( v (cid:48) a , v (cid:48) b , v (cid:48) c ) ) can be written as a linear combi-nation of the involved vectors (cf. Eq. (21) and Eq. (22), respectively). Therefore we are only left with the partial derivativeof the elastic strain energy of a bar ex ∈ G with V e ∈ B i , i.e. v e = v a + ξ ( v b − v a ) + υ ( v c − v a ) + ζ ( v d − v a ) (27) Note that the isostaticity of a spatial framework is a problem for its own and not treated within this article (see e.g. [7, 14]) here ζ = ∇ a U ex ( v (cid:48) a , v (cid:48) b , v (cid:48) c , v (cid:48) d , v (cid:48) x ) = Area ex ( L (cid:48) ex − L ex ) L ex (cid:2) ξ ( v (cid:48) b − v (cid:48) a ) + υ ( v (cid:48) c − v (cid:48) a ) + ζ ( v (cid:48) d − v (cid:48) a ) + v (cid:48) a − v (cid:48) x (cid:3) ( − ξ − υ − ζ ) . (28)This shows that ∇ a u ( (cid:101) V (cid:48) ) can be written as a linear combination of the partial derivatives of the squared distances of verticeslinked by tetrahedral (triangular) edges or green edges (edges ∈ G ).As a consequence, each solution of the ˜ w equations of the form ∇ i u ( (cid:101) V (cid:48) ) = wd -dimensionalstress-vector ω . If this vector differs from the zero vector then the solution has to imply a rank defect of the square matrix,whose columns are the gradient vectors of the realization equations c , . . . , c ˜ wd . This rank defect either corresponds with ashaky configuration of the framework G ( V (cid:48) ) or arise from the shakiness of a substructure (i.e. a tetrahedron or a triangle)substituting a body. A tetrahedron (triangle) can only get infinitesimal flexible if its dimension is reduced to at least a plane(line). (cid:3) An advantage of Theorem 4 over Theorem 2 is that the property of the realization G ( V (cid:48) ) concerns the pin-jointed body-bar framework and not the equivalent bar-joint framework (assumed that the given pin-jointed body-bar framework is not abar-joint framework). Is the assumption of affine deformations really a restriction?
For our preferred choice of ν = / • Assume a polyhedron B j ( n j ) with n j > ( b , c , d , e ) , ( a , c , d , e ) , ( a , b , d , e ) and ( a , b , c , e ) belong to the index set C j of non-degenerated tetrahedra. Therefore one can compute the barycentric coor-dinates of V e with respect to the tetrahedron V a , . . . , V d , which equal the ratio of the oriented volumes ( Vol bcde : Vol acde :Vol abde : Vol abce ) according to [35].Now we assume that the tetrahedron V a , . . . , V d was deformed isochoricly into V (cid:48) , . . . , V (cid:48) . Therefore not only the volumeremains constant under the deformation, but also its orientation . As a consequence the mapping from V , . . . , V to V (cid:48) , . . . , V (cid:48) can be written as v (cid:48) i = Av i + a with det ( A ) = i = a , b , c , d . (29)Then V (cid:48) e is uniquely determined by its barycentric coordinates ( Vol (cid:48) bcde : Vol (cid:48) acde : Vol (cid:48) abde : Vol (cid:48) abce ) with respect to thetetrahedron V (cid:48) a , . . . , V (cid:48) d which have to equal the above given homogenous 4-tuple. This already implies that V (cid:48) e and V e arealso in the affine correspondence of Eq. (31) (e.g. [1, page 61]). • Assume a polygonal panel B i ( n i ) with n i > ( b , c , d ) , ( a , c , d ) and ( a , b , d ) belong tothe index set C i of non-degenerated triangles. According to Section 3.2.2 the panel height h (cid:48) abc of the deformed triangularsubpanel can be computed as Vol abc / Area (cid:48) abc . According to the second item of Remark 2 this height can be identified withthe height h (cid:48) i of the deformed panel. In order that the heights h (cid:48) bcd , h (cid:48) acd and h (cid:48) abd of the other three triangular subpanels areequal to h (cid:48) i the following ratio has to hold:Area (cid:48) abc : Area (cid:48) bcd : Area (cid:48) acd : Area (cid:48) abd = Area abc : Area bcd : Area acd : Area abd . (30)By means of planar barycentric coordinates it can be seen that all four points have to be mapped by an affine transforma-tion: v (cid:48) i = Av i + a for i = a , b , c , d . (31)Note that in this case the matrix A is not restricted to det ( A ) = (cid:48) abc (cid:54) =
0. For d = (cid:48) abc (cid:54) = ( A ) (cid:54) = R (and still open for R ; cf. Section 1.2) and characterized by 3-connectivity and redundantrigidity of the graph [22], this assumption is maybe restrictive. Taking these possible minor restrictions into account, wecan compute the local snappability in an efficient way as follows. A change in orientation can only happen in a flat pose of the tetrahedron, which has zero volume. To clarify this open problem, one has to study in more detail the properties of globally rigid inner graphs (cf. Definition 1). lgorithm for computing the local snappability.
Given is an undeformed realization G ( (cid:101) V ) ∈ M and we want to deter-mine s ( (cid:101) V ) . To do so, we consider the saddle realization G ( (cid:101) V (cid:48) ) ∈ S which has the minimal value for d ( (cid:101) L , (cid:101) L (cid:48) ) and define thetransformation (cid:101) Q t : = (cid:101) Q + t ( (cid:101) Q (cid:48) − (cid:101) Q ) with t ∈ [ , ] . (32)This gradient flow in the space of squared leg lengths (cf. Remark 4) implies a path (cid:101) L t in R a between (cid:101) L and (cid:101) L (cid:48) . Along thispath the deformation energy of each tetrahedron U abcd , triangular panel U abc as well as bar U ab is monotonic increasing withrespect to the path parameter t . This ensures that only the minimum mechanical work needed is applied on the framework toreach G ( (cid:101) V (cid:48) ) . This results from Lemma 1, as U abcd ( (cid:101) L t ) , U abc ( (cid:101) L t ) as well as U ab ( (cid:101) L t ) are quadratic functions in t , which are attheir minima for t =
0. The path (cid:101) L t corresponds to different 1-parametric deformations of realizations in R d . If among thesea deformation G ( (cid:101) V t ) with the property G ( (cid:101) V t ) (cid:12)(cid:12) t = = G ( (cid:101) V ) , G ( (cid:101) V t ) (cid:12)(cid:12) t = = G ( (cid:101) V (cid:48) ) (33)exists, then the given realization G ( (cid:101) V ) is deformed into G ( (cid:101) V (cid:48) ) under (cid:101) L t . Computationally the property (33) can easily bechecked as follows: We consider the set of algebraic realization equations c , . . . , c n implied by the framework (cf. Section1.2). Due to Eq. (32) the equations, which correspond to bar constraints, depend linearly on t . We have to track the pathof the solution (cid:101) V of this algebraic system while t is increasing from zero to one. This is a homotopy continuation problemwhich can be solved efficiently e.g. by the software Bertini [3, Section 2.3]. Remark 8.
Note that this approach has to be adapted in the special case that the undeformed realization G ( (cid:101) V ) ∈ M isshaky, as (cid:101) V is a singular solution of the algebraic system for t = . In this case one has to solve the set of algebraic equationsc , . . . , c n for a random value t ∗ ∈ ( , ) . The resulting solutions are then tracked by homotopy continuation back to the valuet = . At least two paths have to lead down to (cid:101) V . The corresponding solutions at t = t ∗ of these paths are then tracked byhomotopy continuation up to the value t = to check if one of them ends up at (cid:101) V (cid:48) . (cid:5) If such a deformation does not exist then we redefine S as S \ (cid:110) G ( (cid:101) V (cid:48) ) (cid:111) and run again the procedure explained in thisparagraph until we get the sought-after realization implying s ( (cid:101) V ) . If we end up with S = { } then we set s ( (cid:101) V ) = ∞ . Remark 9.
Even if G ( (cid:101) V ) and G ( (cid:101) V (cid:48) ) have the same volume, the deformation given by Eq. (32) is in general not isochoric. (cid:5) In the following we want to determine the real point V (cid:48)(cid:48)(cid:48) of the shakiness variety V ( J ) (cf. end of Section 1.2) minimizingthe value d ( L , L (cid:48)(cid:48)(cid:48) ) , where G ( V ) is the given undeformed realization of a pin-jointed body-bar framework G ( K ) . In additionthere should again exist a 1-parametric deformation of G ( V ) into G ( V (cid:48)(cid:48)(cid:48) ) such that the deformation energy density has toincrease monotonically. If this is the case we call ς ( V ) = d ( L , L (cid:48)(cid:48)(cid:48) ) = u ( L (cid:48)(cid:48)(cid:48) ) / E the local singularity-distance . By taking theminimum of all local singularity-distances of possible undeformed realizations of a framework we get the global singularity-distance ς ( L ) ; i.e. ς ( L ) : = min { ς ( V ) , . . . , ς ( V k ) } .In the general case one has to compute the local minima of the Lagrangian F ( V (cid:48) , λ ) = u ( V (cid:48) ) − λ f − . . . − λ ϕ f ϕ − λ ϕ + g − . . . − λ ϕ + γ g γ with λ : = ( λ , . . . , λ ϕ + γ ) , (34)where we recall that g , . . . , g γ are the generators of the ideal J of the shakiness variety A ( J ) and f , . . . , f ϕ denote the sideconditions for an isochoric deformation if this is desired. Starting from the corresponding realizations one can apply againgradient descent algorithms with respect to the function of the deformation energy density in order to find the neighboringstable realizations. Clearly, this strategy is faced with the same problems as already mentioned in Section 4.1. But forpin-jointed body-bar frameworks with minimal rigidity under affine transformations we are able to prove the followingstatements (Theorem 5 and Corollaries 3 and 4). Theorem 5.
If the undeformed realization G ( V ) is not shaky, then the local snappability s ( V ) is a lower bound on the localsingularity-distance ς ( V ) ; i.e. s ( V ) ≤ ς ( V ) . If the realization G ( V (cid:48) ) , which implies the snappability s ( k ) , is shaky, then theequality holds.roof. We show the relation s ( V ) ≤ ς ( V ) indirectly by assuming ς ( V ) < s ( V ) . We denote the shaky realization implying ς ( V ) by G ( V (cid:48)(cid:48) ) which corresponds to (cid:101) L (cid:48)(cid:48) ∈ R a . In analogy to Eq. (32) we consider the relation (cid:101) Q t : = (cid:101) Q + t ( (cid:101) Q (cid:48)(cid:48) − (cid:101) Q ) with t ∈ [ , ] (35)defining a path (cid:101) L t in R a between (cid:101) L and (cid:101) L (cid:48)(cid:48) , which corresponds to a set of 1-parametric deformations (cid:8) G ( V t ) , G ( V t ) , . . . (cid:9) .A subset D of this set has the property G ( V it ) | t = = G ( V (cid:48)(cid:48) ) where D > G ( V (cid:48)(cid:48) ) is shaky [50, 45]. Therefore theframework can snap out of G ( V ) over G ( V (cid:48)(cid:48) ) which contradicts ς ( V ) < s ( V ) ( = ⇒ s ( V ) ≤ ς ( V ) ). (cid:3) In the case of Theorem 5 the local snappability gives the radius of a guaranteed singularity-free sphere in the spaceof intrinsic framework metrics for a non-shaky realization. Note that in the space R b of squared edge lengths Q i j thissingularity-free zone is bounded by a hyperellipsoid due to the third item of Remark 2.Moreover, Theorem 5 implies the following statement: Corollary 3.
If non of the undeformed realizations of a framework G ( V ) , . . . , G ( V k ) is shaky, then the global snappabilitys ( L ) is a lower bound on the global singularity-distance ς ( L ) . Note that in case of Corollary 3 also o ( L ) ≤ ς ( L ) has to hold due to Lemma 3. Therefore ς ( L ) as well as o ( L ) are radiiof guaranteed singularity-free spheres in the space of intrinsic framework metrics for any of the undeformed realizations.Moreover, we can make the following statement on the reality of deformations: Corollary 4.
The deformation associated with the local snappability s ( V ) = d ( L , L (cid:48) ) , which is implied by Eq. (32), isguaranteed to be real, if not both realizations G ( V ) and G ( V (cid:48) ) are shaky.Proof. A real solution of an algebraic set of equations can only change over into a complex one through a double root, whichcorresponds either to a (1) shaky realization or to a (2) body of reduced dimension. Case (1) is impossible due to Theorem5. Moreover, case (2) can also not hold, which can be shown in the same way as in the proof of Theorem 5.Therefore the entire path has to be real if at least one of the two realizations is not shaky. (cid:3)
In the following example we want to demonstrate the results obtained so far.
Example 2.
We consider a closed serial chain composed of four directly congruent tetrahedral chain elements, which arejointed by four hinges. The studied example was given by Wunderlich [52] and is illustrated in Fig. 6. It has a threefoldreflexion symmetry with respect to three copunctal lines, which are pairwise orthogonal. Using them as axes of a Cartesianframe, the vertices can be coordinatised as follows:A = ( u , v , w ) T A = ( − u , v , w ) T A = ( − u , − v , w ) T A = ( u , − v , w ) T (36) B = ( u , − v , − w ) T B = ( u , v , − w ) T B = ( − u , v , − w ) T B = ( − u , − v , − w ) T (37) The intrinsic metric of the framework is given by the following assignment:A A = A A = A A = A A = B B = B B = B B = B B = √ − ( √ + ) √ − √ + A B = A B = A B = A B = + √ ( √ + ) √ − √ + A B = A B = A B = A B = − √ ( √ + ) √ − √ + A B = A B = A B = A B = √ ( √ − ( √ − ) √ + √ + (38) which has the property that the average edge length equals 1. We consider the two undeformed realizations G ( V ) andG ( V ) of the chain illustrated in Fig. 6, which can be computed according to the procedure given in [52]. The vector ( u , v , w , u , v , w ) which corresponds to V is given by: ( . , . , . , . , . , . ) (39) The coordinates of V are obtained by the following exchange of the coordinate entries of V : u ↔ v , v ↔ u and w ↔ w .Note that the framework can snap between the two realizations G ( V ) and G ( V ) . The shaky saddle realization, which hasto be passed during the snap, is denoted by G ( V (cid:48) ) . B B B A A A A x yz Fig. 6. Left: Illustration of the realization G ( V ) as a bar-joint framework, where bars of equal length have the same color. The fourtetrahedra are hinged along the yellow bars. Moreover, the coordinate frame is displayed where the axes have a length of 1. Center: Thesame configuration as on the left side but illustrated with panels instead of bars. Congruent triangular panels are again colored the same(either red or green). Right: At the top the second realization G ( V ) is visualized and at the bottom the shaky realization G ( V (cid:48) ) . Chain elements R S ς ( L ) = s ( L ) bar-joint 729 113 96 6 . · − panel-hinge ν = / . · − ν = / . · − ν = . · − tetrahedra ν = / . · − ν = / ν = / . · − ν = . · − Table 2. Computational data: Note that the computation of the set R was done by a total degree homotopy using Bertini. In general a closed chain composed of four tetrahedra results in an overbraced bar-joint framework, as one can removee.g. the bars A B and B B to get a minimal rigid structure. But under the assumed threefold symmetry resulting in directlycongruent chain elements the framework is minimal rigid, as the input of the six edge lengths A B , A B , A A , A B ,A B and B B already determine the six values u , u , v , v , w , w coordinatising the four involved points and thereforethe complete structure.We can interpret the chain elements as bar-joint frameworks, panel-hinge frameworks or as tetrahedra. Moreover,in the case of triangular panels and tetrahedra we compute the snappability with respect to three different Poisson ratios ν = , , . The computational data for the different cases is summarized in the Tables 2 and 3. Note that independent of theinterpretation we get s ( L ) = s ( V ) = s ( V ) . The corresponding saddle realizations G ( V (cid:48) ) ∈ S (cf. Table 3) are all shakyas they fulfill the equation u v w − u v w = indicating that the Plücker coordinates of the four lines A i B i are linearlydependent (cf. [39]). Note that for the interpretation as bar-joint framework or panel-hinge framework the singularitycondition consists of a second factor u v w + u v w = which implies the coplanarity of the vertices A i , B i , A i + , B i + (mod 4) for i = , . . . , . According to Theorem 5 we get s ( L ) = s ( V ) = s ( V ) = ς ( V ) = ς ( V ) = ς ( L ) .Concerning the interpretation of the chain as bar-joint framework we can give the maximal absolute and relative varia-tion of a bar in length during the deformation which equals . and . , respectively. Moreover,each edge must change its length in average absolutely by . and relatively by . . Remark 10.
In the case of interpreting the chain elements as panel-hinge frameworks or as tetrahedra, we also put ν asa unknown in the optimization process as stated in Remark 3. Computing the critical points with Bertini resulted int
24 576 paths. For panel-hinge frameworks non of the local extrema for ≤ ν < / has a value less than . · − and for tetrahedra no local extrema exists within this interval. Thus in both cases we get the local minimum at the boundary ν = / . (cid:5) hain elements u , v v , u w , w bar-joint 0.733113570223 0.186762548180 0.226463240099panel-hinge ν = / ν = / ν = ν = / ν = / ν = Table 3. Coordinates of the shaky saddle realization G ( V (cid:48) ) ∈ S for the different interpretations of the tetrahedra. . . (cid:48) T G ( V ) G ( V (cid:48) ) G ( V ) . . x . y .
20 0 . . zA ∈ G ( V (cid:48) ) Fig. 7. Left: The change of the volume of a tetrahedron under the transformation implied by the gradient flow in the space of squared leglengths (cf. Eq. (32) and Remark 9). The corresponding trajectory of the point A is illustrated by the red curve in the right figure. The greencurve corresponds to an isochoric deformation, which was computed with a projected gradient algorithm. We close this example with some remarks on the case, where the chain is assembled by four tetrahedra of Poisson ratio ν = / : Due to the symmetry of the chain we only have to consider one isochoric constraint Vol (cid:48) T − Vol T = , where Tstands for one of the tetrahedra with vertices A i , B i , A i + , B i + (mod 4) for i = , . . . , . The computation of critical points ofu ( V ) under this constraint (cf. Eq. (34)) results in the tracking of
279 936 paths (cf. Table 2). If we also invest the information,that the orientation of the tetrahedra has to remain constant under an isochoric deformation (cf. footnote 12), we can usethe simplified condition Vol (cid:48) T − Vol T = of oriented volumes, which has the half degree. This approach reduces the numberof paths to for the computation of G ( V (cid:48) ) given in Table 3. But we are still lacking for an efficient determination of anisochoric deformation from G ( V ) into G ( V (cid:48) ) with a monotonically increasing deformation energy density. Until now we canonly achieve such a deformation by a projected gradient descent approach resulting in Fig. 7. Further demonstration/verification of our method is done in Appendix A, where two geometric structures with a knownmodel flexibility are studied. Moreover, the obtained results are compared with those reported in the literature.
A Stewart-Gough (SG) manipulator is a parallel robot consisting of a moving platform, which is connected over sixtelescopic legs to the base. These legs are anchored by spherical joints to the platform and the base (cf. Fig. 8). If theprismatic joints of the legs are fixed, then the pin-jointed body-bar framework is in general rigid. It is well-known, that it hasan infinitesimal flexibility if and only if the carrier lines of the six legs belong to a linear line complex [33].A detailed literature review on works dealing with the determination of the closest singular configuration to a given non-singular one, which is of interest for singularity-free path-planning and performance optimization of the robot, was doneby the author in [36]. Most of these approaches (also the one presented in [36]) evaluate the closeness extrinsically (i.e. inthe 6 dimensional configuration space) and not intrinsically (i.e. in the 6 dimensional space of prismatic joints). Up to theknowledge of the author only one work of Zein et al. [59] determines a singularity-free cube in the joint space of a 3-RPRmanipulator, which is the planar analogue of a SG platform. For a detailed comparison of extrinsic and intrinsic singularitydistance measures for planar 3-RPR manipulators we refer to [24].n the following we want to compute the singularity-distance within the 6-dimensional joint space of the manipulator.As a SG manipulator is an isostatic body-bar framework, this computation can be based on Theorem 4 under the additionalcondition that the affine deformations of the platform and the base are restricted to direct isometries. In this case the functionof the strain energy density of a SG manipulator simplifies to: u ( L (cid:48) ) = ∑ i = L i ∑ i = (cid:16) L (cid:48) i − L i (cid:17) L i (40)where L i (resp. L (cid:48) i ) denotes the length of the undeformed (resp. deformed) i th leg spanned by the platform anchor point V i and the corresponding base anchor point V i + . The base can be pinned down , i.e. v (cid:48) i = v i for i = , . . . ,
12, and for theplatform we set up an affine moving frame with origin V and the three vectors v − v , v − v and ( v − v ) × ( v − v ) under the assumption that V , V , V are not collinear. Then one can compute the affine coordinates ( ξ j , υ j , ζ j ) of the points V j with respect to this frame for j = , ,
6, which can be used for writting down the coordinate vector of V (cid:48) j as follows: v (cid:48) j = v (cid:48) + ξ j ( v (cid:48) − v (cid:48) ) + υ j ( v (cid:48) − v (cid:48) ) + ζ j [( v (cid:48) − v (cid:48) ) × ( v (cid:48) − v (cid:48) )] . (41)This affine transformation is an orientation preserving isometry if the three side conditions e i = e : = (cid:107) v (cid:48) − v (cid:48) (cid:107) − (cid:107) v − v (cid:107) , e : = (cid:107) v (cid:48) − v (cid:48) (cid:107) − (cid:107) v − v (cid:107) and e : = (cid:107) v (cid:48) − v (cid:48) (cid:107) − (cid:107) v − v (cid:107) (42)hold true. Under consideration of Eq. (41) one can write Eq. (40) in dependence of v (cid:48) , v (cid:48) , v (cid:48) , which is part of the Lagrangian F ( v (cid:48) , v (cid:48) , v (cid:48) , η , η , η ) = u ( v (cid:48) , v (cid:48) , v (cid:48) ) − η e − η e − η e . (43) Remark 11.
By using the linear combination given in Eq. (41) instead of the formulation of Eq. (27) the number of sideconditions forcing an isometric transformation is reduced from 6 to 3; i.e. e = e = e = . In addition this formulationalready restricts to direct isometries. Clearly, one can also use a parametrization of the Euclidean motion group (e.g. Studyparameters) for the representation of v (cid:48) , . . . , v (cid:48) . We use here this so-called point-based formulation as it turned out to havecertain computational advantages (cf. [23]). (cid:5) The formulation given in Eq. (40) rely on the change of the leg lengths relative to its initial length. As we are nowworking in the joint space of the SG manipulator also the following function of absolute changes in the leg lengths makessense: l ( L (cid:48) ) = ∑ i = (cid:16) L (cid:48) i − L i (cid:17) . (44)Therefore one can also substitute u ( v (cid:48) , v (cid:48) , v (cid:48) ) by l ( v (cid:48) , v (cid:48) , v (cid:48) ) in in Eq. (43). For both Lagrangians, analogous considerationsas done in the proof of Theorem 4 show that the saddle realizations have to be shaky (as the bodies cannot reduce in dimensiondue to the enforced direct isometries). We demonstrate this in the following example. Example 3.
We study a SG manipulator, which is of interest for practical applications, as the positioning and orientationof the relative pose of the platform and the base is decoupled (cf. [4, Section VI]). The moving platform has a semihexagonalshape (with central angles of π / and π / , respectively) and the base is a truncated triangular pyramid (cf. Fig. 8). Thecoordinates of the base anchor points V , . . . , V with respect to the fixed frame are given by: v : = , v : = √ , v : = √ , v : = √ , v : = √ , v : = √ . (45) In this context it should be mentioned that the results of the paper also hold for pinned frameworks (cf. [37, Section 3.3]). yz yz yx
Fig. 8. Axonometric view (left), front view (center) and top view (right): G ( V ) is illustrated in yellow, G ( V ) is displayed in green and G ( V (cid:48) ) w.r.t. u ( L (cid:48) ) is shown in red. Note that the realization G ( V (cid:48) ) w.r.t. l ( L (cid:48) ) is not shown here as it is too close (cf. Table 5) to G ( V (cid:48) ) w.r.t. u ( L (cid:48) ) . The vertices V , . . . , V of the moving platform are determined by the three conditions (cid:107) v − v (cid:107) = , (cid:107) v − v (cid:107) = , (cid:107) v − v (cid:107) = − √ and the affine coordinates ( ξ j , υ j , ) for j = , , given by ξ : = − √ + , υ : = √ + , ξ : = − ( √ + ) , υ : = √ + , ξ : = −√ − , υ : = . (47) The input data is completed by the following six leg lengths:L : = / , L : = / , L : = / , L : = / , L : = / , L : = / . (48) We compute the closest singularity with respect to both intrinsic metrics in the 6-dimensional joint space, which aregiven in Eq. (40) and Eq. (44), respectively. The computational data is summarized in Table 4.
Intrinsic metric R S ς ( L ) = s ( L ) u ( L (cid:48) ) . · − l ( L (cid:48) ) . · − Table 4. Computational data: Note that the computation of the set R was done by a regeneration homotopy performed with Bertini. For this input data the SG manipulator has 4 real solutions for the direct kinematics, whereby two solutions are out ofinterest as the platform is below the base. The other two undeformed realizations are illustrated in Fig. 8 and the corre-sponding vectors v i = ( x i , y i , z i ) T (with respect to the fixed frame) of the platform anchor points V i for i = , , are given inTable 5. This table also contains the corresponding coordinate entries of the two saddle realization G ( V (cid:48) ) , which implyingthe singularity-distance with respect to the different metrics. It can easily be checked that in both realizations G ( V (cid:48) ) the linesof the six legs belong to a linear line complex [33]. Remark 12.
Note that the snapping octahedra of Wunderlich [51] imply further examples of octahedral hexapods with ahigh snapping capability. (cid:5) V V (cid:48) w.r.t. u ( L (cid:48) ) V (cid:48) w.r.t. l ( L (cid:48) ) x x x y y y z z z Table 5. Coordinates of the undeformed realizations G ( V ) and G ( V ) of the SG manipulator and of the shaky saddle realization G ( V (cid:48) ) with respect to both intrinsic metrics given in Eq. (40) and Eq. (44), respectively. The first considerations in the design process of a pin-jointed body-bar frameworks concern its geometry. In this paperwe presented an index, which evaluates the framework geometry with respect to its capability to snap. As this so-called snappability only depends on the intrinsic framework geometry, it enables a fair comparison of frameworks differing inthe combinatorial structure, inner metric and types of structural elements (bars, polygonal panels or polyhedral bodies).Therefore it can serve engineers as a criterion in an early design stage for the geometric layout of frameworks without orwith the capability to snap, depending on the application. In the context of multistability the index can for example be used forthe geometric layouting of unit-cells/building-blocks of periodic metamaterials (e.g. [18,44,58,8,27]) and origami structures(e.g. hypar tessellation [31], waterbomb cylinder tessellation [10] or the Kresling pattern with a circular arrangement toclosed strips, which correspond to snapping antiprisms [53] and can be composed to cylindrical towers [28, 5, 30], or ahelical arrangement according to C.R. Calladine studied in [15, 16, 17, 49]). For the resulting framework geometry one canspecify in a later design phase the material and dimensioning (e.g. profile of bars) of the framework elements in such a waythat the wanted effects are even increased.We also demonstrated on basis of parallel manipulators of Stewart-Gough type, that our approach can be used by thekinematics community for the computation of intrinsic singular-distances. The computational aspects of the index were alsoilluminated and an allover algorithm was presented for isostatic frameworks (under affine deformation of the bodies). Notethat this algorithm does not take the bending of panels into account and also ignores the collision of bars and/or bodies duringthe deformation. For overbraced frameworks we gave a strategy for the snappability computation but the result is withoutguarantee. In this case one can use the proposed lower bounds, which always hold true.
Acknowledgments.
The research is supported by grant P 30855-N32 of the Austrian Science Fund FWF as well as byproject F77 (SFB “Advanced Computational Design”, subproject SP7). Moreover, the author thanks Miranda Holmes-Cerfon for providing the data of the four-horn example discussed in Section A.2. Further thanks to Aditya Kapilavai for hishelp in outsourcing some of the Bertini computations.
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Within this appendix we discuss the examples of the Siamese dipyramid and the four-horn in detail and compare theobtained results with existing ones reported in the literature.
A.1 Siamese dipyramid
The original Siamese dipyramid (SD) introduced by Goldberg [12, page 167] consists of 20 equilateral triangles withan edge lengths of 1, which are arranged in two dipyramids with a hexagonal equatorial polygon (see Fig. 9, left). Note thatwe assume that the SD has a reflexion-symmetry with respect to two orthogonal planes. We can insert a coordinate frame insuch a way that these planes are the xy -plane and the yz -plane, respectively.Then the vertices, which are noted according to Fig. 9, can be coordinatised as follows: A = ( x , y , ) T A = ( − x , y , ) T A = ( , u , v ) T A = ( , u , − v ) T (49) B = ( x , y , ) T B = ( − x , y , ) T B = ( , u , v ) T B = ( , u , − v ) T (50) C = ( x , y , ) T C = ( − x , y , ) T C = ( , u , v ) T C = ( , u , − v ) T (51)In addition we can assume without loss of generality that u = − y holds; i.e. the vertices C and C have the same distancefrom the xz -plane as the points C and C . Therefore the total number of unknowns is 11.It is well-known [12, 13] that the SD can snap out of the symmetric realization G ( V ) (cf. Fig. 9, left/center) into oneof the two asymmetric realizations G ( V ) and G ( V ) , respectively (cf. Fig. 9, right). A simple procedure for the computationof these three undeformed realizations is given in [13]. We only give the numerical values of these configurations in Table 6. With respect to the height of the two dipyramids. A C B C B zx yB B A A C C Fig. 9. Left: Illustration of the realization G ( V ) as a bar-joint framework together with the coordinate frame, where the axes are of lengthone. Center: The same configuration as on the left side but illustrated with panels instead of bars. Right: At the top the second realization G ( V ) is visualized and at the bottom the third one G ( V ) .Fig. 10. On the left (resp. right) side the shaky saddle realizations G ( V (cid:48) ) (resp. G ( V (cid:48)(cid:48) ) ) is illustrated, which is passed during the snapbetween G ( V ) and G ( V ) (resp. G ( V ) ). A.1.1 Isostaticity and shakiness
The bar-joint framework of the SD is isostatic, because every closed polyhedral surface of genus 0 with triangularfaces has this property . This isostaticity remains intact under the assumption of the 2-fold reflexion-symmetry, as it onlycorresponds to the identification of some of the coordinates within the structure.The SD is in a shaky configuration if the rank of its rigidity matrix R G ( V ) is less than 30. From this one can compute thealgebraic characterization, which corresponds to the vanishing of the following polynomial v ( x y + x y − x y − x y ) (cid:124) (cid:123)(cid:122) (cid:125) copl ( C , C , A , B ) v ( x y − x y − x y ) (cid:124) (cid:123)(cid:122) (cid:125) copl ( C , C , B , C ) x ( u v − u v + v y − v y ) (cid:124) (cid:123)(cid:122) (cid:125) copl ( C , C , A , B ) x ( u v + v y − v y ) (cid:124) (cid:123)(cid:122) (cid:125) copl ( C , C , B , C ) S (52)where copl indicates the coplanarity of the vertices given in the round bracket. For the condition x = v = S ,which denotes an algebraic expression with 374 terms and a total degree of 9. Interestingly S is only quadratic with respectto the two non-zero coordinates of the following points: A i , B i , A i and B i for i = ,
2. For the points A i and A i it is even linearin x (for i =
1) or v (for i = This can easily be followed from Euler’s polyhedral formula. This factor can be downloaded from: V V (cid:48) (bar-joint) V (cid:48) (panel-hinge) x -0.5 -0.5 -0.501499108259 -0.501518680610 x -0.940024410925 -0.997453425271 -0.979262620688 -0.979200605399 x -0.327267375345 -0.492373245899 -0.432379113707 -0.432385909548 y -1.245032582350 -1.296828963170 -1.282364611843 -1.282380966624 y -0.347046770776 -0.429338277522 -0.400464708308 -0.400381868195 y u u v v v Table 6. Coordinates of the undeformed realizations G ( V ) and G ( V ) , respectively, and of the shaky saddle realization G ( V (cid:48) ) withrespect to the two different interpretations. The coordinates of G ( V ) and G ( V (cid:48)(cid:48) ) can be obtained from G ( V ) and G ( V (cid:48) ) by the followingexchange of coordinate entries: x i ↔ − v i and y i ↔ − u i for i = , , . A.1.2 Interpretation as a bar-joint structure
We set up our formulation of the deformation energy density u under the assumption that the SD keeps the 2-foldreflexion-symmetry during the deformation. The obtained system of 11 equations ∇ u results in 177 147 paths within a totaldegree homotopy (cf. [3]). The path tracking done by the software Bertini ends up in 22 153 finite real solutions (set R ).After reduction to the set S we remain with 21 904 solutions. This set is the input for the algorithm described in Section 4.2,which outputs the two shaky saddle realizations G ( V (cid:48) ) and G ( V (cid:48)(cid:48) ) , respectively, displayed in Fig. 10. The numerical valuesof these realizations are also given in Table 6.We get s ( L ) = s ( V ) = s ( V ) = s ( V ) = . · − and due to Theorem 5 (under consideration of Corollary1) this value also equals ς ( L ) = ς ( V ) = ς ( V ) = ς ( V ) . Comparison with the results obtained in [19]:
According to [19] there exists a realization within the deformation pathbetween two snapping realizations G ( V ) and G ( V / ) , where the value for e : = (cid:113) ∑ i j ( L i j − L (cid:48) i j ) is greater or equal to avalue e ∗ min given by 2 . · − for G ( V ) and 3 . · − for G ( V / ) , respectively. As noted in [19] the value e ∗ min is aminimum bound and does not say how close this bound is to the true barrier.By our approach we can determine this true barrier value numerically as e min = . · − , which has tobe the same for the three realizations G ( V i ) for i = , , e ∗ min value.From e ∗ min one can also approximate the length ∆ L ∗ (cid:31) , which an edge must change in average according to [19, Example2], yielding the values 2 . · − and 3 . · − , respectively. Based on G ( V (cid:48) ) we can also com-pute the absolute average change ∆ L abs (cid:31) = . · − (which equals also the relative average change ∆ L rel (cid:31) as theinitial length of the edges is 1). Therefore it is 62 times and 55 times, respectively, larger than the values resulting from thedata given in [19]. Comparison with the results obtained in [13]:
The intrinsic index given in [13] equals ∆ L ∗ max = . · − and corre-spond to the maximal relative (with respect to the initial length of 1) change in the length of an edge during the deformation.But it should be noted that the setup of the pyramids in [13] is more restrictive than ours, as all edges through the vertices C , C and C , C cannot be deformed and have a fixed length of 1; all other edges are restricted to have the same length.Based on G ( V (cid:48) ) we can also compute this maximal relative change ∆ L relmax of an edge as 2 . · − (whichequals also the maximal absolute change ∆ L absmax as the initial length of the edges is 1). Therefore the value reported in [13] is31% larger than ours. Remark 13.
In [19] and [13] also some indices are given to estimate/quantify the variations of the spatial shape of thesnapping framework. Due to their extrinsic nature they cannot give information about the snappability of a framework,which only depends on the intrinsic geometry. (cid:5) A A A B B C C C z C yx Fig. 11. Left: Illustration of the realization G ( V ) of the original four-horn as a bar-joint framework together with the coordinate frame, wherethe axes are of length one. Center: The same configuration as on the left side but illustrated with panels instead of bars. Right: The two flatrealizations are visualized, where G ( V ) is displayed at the top and G ( V ) at the bottom.Fig. 12. On the left (resp. right) side the shaky saddle realizations G ( V (cid:48) ) (resp. G ( V (cid:48)(cid:48) ) ) of the design FH is illustrated, which is passedduring the snap between G ( V ) and G ( V ) (resp. G ( V ) ). A.1.3 Interpretation as a panel-hinge structure
The same study can also be done by considering the SD as a polyhedral surface composed of triangular panels witha Poisson ratio of ν = /
2. The tracking of the 177 147 paths of a total degree homotopy using Bertini ends up in 20305real solutions, which can be reduced to 20 056 solutions of the set S . In this case we get s ( L ) = s ( V i ) = ς ( V i ) = ς ( L ) = . · − for i = , , A.2 Four-horn
The original four-horn (FH) was introduced by Casper Schwabe at the
Phänomena exposition 1984 in Zürich, Swiz-erland (cf. Fig. 11). From the combinatorial point of view the FH equals a SD with pentagonal equatorial polygons. Incontrast to a SD, a FH does not consist of congruent equilateral face-triangles but of congruent isosceles ones where α denotes the angle enclosed by the base of length b > a >
0. Under consideration of the two-foldreflexion-symmetry with respect to two orthogonal planes, we can insert a Cartesian frame in such a way, that the vertices,which are noted according to Fig. 11, are coordinatised as follows: A = ( x , y , ) T A = ( − x , y , ) T B = ( , y , ) T C = ( x , y , ) T C = ( − x , y , ) T (53) A = ( , u , v ) T A = ( , u , − v ) T B = ( , u , ) T C = ( , u , v ) T C = ( , u , − v ) T (54)As in Section A.1 we can assume without loss of generality that u = − y holds.It is well-known that the FH can snap out of the symmetric realization G ( V ) into one of the two flat realizations G ( V ) and G ( V ) , respectively, which are both shaky due to their planarity. According to [57] these three undeformed realizations,which are displayed in Fig. 11, exist for all choices of a and b with 2 a > b . tracked paths R S s ( L ) = ς ( L ) e min FH . · − . · − FH . · − . · − FH . · − . · − Table 7. Computational data for the three designs FH , FH and FH . Note that the computation of the set R was done by a total degreehomotopy using Bertini. ∆ L abs (cid:31) ∆ L rel (cid:31) ∆ L absmax ∆ L relmax FH . · − . · − . · − . · − FH . · − . · − . · − . · − FH . · − . · − . · − . · − Table 8. Continuation of Table 7.
As done in [57] we will distinguish three different designs FH i i = , , a i and base b i with a = √ + − (cid:113) + √ a = − √ a = √ + − ( √ + ) √
22 (55) b = (cid:113) + √ − √ − b = √ − b = ( √ + ) √ − √ −
21 (56)For these values, which result in an average edge length of 1, we get the angles α = . ◦ (the original design of Schwabe), α = ◦ and α = ◦ , respectively.How the coordinates of the vertices can be computed for the two flat realizations G ( V ) and G ( V ) and the symmetricrealization G ( V ) of these three designs FH i can be looked up in [57]. We only give the numerical values of these realizationsin the Tables 9–11. A.2.1 Isostaticity and shakiness
The FH is isostatic for the same reasons as pointed out in Section A.1.1. Moreover, we can also determine the algebraiccondition of shakiness in an analogous way, which yields: x x v v v ( x y − x y − x y ) (cid:124) (cid:123)(cid:122) (cid:125) copl ( C , C , A , C ) x ( u v + v y − v y ) (cid:124) (cid:123)(cid:122) (cid:125) copl ( C , C , A , C ) S = x = ( A , B , C ) and ( A , B , C ) coincide with ( A , B , C ) and ( A , B , C ) , respectively.The same holds for the condition v = x = v = S , which denotes an algebraic expression with 110 terms and a total degree of 9. Again S is onlyquadratic with respect to the two non-zero coordinates of A i and A i , respectively, for i = , A.2.2 Interpretation as a bar-joint structure
For the three designs FH i for i = , , G ( V (cid:48) ) and G ( V (cid:48)(cid:48) ) , respectively, which are displayed in Fig. 12 for FH . The numerical values of these realizationsare also given in the Tables 9–11. For all three designs the relation s ( L ) = s ( V i ) = ς ( V i ) = ς ( L ) holds true for i = , , i the additionalvalues e min , ∆ L abs (cid:31) , ∆ L rel (cid:31) , ∆ L absmax and ∆ L relmax as in den case of the SD. For a better comparison they are arranged in the Tables7 and 8, respectively. This factor can be downloaded from: V V (cid:48) (bar-joint) V (cid:48) (panel-hinge) x -0.439833121345 -0.551313194956 -0.514676938265 -0.513910947926 x -0.402578359944 -0.551313194956 -0.496123528337 -0.495616858966 y -1.045126760122 -1.055331194900 -1.049041632768 -1.047987400977 y -0.401357155967 -0.504017999944 -0.463165051665 -0.461829297096 y -0.266342733180 -0.275656597478 -0.269426558812 -0.269389463808 u u v v Table 9. Coordinates of the undeformed realizations G ( V ) and G ( V ) of the design FH and of the shaky saddle realization G ( V (cid:48) ) ofFH with respect to the two different interpretations. The coordinates of G ( V ) and G ( V (cid:48)(cid:48) ) can be obtained from G ( V ) and G ( V (cid:48) ) by thefollowing exchange of coordinate entries: x j ↔ − v j for j = , and y i ↔ − u i for i = , , . Comparison with the methods presented in [19]:
According to [20] the minimum bound e ∗ min of FH ’s realization G ( V ) equals 4 . · − , which is approximately 1/6 of the true barrier e min (cf. Table 7). Moreover, e ∗ min implies an approxi-mation of the absolute length ∆ L ∗ (cid:31) = . · − an edge must change in average, which is about 23% of the value ∆ L abs (cid:31) (cf. Table 7). Remark 14.
For the flat realizations the method of [19] does not work, as they are not pre-stressed stable. Therefore wecannot compare the method of [19] with the values obtained by our method regarding G ( V ) and G ( V ) , respectively. (cid:5) Comparison with the results obtained in [57]:
The authors of [57] sliced the four-horn along the polylines C C C and C C C with exception of the points C and C . In this way they get two two-horns, which are linked over the points C and C . Maintaining the 2-fold reflexion-symmetry, the resulting structure has a one-parametric mobility. Apart from theconfigurations G ( V ) , G ( V ) and G ( V ) the points on both two-horns, which correspond to the point C do not coincide.This mismatch of points is measured by the relative error in the z-coordinate. The maximum of this relative error during theflexion of the two two-horns between the two flat configurations equals the index given in [57]. Therefore this index is alsoof extrinsic nature, but the authors of [57] also noted an empirical rule of thumb without further explanation, which reads as δ ∗ = . · ( α / ) % where α has to be inserted in degree. This formula only depends on the intrinsic geometry of theframework.For our considered values α i for i = , , δ ∗ = . δ ∗ = . δ ∗ = . δ ∗ cannot be compared one-to-one with any of our given values, we can evaluate the index δ ∗ by considering therelation δ ∗ : δ ∗ : δ ∗ . It can easily be seen that this relation does not go along with the corresponding relation of any of thevalues s ( L ) = ς ( L ) , e min , ∆ L abs (cid:31) , ∆ L rel (cid:31) , ∆ L absmax and ∆ L relmax , respectively, given in Tables 7 and 8. A.2.3 Interpretation as a panel-hinge structure
The same study can also be done by considering the FH as a polyhedral surface composed of triangular plates with aPoisson ratio of ν = /
2. We track for each of the three designs 19 683 paths of a total degree homotopy performed withBertini. The computations end up in 1 259 real solutions for FH (1 457 for FH and 1 324 for FH ). After reduction to the set S we remain with 1 242 realizations for FH (1 360 for FH and 1 238 for FH ). Also for the interpretation as a panel-hingestructure the relation s ( L ) = s ( V i ) = ς ( V i ) = ς ( L ) ( i = , ,
3) holds true for all three designs. The corresponding valueequals 1 . · − for FH (2 . · − for FH and 6 . · − for FH ). V V (cid:48) (bar-joint) V (cid:48) (panel-hinge) x -0.610560396069 -0.696152422706 -0.674892191647 -0.673709307095 x -0.514152040259 -0.696152422706 -0.629718504095 -0.630052932256 y -0.969412109993 -1.004809471616 -0.984812341395 -0.981100971148 y -0.446547949553 -0.602885682969 -0.546021679456 -0.541654138725 y -0.171366775159 -0.200961894323 -0.181083746571 -0.180980608226 u u v v Table 10. The analogous table to Table 9 but with respect to the design FH . V V V (cid:48) (bar-joint) V (cid:48) (panel-hinge) x -0.284308975844 -0.381499642545 -0.346332965123 -0.345941194845 x -0.274123668705 -0.381499642545 -0.341068732408 -0.340681988409 y -1.091519316228 -1.093387667069 -1.092180085853 -1.092027657175 y -0.383468247941 -0.432610903111 -0.412298335661 -0.412002561558 y -0.328588016197 -0.330388381978 -0.329187289575 -0.329179948495 u u v v Table 11. The analogous table to Table 9 but with respect to the design FH3