And Yet It Moves: Paradoxically Moving Linkages in Kinematics
AAnd Yet It Moves:Paradoxically Moving Linkages in Kinematics
Josef Schicho, JKU Linz, AustriaApril 28, 2020
Look at Figure 1: you see a mechanism that is able to draw an ellipse. If you press gentlyon the green bar (connected to the right endpoint of the grey segment which is fixed), then thewhole vehicle will start to move and bounce so that the red point traces the ellipse. Historically,it was a famous challenge in the 19th-century to find a mechanism that draws a straight linesegment. Mathematicians even tried to prove the non-existence of an exact solution. But thenthe French engineer Peaucellier and the Russian mathematician Lipkin independently found anexact solution. Starting from the mechanism in Figure 1, we can do the same thing as well (eventhough this was not the solution of Peaucellier/Lipkin): you can change some of the lengths sothat the ellipse degenerates into a line segment traced twice in a full round.Figure 1: A mechanism which is able to draw an ellipse. The short gray horizontal bar is fixedon the x-axis, whereas all the other bars are allowed to move, according to the rotational jointswhich link them one to another.
Kempe’s Universality Theorem.
A few years after the invention of the “straight line mecha-nism” by Peaucellier and Lipkin, Kempe [27] proved that every plane algebraic curve can be drawnby a mechanism moving with one degree of freedom! His construction uses the implicit equationof the algebraic curve, and the linkage draws a bounded subset of the curve. Kempe himselfadmits that the mechanisms constructed by his general construction are quite complicated. Oneof the objectives in this article is to explain how to construct a mechanism that draws a givenrational curve, i.e., a curve that it is given by a parameterization by rational functions. Comparedwith Kempe, this construction gives simpler results when it applies (not every algebraic curve isrational).
Unexpected Mobility.
Most of the mechanisms in this paper will be paradoxical , in the fol-lowing sense: by a systematic counting of degrees of freedom and constraints, one can estimate if1 a r X i v : . [ m a t h . M G ] A p r given mechanism moves. For a paradoxical mechanism, this estimate predicts that the mecha-nism is rigid: there are sufficiently many constraints so that there should be no freedom left formotion, except moving the mechanism as a whole like a rigid body. Still, the mechanism doesmove non-trivially. We discuss five mathematical tools that somehow “explain” the unexpectedmobility: • edge colorings of graphs; • factorization of polynomials over skew coefficient rings; • symmetry as a rule changer for counting variables and constraints; • a projective duality relating a set of relative positions to a set of geometric parameters; • compactification, i.e., a closer analysis of “limit configurations at infinity”. Links and Joints.
We need to introduce a few concepts from kinematics (please do not worry,we will keep the amount of definitions at a minimal level). A linkage (or mechanism) in 3-spaceis composed of rigid bodies called links (or bars, rods) that are connected by joints (e.g., hingesor spherical joints); examples occur in mechanical engineering and robotics, but also in sportsmedicine – the human skeleton may be considered as a quite complex linkage – and in chemistry,at a microscopic scale. If two links are connected by a joint, then the type of joint determines aset of possible relative positions of one link with respect to the other. A revolute joint (or R-jointor hinge) allows a one-dimensional set of rotations around an axis which is fixed in both links;this set is a copy of SO . This type of joint appears most frequently, for example in doors andwindows or in connection with wheels (see also Figure 2, left). A spherical joint (or S-joint) allowsa three-dimensional set of rotations around a point which is fixed in both links; this set of motionsis a copy of SO . An example is the hip joint of the human skeleton (see Figure 2, middle). Anda prismatic joint (or P-joint) allows a one-dimensional set of translations in a fixed direction; thisset is theoretically a copy of R , but in reality, it is a bounded interval. Teachers and students inmathematics often operate such a joint when moving a blackboard up and down (see Figure 2,right, for a different example).Figure 2: A hinge, the hip joint (spherical), and a prismatic joint on a crane. Configurations.
If two links are not directly connected by a joint, then the set of possiblerelative positions of one with respect to the other is determined by other links and joints formingchains that connect the two given links. In general, the description is more complicated, and itis one of the main tasks of kinematics to determine these sets. In any case, they are subsets ofthe group SE of direct isometries, also known as Euclidean displacements. The set of all possiblerelative positions of any pair of rigid bodies of a linkage L is called the configuration space of L .It is possible to express the constraints coming from the joints by algebraic equations in the jointparameters. Therefore, the configuration space is an algebraic variety. Its dimension is called the mobility of L .A linkage is given by combinatoric data, namely the graph indicating which rigid bodies areconnected by joints and the type of joints such as revolute, spherical, prismatic; and by geometric2arameters determining the fixed position of the joint axis in each of the two links attached toany R-joint and the fixed position of the anchor point in each of the two links attached to anyS-joint. The computation of the configuration space of a given linkage can be reduced to solvinga system of algebraic equations with parameters, with the size of the system determined by thecombinatorics. These systems form a rich source of computational problems in computer algebraand polynomial system solving (see [36] and the references cited there). Structure of the Paper.
The paper has 6 sections. In Section 1, we discuss combinatoricmethods for estimating the dimension of the configuration space, based on counting variables andequational conditions; this is necessary to make precise what “paradoxical” means. Section 2deals with planar linkages whose links are line segments joined by revolute joints, also known asmoving graphs; we discuss graphs that should be rigid but actually move. Section 3 deals withspatial linkages in the plane with revolute joints, and uses dual quaternions to construct examplesof simply closed linkages that are paradoxically movable. Section 4 deals with symmetries andexplains how they can change the counting rules. Section 5 deals with a particular type of linkagecalled multipods or Stewart platforms; here, projective duality is a powerful mathematical toolthat allows us to construct paradoxical examples. Section 6 is concerned with the problem offinding necessary conditions for mobility, based on the idea to analyze the “configurations atinfinity” of a mobile linkage. In the three subsections of Section 6, moving graphs, simply closedloops with revolute joints, and multipods are revisited from this point of view what happens atinfinity.
Acknowledgements.
Matteo Gallet, Georg Grasegger, Christoph Koutschan, Jan Legersky,Zijia Li, Georg Nawratil, and Hans-Peter Schr¨ocker are coauthors of papers of which I took pictures- thanks for allowing me to use their work. I also would like to thank Matteo Gallet, Zijia Li, andJiayue Qi for helping to improve the narration. This work has been supported by the AustrianScience Fund (FWF): P31061.
Given the combinatorics of a linkage, i.e., the number of its rigid bodies and the information whichof them are connected by joints, it is possible to estimate the mobility by counting free variablesand equational constraints. In kinematics, this is called the
Chebychev/Gr¨ubler/Kutzbach (CGK)formula . Moving Graphs.
In this section, we start with the two-dimensional situation. Every link is aline segment in the plane R . In the plane, it does not make sense to distinguish revolute jointsand spherical joints, and we do not consider prismatic joints. All joints in the linkages we considerallowing rotations around a fixed point. The combinatorics of the linkage is conveniently describedby a graph G = ( V, E ), with vertices corresponding to joints and edges corresponding to links. Ifa line segment has three or more (say k ) joints connecting to other links, then we have to “splitit up” into several edges: we get k vertices corresponding to joints and we connect them by (cid:0) k (cid:1) edges. For instance, the green link in Figure 1 will correspond to a triangle in the graph, which isgeometrically degenerate because its three vertices are collinear. We assume that the linkage hasno “dangling links”, i.e., no vertices of degree 1, because they would obviously rotate around theconnected vertex.For a graph G = ( V, E ), an “edge length assignment” is a vector λ ∈ R E indexed by the edgeswith positive real coordinates λ e , e ∈ E . A configuration of ( V, E, λ ) is a collection ( ρ v ) v ∈ V with ρ v ∈ R , such that for any edge e = ( u, v ), we have || ρ u − ρ v || = λ e . Two configurations ρ, ρ (cid:48) areequivalent if there is a direct isometry σ : R → R of the plane such that σ ( ρ v ) = ρ (cid:48) v for all v .If we choose two vertices v, w ∈ V such that ρ v (cid:54) = ρ w , then there is a unique representative ρ (cid:48) inthe equivalence class of ρ such ρ v = (0 ,
0) and ρ w = (0 , c ) for some c >
0; we then say that ρ (cid:48) is a normalized configuration. 3or a given graph G = ( V, E ) with edge length assignment λ , its normalized configurations arethe solutions of a system of algebraic equations of the form( x a − x b ) + ( y a − y b ) = λ ab for each edge { a, b } ∈ E , and the normalization conditions x v = y v = x w = 0 , y w > . The number of nonzero variables is 2 | V | −
3, and the number of equations is | E | . We leave out theinequality, because it is inessential for the dimension count. Now the CGK formula predicts thatthe linkage is rigid if 2 | V | − | E | . If this number is nonnegative, then we call 2 | V | − − | E | the CGK estimate for the dimension of equivalence classes of configurations. In kinematics, thisdimension is called the mobility of the linkage.Figure 3: Two planar linkages with 6 joint and 9 links with the same underlying graph. The leftone is rigid, the right one is mobile. Generic Mobility.
For a concrete instance, the CGK estimate comes without any warranties.But we can say something definite for the “generic case”. Here we use the word “generic” inthe following sense. Assume that a certain statement depends on instances parametrized by anopen subset of an irreducible algebraic variety P (in most cases, P is an open subset of a vectorspace). Then we say that the statement is generically true if the subset of instances such that thestatement is false is contained in an algebraic subvariety of P of strictly smaller dimension. Proposition 1.1.
Let G = ( V, E ) be a graph. Let λ ∈ R E be a generic length assignment. Let X λ ∈ R | V |− be set of normalized configurations of ( V, E, λ ) . If | V | − − | E | ≥ , then X λ iseither empty or a real manifold of dimension | V | − − | E | . In particular, if | V | − − | E | = 0 ,then a generic length assignment allows only finitely many normalized configurations.Proof. Let f : R | V |− → R | E | be the map ( x a , y a ) a ∈ V (cid:55)→ (( x a − x b ) + ( y a − y b ) ) { a,b }∈ E (in thedomain, remove the three coordinates known to be zero). This is a differential map, which assignsto each normalized configuration of points in R the square of the lengths of edges. Therefore X λ = f − ( λ ).If the image of f does not contain an open neighborhood of λ , then it also does not contain λ because λ is chosen generically. Hence X λ is empty and there is nothing left to prove.Otherwise, let U be an open neighborhood of λ and apply Sard’s theorem to the map f | f − ( U ) .It implies that the set of critical values does not contain λ . Hence the Jacobian of f has rank E at every point of f − ( U ), and this shows the claim. Generic Rigidity. If | E | = 2 | V | −
3, then two cases are possible: either the image of the map f : R | V |− → R | E | in the proof contains an open subset. Then the graph is rigid: a genericconfiguration cannot move continuously, by Proposition 1.1. Or the image of the map is containedin a subset of lower dimension. The following theorem determines which of the two cases holds.4 heorem 1.2. Let G = ( V, E ) be a graph such that | E | = 2 | V | − . Then there is an open set ofedge assignments λ with a finite and positive number of configurations if and only if | E (cid:48) | ≤ | V (cid:48) |− for every subgraph G (cid:48) = ( V (cid:48) , E (cid:48) ) of G . This theorem was proven by Pollaczek-Geiringer [33] and rediscovered 40 years later by Laman[28]. The graph that satisfy the necessary and sufficient condition above are called
Laman graphs .The necessity is easy to see: if there is a subgraph G (cid:48) = ( V (cid:48) , E (cid:48) ) with | E (cid:48) | > | V (cid:48) | −
3, then thealgebraic system describing normalized configurations of the subgraph is overdetermined. So, forgeneric edge length assignments, there is no configuration for the subgraph, and therefore also noconfiguration for the graph G itself.In dimension 3, the CGK estimate for the mobility of a graph G = ( V, E ) is equal to 3 | V | − − | E | . Proposition 1.1 holds with that bound: if λ ∈ R | E | is a generic edge assignment, andthe normalized configuration space X λ is not empty, then it has dimension 3 | V | − − | E | . Thecondition | E (cid:48) | ≤ | V (cid:48) | − V (cid:48) , E (cid:48) ) is still necessary for the statement that X λ is generically not empty, but it is not sufficient: Figure 4 shows the “double banana”, a graphwith 8 vertices and 18 edges, such that a generic assignment of its vertices to points in R isflexible. The Jacobi matrix of the map f mapping normalized configurations to edge assignments(see Proposition 1.1) is quadratic and singular. So the 3-dimensional analogue of Theorem 1.2 isnot true, and the search for another combinatoric analogue is an active research topic in rigiditytheory (see [24]). 123 4 5 678Figure 4: The smallest graph that is generically mobile and still fulfills the 3D-analogue of Laman’scondition for generic rigidity: 3 | V | − | E | , and 3 | V (cid:48) | − ≥ | E (cid:48) | for every subgraph ( V (cid:48) , E (cid:48) ).The blue part may revolve around the line through two vertices. Molecules.
For some classes of graphs, the 3-dimensional analogue of Theorem 1.2 is true. Themost interesting class appears in a statement which used to be called the “Molecular Conjecture”,until it was proven in [26]. It is of special interest because it makes a statement on linkages thatappear as models of molecules: atoms are modeled as balls with cylinders attached. A molecularjoint is a cylinder who is joined to an atom at both of its ends (see Figure 5). From a kinematicpoint of view, a molecule model is a linkage with R-joints, such that for each link, all axis of jointsattached to this link meet in a fixed point (the center of the atom).The following equivalent re-formulation appear in [23]. For any graph, we can define its square by drawing an edge between any two vertices of graph distance two. A graph is called a squaregraph if it can be obtained as the square of a subgraph.
Theorem 1.3 (KatohTanigawa) . Assume that G = ( V, E ) is a square graph such that | E | =5igure 5: A kinematic model of the Methoxyethanol molecule C H (OH) . The cylinders arejoints allowing a rotation around the central axis of the cylinder. Note that the axis always passesthrough the centers of the joined atoms.3 | V | − . Assume that | E (cid:48) | ≤ | V (cid:48) | − for every subgraph of G . Then a generic assignment of thevertices by points in R defines a rigid embedding. To see the equivalence with the molecule conjecture, start with a molecule and draw a graph G with vertices corresponding to atoms and edges corresponding to cylinders in the molecularmodel. It is clear that every motion of the molecule fixes the length of each edge. However, everysuch motion also fixes the angle between two cylinders attached to the same atom. But this isequivalent to the statement that the motion fixes the length between the two atoms that are onthe other end of the two cylinders. If you add an edge for any two such atoms, then you getexactly the square of G . Let us call a linkage paradoxical if a generic linkage with the same combinatoric structure is rigid,but the linkage itself is moving. For instance, an instance of a Laman graph which is mobile inthe plane is paradoxical.
Should we Expect Paradoxical Linkages?
Let us do a simple variable counting, as in theCGK formula, to see if we should be surprised by the existence of paradoxical linkages. Fix acombinatorial structure, for instance a Laman graph G = ( V, E ). For a generic instance, thenumber of non-equivalent configurations is finite. These configurations are real solutions of asystem of algebraic equations; let N G be the number of complex solutions of these system. Notethat the number of complex solutions does not depend on the choice of the generic instance, aslong as the choice is generic, in contrast to the number of real solutions, which would depend onthe choice of a generic instance.For any system of equations that has finitely many solutions, it is possible to compute a singleunivariate polynomial, such that the solutions of the system are in bijection with the zeroes of thepolynomial. In theory, it is possible to compute such a polynomial by introducing a new variabletogether with a generic linear equation between the new variable and the old variables, and thenby eliminating all old variables. (In practice, it turns out that the elimination is quite costly.) Theprocess can even be carried out in the presence of parameters, which will then also appear in thecoefficients of the univariate polynomial. Let us therefore assume that we have now, for each graph G = ( V, E ), such a polynomial F G , with coefficients depending on an edge length assignment λ .The degree of F G would then have to be equal to N G , because it has N G complex solutions andwe may assume that F G is squarefree.Now, a labeled graph ( V, E, λ ) is mobile if and only of all N G + 1 coefficients of the polynomialare zero, i.e. the polynomial F G vanishes identically and there are infinitely many configurations.(We have to take non-real configurations into account, but let us ignore this point for the moment.)6he instances of the graph form a family of dimension | E | parametrized by the edge lengths. Inorder to find a paradoxical linkage, we need to find a solution of a system in | E | variables with N G + 1 equations. So we need to compare these two numbers. If the number | E | of variables isbigger than or equal to the number N G + 1 of equations, then we should not be surprised by theexistence of paradoxical linkages.Currently, we do not know any lower bounds for N G , but there are conjectured lower boundswhich are exponential in | E | , so the system of equations that would have to be fulfilled for theparameters of a paradoxical linkage would be highly overdetermined. This is also true for smallgraphs: for 5 ≤ | V | ≤
12, the numbers N G are all known [8], and we always have | E | < N G + 1.Consequently, the very existence of paradoxical linkages is itself paradoxical! At least, this is sofor the type of linkages we considered in this counting, namely moving graphs in the plane. Bipartite Graphs.
The smallest mobile Laman graphs have 6 vertices. One is the completebipartite graph K , . In [13], Dixon describes a construction to make arbitrary bipartite graphsmobile. The set V of vertices is partitioned into two disjoint subsets V , V . Put all vertices in V on the x -axis and all vertices of V on the y -axis. An easy exercise using Pythagoras’ Theoremshows that the linkage is actually moving.Figure 6: A mobile complete bipartite graph K , . Its points form two rectangles sharing theirsymmetry axes.Using computer algebra, Husty/Walter[38] proved that Dixon’s construction is one of twopossible mobile K , ’s; in all other cases, K , is rigid. The second mobile K , , also found in [13],is a mobile K , with two points removed – see Figure 6. The configuration has a finite symmetrygroup, namely the symmetry of a rectangle. Indeed, the points form two rectangles sharing theirsymmetry axes.Note that Dixon I applies to arbitrary bipartite graphs. In contrast, the symmetric constructionDixon II does not scale, it just applies to K , and to its subgraphs. NAC colorings.
Another construction that does scale is based on the possibility of partitioningthe set E of edges into two non-empty subsets E r , E b of red and blue edges. We assume that everycycle in G is either unicolored or has at least two edges of both colors; especially, triangles arealways unicolored. Such a partition is called a NAC – “no almost (unicolored) cycle” – coloring.For each connected component of the subgraph R i of ( V, E r ) we assign a complex number z i , andfor each vertex of the subgraph B j of ( V, E b ), we assign a complex number w j . Then we choosea real parameter t parametrizing a periodic motion, as follows: map any vertex in R i ∩ B j to thepoint z i + e it w j ∈ C . But C is a model for the plane R . Hence we have constructed, for any realvalue of t , a configuration of the graph in R . The construction is continuous in t , so we may callit a motion. The blue edges always keep their orientation while the red edges are rotated withuniform speed, as in Figure 7.A partition of E into E r ∪ E b is a NAC coloring if and only if we can map the vertices into theplane so that all red edge are parallel to the first coordinate axis and all blue edges are parallelto the second coordinate exists. It is obvious that this map defines a flexible embedding. Such amoving graph can be constructed by taking a very small moving graph, with three vertices and7igure 7: A mobile graph with a NAC coloring. The blue edges remain parallel to the originalorientation, and the orientation of the red edges rotates with speed that is independent of theedge, as long as it is red.two edges, and enlarging it by parallel copies of edges. But wait - we can do the same with othergraphs as well! Let us start with a moving quadrilateral. Then we add more edges that are parallelto one of the four edges of the quadrilateral. We get a bigger graph with the property that everymotion of the quadrilateral induces a motion of the bigger graph – see Figure 8 for an example.Figure 8: A moving Laman graph with 8 vertices and 13 edges. The two figures – 2D, not 3D! Tosee this picture correctly, please switch off your spatial perception for a moment! – show two ofinfinitely many possible configurations of the graph in R , with the same edge lengths. Every edgeis parallel to one of the four sides of the red quadrilateral. The red quadrilateral has obviouslyinfinitely many configurations; and any configuration of the red quadrilateral can be extended toa configuration of the whole graph.In Section 6, we will see that the existence of a NAC coloring is not only sufficient, but alsonecessary for the existence of a length assignment that makes a given graph mobile in R . Thisresult requires a few tools from algebraic geometry. More examples of graphs moving in the planeand NAC-colorings can be found in https://jan.legersky.cz/project/movablegraphs/ . Let n ≥
4. An n R chain is a linkage consisting of n + 1 links connected by n revolute joints.In robotics, the first link is called the base and the last link is called the hand or end effector .Each joint can is controlled by an electric motor in such a way that the end effector performs aparticular task.If we firmly connect the first and the last link of an n R chain, then we get an n R loop: alinkage with n links connected cyclically by n revolute joints. According to the CGK formula, themobility is max(0 , n − n ≥
7, then a generic n R loop is generically mobile. A generic 6Rlinkage is rigid; the number of configurations, including complex solutions, is 16 (see [35, 11.5.1]).For n = 5 and n = 4, we obtain an overdetermined system of equations. Remark 3.1.
Revolute loops may be considered as special cases of linkages of graph type, inthe following way: we pick two points on each joint axis and connect them by an edge. For eachlink, we draw 4 additional edges connecting the points on the two axes that belong to the link, sothat every link carries a complete graph K , which is geometrically a tetrahedron. This graph has2 n vertices and 5 n edges, and it is apparent that the linkage of graph type has exactly the samemobility as the revolute loop. See Figure 9 for an example of a tetrahedral 6R loop.8igure 9: A thumbnail movie of a mobile 6R loop. Each of the 6 link is realized as a tetrahedron.Each tetrahedron has two edges, opposite to each other, playing the role of R-joints connectingthe link to its to two neighbors.But even though revolute loops may be considered as a subclass of linkages of graph type, itis advantageous to introduce new techniques especially suited for them.
4R Loops.
The classification of mobile 4R loops is due to Delassus [10]. He proved that thereare three types of mobile 4R linkages: planar: all rotation axes are parallel. Essentially, this is a quadrilateral moving in the plane. Thethird coordinate is not changed in any of the moving links. spherical: all rotation axes pass through a single point. Essentially, this is a moving sphericalquadrilateral. The planar case may be considered as a limit case of the spherical case. skew isogram:
Bennett [2] discovered a mobile 4R linkage such that the axes of joints attachedto the same link are skew, for all four links. We describe it below in more detail.Let L , . . . , L n = L be the rotation axes of in some configuration of an n R loop. For i =0 , . . . , n −
1, we assume that the lines L i and L i +1 belong to the i -th link. Since the link isassumed to be a rigid body, the normal distance d i and the angle α i between L i and L i +1 doesnot change as the linkage moves: they are invariant parameters. Assume that none of the anglesis zero, i.e., L i and L i +1 are not parallel. Then there is a unique line N i intersecting both L i and L i +1 at a right angle. The distance s i between N i ∩ L i and N i ∩ L i +1 is called offset. Theangles, normal distances, and offsets are 3 n invariant geometric parameters of the linkage; inrobotics, they are called the invariant Denavit-Hartenberg parameters [11]. A configuration isdetermined by n angles, and the 3 n invariant Denavit-Hartenberg parameters together with the n configuration parameters determine the positions of the n rotation axes and the position of thelinks uniquely up to SE . These 4 n parameters fulfill a condition of codimension 6, called the closure equation : we attach an internal coordinate system to each link, with the axis L i being the x and the common normal N i being the z -axis. Then the transformation of the i -th coordinatesystem to the ( i + 1)-th coordinate system is the composition of the translation by a vector oflength d i parallel to the z -axis, the rotation around the z -axis by the angle α i , the translation bya vector of length s i parallel to the x axis, and a rotation around the x -axis determined by the9 -th configuration parameter. The product of all these 4 n direct isometries is equal to the identity,and this statement gives the closure equation.Figure 10: The skew isogram is a mobile linkage of type 4R-loop with four rotation axes, so thataxes in the same link are always skew. It is the only mobile 4R-loop which is neither planar (allaxes are parallel) nor spherical (all axes are concurrent).A skew isogram is a 4R linkage such that the invariant Denavit-Hartenberg parameters d , . . . , s satisfy the conditions d = d , d = d , α = α , α = α , d sin( α ) = d sin( α ) , s = s = s = s = 0 . (1) Dual Quaternions.
In order to prove that the skew isogram is mobile, we use an algebraic waysuggested by [37] to parametrize SE . The algebra DH of dual quaternions is the 8-dimensionalreal vector space generated by 1 , i , j , k , (cid:15), (cid:15) i , (cid:15) j , (cid:15) k . Its multiplication is R -linear, associative, theelement (cid:15) – the dual unit – is central and satisfies (cid:15) = 0. The symbols i , j , k are Hamiltonianquaternions: i = j = k = ijk = −
1. The center, generated by 1 and (cid:15) , is called the algebra ofdual numbers. Conjugation is a D -linear antihomomorphism from DH to itself: it maps 1 to itself, i to − i , j to − j , and k to − k . For any dual quaternion h ∈ DH , the element N ( h ) := hh = hh is a dual number, called the norm of h . The norm is a semigroup homomorphism with respectto multiplication; its image is the subsemigroup consisting of all dual numbers with nonnegativeprimal part.The set S of dual quaternions with norm in R ∗ is a multiplicative group. Its center is R ∗ . Thequotient group S / R ∗ happens to be isomorphic to SE . The isomorphism is determined by theaction of S / R ∗ on R . We may regard R as the abelian normal subgroup T of S / R ∗ of classesrepresented by dual quaternions of the form 1 + x(cid:15) i + y(cid:15) j + z(cid:15) k (this subgroup is going to be thesubgroup of translations in SE ). The substitution of (cid:15) by − (cid:15) is an outer automorphism of S / R ∗ of order 2 – lets call it τ – which fulfills the following property: if h ∈ S / R ∗ , then h − τ ( h ) ∈ T .This implies that for all h ∈ S / R ∗ and v ∈ T , the element h − vτ ( h ) = ( h − vh )( h − τ ( h )) is in T ,and this defines a right action of S / R ∗ on T . The bijections of T in the image of this action aredirect isometries, and this defines a group isomorphism S / R ∗ ∼ = SE . At the same time, we haveconstructed an embedding of SE into the projective space P ( DH ) ∼ = P , as the subset defined bya quadratic form S = 0, namely the dual part of the norm, and by a quadratic inequation N (cid:54) = 0,namely the primal part of the norm.There is a bijection between elements of order 2 in SE and lines in R : every line correspondsto a half turn round that line (a rotation by the angle π ). A point in SE ⊂ P ( DH ) has order 2if and only if its scalar part is zero. Here we have two linear equations, namely the coefficient of1 and the coefficient of (cid:15) , defining a P in P ( DH ). The intersection of this P with the quadrichypersurface defined by S (a.k.a. the Study quadric ) is isomorphic to the Pl¨ucker quadric, andthe remaining six coefficients are the Pl¨ucker coordinates of lines.Let l ∈ DH be a dual quaternion representing an element of order 2 in SE . Then l = − N ( l )is a negative real number; without loss of generality, we may assume l = −
1. The line connecting101] and [ l ] is contained in the Study quadric: its elements are the rotations around the line L corresponding to l . (Note that [1] denotes the equivalence class of the dual quaternion 1 in P anddoes not indicate a reference to the bibliography.) These elements form a group; indeed, the vectorspace generated by 1 and l is a subalgebra isomorphic to C over R , and the projectivization of thistwo-dimensional real algebra is a Lie group isomorphic to SO . We call this group the revolution with axes L . A parametric representation of the revolution is ( t + l ) t , where the parameter t rangesover the real projective line; the parameter t = 0 corresponds to [ l ], and the parameter t = ∞ corresponds to [1]. In general, the parameter t corresponds to the cotangent of half of the rotationangle. Remark 3.2.
Conversely, assume that we have a line in S passing through [1]. Then we canparametrize it by a linear polynomial in t with leading coefficient 1, i.e., by a polynomial ( t + h )with h ∈ S . Because N ( t + h ) = t + ( h + ¯ h ) t + N ( h ) has to be real for all t ∈ R , it follows that h + ¯ h ∈ R : the scalar part of h is real (its dual part is zero). Then a reparametrization of theline is ( s + h − ¯ h ) s , setting s = t + h +¯ h . This reparametrization shows that the line parametrizesa revolution with axis corresponding to [ h − ¯ h ], except in the case when N ( h − ¯ h ) = 0. In theexceptional case, the line will parametrize a translation along a fixed direction.Let us now study conics passing through [1] and contained in the Study quadric. Any suchconic has a quadratic parametrization ( t + at + b ) t where a, b ∈ DH . Does this quadric polynomialfactor into two linear polynomials? And if yes, do the linear polynomials parametrize revolutions?To answer these questions, we study DH [ t ], the non-commutative algebra of univariate polynomialswith coefficients in DH , where the variable t is supposed to be central, i.e., it commutes with thecoefficients. Quaternion Polynomials.
As a preparation, let us ask the analogous question for the non-commutative algebra H [ t ]. We will show that here, every polynomial can be written as a productof linear factors; in other words, the skew field of quaternions is algebraically closed! The proof istaken from [18]. Lemma 3.3 (polynomial division) . Let
A, B ∈ H [ t ] , B (cid:54) = 0 . Then there exist unique polynomials Q, R ∈ H [ t ] such that A = QB + R and either deg( R ) < deg( B ) or R = 0 .Proof. We start with uniqueness. Assume Q B + R = Q B + R for deg( R ) < deg( B ) anddeg( B ) < deg( R ). We obtain ( Q − Q ) B = R − R . If the left side of this equation is not zero,then its degree is at least deg( B ). If the right side s not zero, then its degree is less than deg( B ).Hence both sides must be zero.For the existence, we proceed by induction on the degree of A : if deg( A ) < deg( B ), then weset Q := 0 and R := A . If deg( A ) ≥ deg( B ), then we can write A = ht deg( A ) − deg( B ) B + A (cid:48) for asuitable h ∈ H and A (cid:48) ∈ H [ t ] with deg( A (cid:48) ) < deg( A ). By induction, we get A (cid:48) = Q (cid:48) B + R (cid:48) . Butthen we can set Q := Q + ht deg( A ) − deg( B ) and R := R (cid:48) .If deg( B ) = 1 in the Lemma 3.3, say B = t − h , then R is a constant in H . The constant iszero if and only if ( t − h ) is a right factor of A . If this is true, then we also say “ h is a right zeroof A ”. So, the questions is: does every polynomial A of positive degree have a right zero? Andmaybe we are also interested in the question how to find it.A right zero of A is also a right zero of the norm polynomial N ( A ) = ¯ AA . We know that thatthe norm polynomial is in R [ t ]. It is also the sum of four squares – if A = A + A i + A j + A q k ,then N ( A ) = A + A + A + A . If N ( A ) has a real zero r , then this real zero is also a zero of A , A , A , A ; hence it is a zero of A , and we have found what we wanted to find.What do we do if N ( A ) has no real zeroes? In this case, we choose a quadratic irreduciblefactor M ∈ R [ t ]. By Lemma 3.3, there are Q, R ∈ H [ t ], with deg( R ) < R = 0, such that A = QM + R . We distinguish three cases.1. If R = 0, then M is a right factor of A . Every right zero of M is also a right zero of A . Soit suffices to show that M has a right zero. But we know that M has a complex zero. So,11ssume that z = a + i b is a complex zero of M , for some a, b ∈ R , b (cid:54) = 0. Then we have theequation M = ( t − a − i b )( t − a + i b ) between complex polynomials. But now we can replacethe complex number i by the dual quaternion i , which also fulfills the equation i + 1 = 0 Itfollows that M = ( t − a − i b )( t − a + i b ) = 0, and a − i b is a right zero of M and also a rightzero of A .2. If deg( R ) = 1, say R = ut + v for suitable u, v ∈ H , u (cid:54) = 0, then h := u − v is a right zero of R . Since RR = ( P − QM )( P − QM ) = N ( P ) + M ( − QP − P Q + QQM ) (2)is a multiple of M , and deg( RR ) = deg( M ) = 2, it follows that M is a left multiple of R . Itfollows that h is right zero of M . Hence it is also a right zero of A = QM + R .3. If deg( R ) = 0, then Equation 2 is self-contradictory: the right side is a multiple of M , andthe left side is a nonzero constant. So, this case cannot occur. Theorem 3.4.
Every polynomial in H [ t ] can be written as a product of linear polynomials. The proof is already clear: given A of positive degree, we can find a right h , write A = A (cid:48) ( t − h ),and iterate.How many distinct factorizations do there exist? Starting with one factorization, we may getinfinitely many distinct factorizations by multiplying with constants and their inverses in betweenthe linear factors. In order to get rid of these “essentially same” factorizations, it suffices to assumethat the polynomial A and the linear factors are normed, i.e., they have leading coefficient 1.If A is a multiple of an irreducible real quadric M (the first case in the above case distinc-tion), then A has infinitely many right zeroes (see [18]). But if not, then the number of distinctfactorizations is finite. Indeed, the only non-deterministic step in the iterative procedure sketchedabove is the choice of the sequence of irreducible factors used for factoring out the right zeroes.In particular, we have: Proposition 3.5.
A normed polynomial of degree d with generic coefficients has exactly d ! distinctfactorizations into normed linear factors. The comparison with polynomial factorization in C [ t ] is illuminating: there, the factorizationis unique. But if we consider two factorizations which differ only by the order of the factors asbeing distinct, then we have again d ! distinct factorizations. In the case of H [ t ], permutation offactors would not lead to the same product, because H [ t ] is not commutative; hence permutationis not a method to get more factorizations, and all d ! factorizations are different. Mobility of the Skew Isogram.
Feeling well prepared? Then, let us go back to polynomialsover the dual quaternions. Can we write every polynomial in DH [ t ] that parametrizes a curve inthe Study quadric into a linear factors parametrizing revolutions?Let us assume that we have given such a polynomial P ∈ DH [ t ]. We can try to copy thefactorization strategy that worked in H [ t ]: factorize the norm polynomial N ( P ), choose a quadraticirreducible factor M (lets assume that N ( P ) has no real zeroes for now), compute the remainderof P modulo M ; if this remainder is a linear polynomial R = ut + v for some u, v ∈ DH , computea right zero h := u − v , factor out ( t − h ) from the right, and iterate. This is going to workfor generic coefficients. Moreover, since N ( P ( t )) is in R (and not in D \ R ) for all t ∈ R ,the norm polynomial N ( P ) is in R [ t ]. Therefore it has a factorization into irreducible factors in M r ∈ R [ t ], r = 1 , . . . , deg( P ). The right factors ( t − h r ) produced by our strategy satisfy theequation ( t − h r )( t + h r ) = M r , so by Remark 3.2, the linear factor will generically parametrize arevolution. So, at least generically, everything is fine!The application of our strategy leads to the following characterization of skew isograms. Itwas first found in [7] by different methods. Theorem 3.6.
For a generic conic in the Study quadric passing through [1] , there is a skewisogram such that the conic parametrizes the motion of the second link. (In particular, this skewisogram is mobile.) roof. Let P = t + at + b be a quadratic parametrization of the conic, with a, b ∈ DH . Thenorm N ( P ) is a real polynomial that has only nonnegative values. By genericity, it has no doublezeroes, and can be written as a product M M of two distinct quadratic irreducible factors. For i = 1 ,
2, we construct as above a factorization P = ( t − r i )( t − w i ) such that N ( t − w i ) = M i . (Itfollows that N ( t − r i ) = N ( t − w − i ), for i = 1 , t − r , t − w , t − r , t − w parametrize lines on the Study quadric.Each of them corresponds to a subgroup of rotations around a line in R . Let L , K , L , K bethese four lines, respectively. We construct a mobile 4R loop as follows: the base link containsthe lines L and L , the first link contains the lines L and K , the second link contains the lines K and K , and the third link contains the lines L and K . For each t ∈ ( R ∪ {∞} ), we get aconfiguration of the 4R loop: the relative displacement of the first link with respect to the baselink is the rotation t − r , the relative displacement of the third link with respect to the baselink is the rotation t − r , the relative motion of the second link with respect to the first linkis the rotation t − w , and relative motion of the second link with respect to the third link isthe rotation t − w . The relative position of the second link with respect to the base link canbe computed in two ways, via the first link or via the third link. In both ways, the result is( t − r )( t − w ) = ( t − r )( t − w ) = P .Once the lines are constructed, it is straightforward to compute the invariant Denavit-Hartenbergparameters of the 4R loop – we omit this calculation. The result are exactly the equations 1. Itfollows that the 4R loop is a skew isogram.The paper [7] also contains the converse statement: for any skew isogram, the relative motionof two links that are not connected by a joint is parametrized by a conic curve on the Studyquadric that passes through [1]. In [21], factorizations of cubic polynomials in DH [ t ] are used toconstruct paradoxically moving 5R loops and 6R loops. Drawing Rational Curves.
It is time to lift the veil of the mystery about the ellipse circleshown in Figure 1. This example is taken from [15], which contains a construction of a linkage thatdraws a rational plane curve. In [30], the construction is extended to rational space curves. Anonline illustration with several examples can be found in .The ellipse with implicit equations ( x + a ) a + y b = z = 0 has a rational parametrization( x, y, z ) = p ( t ) := (cid:18) − at + 1 , btt + 1 , (cid:19) . For any t ∈ R , the dual quaternion 1 + (cid:15) ( − at +1 i + btt +1 j ) represents a translation that maps theorigin to p ( t ). The class of a dual quaternion is not changed when we multiply it with t + 1.So we set P := t + 1 + (cid:15) ( − a i + bt j ) and try to factorize. The norm polynomial is ( t + 1) ,hence our only choice of an irreducible factor is M = t + 1. The remainder of P modulo M is R = (cid:15) ( − a i + bt j ). But now something is wrong: even though R has a right zero, namely h = − ba k ,there is no common zero R and M except in the case a = ± b . (If a = ± b , then the ellipse isa circle, and we are not interested.) The argument we used in the quaternion case fails because N ( R ) = 0.There is a way out: instead of factorizing P , we can factorize Q := ( t − i ) P . The displacement[ t − i ] fixes the origin, hence the displacement [ Q ( t )] maps the origin to the point p ( t ), just likethe translation [ P ( t )]. The remainder of Q modulo M is (cid:15) ( b − a )( i t − j ), and this time we do havea common right zero of M and R ! Any dual quaternion of the form − k − (cid:15) ( c j + d j ) is fine. Forsimplicity, we set d = 0. Now we can factor ( t + k + (cid:15)c j ) from the right and proceed. The finalresult is Q = ( t − k + ( a/ b/ (cid:15) j )( t − k + ( − a/ b/ − c ) (cid:15) j )( t + k + (cid:15)c j ) = ( t − h )( t − h )( t − h )(we leave the remaining steps as an exercise – they are not problematic and give a unique result).In order to construct a linkage with mobility one, we could use another factorization with adifferent linear factor on the left. But such a factorization does not exist: the norm polynomial of13 is ( t + 1) , so there is no choice of choosing different factors of the norm polynomial. We needto mix a different quadratic irreducible polynomial into our soup.Let d ∈ R and define h := 2 k + d(cid:15) j . The polynomial ( t − h )( t − h ) has exactly twofactorizations – one we know already, the second one is ( t − h )( t − h ), for some h , h ∈ DH .Then the polynomial ( t − h )( t − h ) also has exactly two factorizations, and we can define twomore dual quaternions such that the second factorization is ( t − h )( t − h ). Finally, let h , h ∈ DH such that ( t − h )( t − h ) = ( t − h )( t − h ). The different factorizations giving the same resultcorrespond to paths in the directed graph G in Figure 11 with equal starting and ending vertex.1 t − h (cid:47) (cid:47) t − h (cid:47) (cid:47) t − h (cid:47) (cid:47) t − h (cid:47) (cid:47) t − h (cid:79) (cid:79) t − h (cid:47) (cid:47) t − h (cid:79) (cid:79) t − h (cid:47) (cid:47) t − h (cid:79) (cid:79) t − h (cid:79) (cid:79) Figure 11: This graph displayes different factorizations of equal products in the polynomial ring ofdual quaternions. For any two directed path between two vertices, the two products of the linearpolynomials appearing as edge labels in each path are equal. The dual quaternions h , . . . , h aredefined as follows: h = 2 k + d(cid:15) j , h = k − ( a/ b/ (cid:15) j , h = k − ( − a/ b/ − c ) (cid:15) j , h = − k − (cid:15)c j , h = k + a + b +4 d (cid:15) j , h = 2 k + − a − b + d (cid:15) j , h = k + − a − b − c +4 d (cid:15) j , h = 2 k + a − b +12 c + d(cid:15) j , h = − k + − a +16 b +3 c − d (cid:15) j , h = 2 k + a − b + d (cid:15) j . Here, a, b, c, d are arbitrary real constants.The ellipse circle consists now of eight links corresponding to the eight vertices of G . Twolinks are connected by a joint if and only if the vertices are connected by an edge. The label ofthe edge – a linear polynmial in DH – parametrizes the relative position of the target link withrespect to the source link. As t varies, the linear polynomials parametrize a revolution. Therefore,the two links are connected by an R-joint. Now we fix the link corresponding to vertex 4. Thenthe relative motion of the link corresponding to vertex 1 maps the origin to the point p ( t ) on theellipse.Note that b = 0 is allowed; in this case, the ellipse degenerates to a line segment traced twice,and we have constructed a linkage that draws this line. The second construction by Dixon of a moving K , is symmetric. Indeed, symmetry may changethe counting rules and can sometimes be the explanation of paradoxical mobility. We discuss heretwo cases in more detail: line symmetry and plane symmetry. Both cases appeared in Bricard’sfamilies of moving octahedra in [4]. Schulze [34] was the first to describe paradoxical movingsymmetric graphs systematically, in every dimension. Line Symmetry.
We assume that we have a graph G = ( V, E ) such that | E | = 3 | V | −
6, andan assignment ( λ ) e ∈ E of a positive real number for each edge. Generically, the configuration set,i.e., the set of all maps V → R respecting edge lengths modulo SE , is finite: we have 3 | V | − | E | equations. Let us now assume that we have a graph automorphism τ : V → V that preserves the edge assignment. Assume also that τ has order 2, does not fix a vertex, anddoes not fix an edge – a priori, an edge could be fixed if τ permutes its two vertices. Then | V | consists of n := | V | pairs of conjugated vertices, and E consists of 3 n − L ⊂ R ; let σ : R → R be the rotation around L by π . For any conjugated pair ( v, τ ( v )) of vertices, we pick one point p v anywhere in R ; the second point is determined by p τ ( v ) := σ ( p v ). The number of variables14o specify all points is 3 v . There is also a two-dimensional subgroup of SE fixing L , generatedby rotations around L and translations into the direction of L . We use two of the variables toget a canonical representative. Hence the number of variables to specify an equivalence class ofconfigurations is 3 n −
2. The number of equations is equal to the number of pairs of conjugatededges, which is 3 n −
3, because conjugated edges always have the same length. Hence the expectedmobility is one.The smallest line symmetric moving graph is the 1-skeleton of an octahedron, with 6 verticesand 12 edges. The group of graph automorphisms is isomorphic to the Euclidean symmetry groupof a regular octahedron, which has 48 elements. There is a unique automorphism of order 2 withoutfixed vertex and fixed edge, corresponding to the point reflection of the regular octahedron. Theconstruction applies, and we get a moving line symmetric octahedron (see Figure 12 left side).Bricard [6] proved that there are three types of moving octahedra, and the line symmetric is oneof the three.Figure 12: Left side: a flexible octahedron that is symmetric by a line reflection; right side: aflexible octahedron that is symmetric by a plane reflection. Corresponding edges are shown in thesame color.More generally, we can take any centrally symmetric convex polyhedron Γ with only triangularfaces and choose as a graph G = ( V, E ) its 1-skeleton. By Euler’s formula, the number of edges is3 | V | −
6. The point reflection acting on Γ defines an automorphism of the graph which satisfiesthe required properties: order 2, no fixed vertex, no fixed edge. The construction applies, and weget, for instance, a line symmetric moving icosahedron with 12 vertices and 30 edges.
Remark 4.1.
Be careful: the point symmetry defines only the graph automorphism! It is ge-ometrically different from the line symmetry in all configurations we allow. Point symmetricconfigurations do also exist, but only finitely many.Another classical example is Bricard’s line symmetric 6R loop. Any 6R loop consists of 6 links,cyclically connected by revolute joints that allow rotations around an axes which is common tothe two attached links; generically, a 6R loop is rigid. In the line symmetric case, we assume thatthe 18 invariant Denavit-Hartenberg parameters d , . . . , s satisfy d i = d i +3 , α i = α i +3 , s i = s i +3 for i = 0 , , , and we are only looking for configurations such that there exists a half turn mapping the i -th linkto the ( i + 3)-rd link.Recall that configurations can be found by solving the closure equation (see Section 3): weattach an internal coordinate system to each link and parametrize the transformation T i from15he i -th link to the ( i + 1)-th link (where the 6-th link is the 0-th link) by the i -th configurationparameter φ i . As mentioned above, T i ( φ i ) is the composition of the translation by a vector oflength d i parallel to the z -axis, the rotation around the z -axis by the angle α i , the translation bya vector of length s i parallel to the x axis, and a rotation around the x -axis determined by the i -th configuration parameter φ i . The configuration set is the set of solutions ( φ , . . . , φ ) of theclosure equation T ( φ ) T ( φ ) T ( φ ) T ( φ ) T ( φ ) T ( φ ) = e, where e is the identity of the group SE . The functions T , . . . , T depend on the invariant Denavit-Hartenberg parameters, and as a consequence we have T = T , T = T , and T = T . Recallthat the closure equation is a codimension 6 condition, because SE is a six-dimensional group,hence the CGK-formula estimates that there are only finitely many solutions.Since we are only interested in line symmetric configurations, we assume φ = φ , φ = φ ,and φ = φ . The closure equation reduces to( T ( φ ) T ( φ ) T ( φ )) = e. We ignore the solutions of T ( φ ) T ( φ ) T ( φ ) = e (these are at most finitely many). This means,we search for configuration parameters such that the transformation of the coordinate system of the0-th link to the coordinate system of the 3rd link is a half turn. This is a codimension 2 condition:as we have already mentioned in Section 3, the set of involutions in SE is a 4-dimensional manifold.Hence there is a one-dimensional set of line symmetric configurations generically. Remark 4.2.
Is there a good reason to explain the mobility of a line symmetric linkage by theclosure equation, instead of just considering them as special cases of line symmetric linkages ofgraph type, as in Remark 3.1. Here is one: we may replace some of the revolute joints by othertypes of joints, like prismatic joints, as in hydraulic telescopes, or helical joints, as commonly seenin the form of screws. In both cases, such a joint allows a one-parameter subgroup of displacementsof the connected links, and exactly the same proof of mobility is valid. On the other hand, a loopwith helical joints cannot be considered as a linkage of graph type, because its closure equation isnot even algebraic.Yet another classical example, the line symmetric Stewart platform, will be explained in Sec-tion 5.
Plane Symmetry.
Plane reflections are involutions in the group E of isometries reversingthe orientation. They are of course not direct isometries, but they still may be responsible forparadoxical mobility of various types of linkages, similar to half turns in the case of line symmetriclinkages. Let us start with 6R loops. In a plane symmetric configuration of a 6R loop, there existsa plane reflection mapping link 0 to link 5, link 1 to link 4, and link 2 to link 3. The existence of aplane symmetric configuration has the following implications on the invariant Denavit-Hartenbergparameters: d = d , d = d , d = d , α = α , α = α , α = α , s = − s , s = − s , s = s = 0 . The relations between the functions in the closure equations are the following: RT ( φ ) R = T ( − φ ) , RT ( φ ) R = T ( − φ ) , RT ( φ ) R = T ( − φ ) , RT ( φ ) R = T ( − φ ) , where R is the reflection by the coordinate plane Π spanned by the first and second axes. Insteadof solving the closure equation, we find all quadruples ( φ , φ , φ , φ ) such that RXR = X , where X := T ( φ ) T ( φ ) T ( φ ) T ( φ ). An element X ∈ SE fulfills the equation RXR = X if and onlyif it is a rotation with an axis orthogonal to Π or a translation by a vector in Π. These rotationsand translations form a manifold of dimension 3 (isomorphic to SE ), hence the condition aboveis a codimension 6 − φ , φ , φ , φ ) of RXR = X , the six-tuple (2 φ , φ , φ , φ , φ , φ ) is asolution of the closure equation: T (2 φ ) T ( φ ) T ( φ ) T (2 φ ) T ( − φ ) T ( − φ ) = T ( φ ) XT ( φ ) RT ( − φ ) T ( − φ ) R = RT ( − φ ) RXRT ( − φ ) T ( − φ ) T ( − φ ) R = RT ( − φ ) RXRX − T ( φ ) R = e. Hence we again get a mobile 6R loop, also known as “Bricard’s plane symmetric 6R linkage”.
Remark 4.3.
As in Remark 4.2, we may replace some of the revolute joints by prismatic orhelical joints – see [1]. Care has to be taken for the special role of the 0-th joint and the 3rd joint,because these two joints are supposed to be mapped to their own inverse by the plane symmetry.This is not possible at all for helical joints. Prismatic joints are fine, but the direction vector hasto be perpendicular to the symmetry plane and not parallel to it.For linkages of graph type, there is also a construction of plane symmetric linkages that areparadoxically mobile. We assume that we have a graph (
V, E ) such that is generically rigid andsatisfies | E | = 3 | V | −
6, for instance the 1-skeleton of a convex polyhedron with triangular faces.Assume that we have a graph automorphism τ : V → V of order 2 that fixes 2 m vertices and2 m − m ≥
1. Choose a generic edge assignment that respects the involutivesymmetry. Fix a plane Π in R , and let R : R → R be the reflection at Π. A configuration( p v ) v ∈ V is symmetric with respect to the plane Π if and only if R ( p v ) = p τ ( v ) holds for all v ∈ V . The number of indeterminates is 3 | V |− m + 4 m − | V | + m −
3: for each 2-orbitin V , the realization is determined by 3 indeterminates, and for each fixed point, we have twoindeterminates because the point must lie in Π. The symmetry group of the plane has dimension 3,which reduces the number of indeterminates of equivalence classes by 3. The number of equationsis | E |− m +22 + 2 m − | V |− m − + 2 m − | V | + m −
4. Again, we obtain a paradoxicallymobile graph.So, how do we find graphs with an automorphism of order 2 fixing 2 m vertices and 2 m − m = 1. This works, for example, for theoctahedron – see Figure 12 right side – and for the icosahedron. Remark 4.4.
In the construction above, there are two geometric symmetries playing entirelydifferent roles: The line symmetry of the convex polyhedron defines a graph automorphism oforder 2 with the right properties; the plane symmetry defines a condition on the configurationsthat we consider. See also Remark 4.1.Here is an example of a generically rigid graph with 12 vertices and 30 edges and with anautomorphism of order 2 that fixes 4 vertices and 2 edges: take a 6R loop and construct a graphas in Remark 3.1, by putting two vertices on each of the four rotation axes. In this case, theplane symmetric construction just reconstructs plane symmetric 6R loops, which we already didin another way.
The Prix Vaillant 1904 asked for curves in the Lie group SE of direct isometries such that “many”points in R move on spheres. Connecting the moving points by sticks with the midpoints of thesespheres, we obtain a multipod , also known as Stewart platform , which is a linkage consisting of afixed base and a moving platform that are connected by legs of fixed length that are attached toplatform and base by spherical joints (see Figure 13). Flight simulators or other linkages that aresupposed to make irregular motions are often manufactured as hexapods with additional prismaticjoints at each leg that change its length; in this section, as already stated, the leg lengths remainconstant. Each leg gives a codimension 1 condition on the displacement of the platform with17espect to the base, hence the CGK-formula gives the estimate 6 − n for the mobility an n -pod.Strictly speaking, each leg may be considered as a link that may also revolve around the lineconnecting its two anchor points, but we disregard this component of the motion. So, pentapodsare generically mobile, and hexapods are generically rigid.A displacement R → R of the platform relative to the base is given by an orthogonal matrix M ∈ SO with determinant 1 and the image y ∈ R of the origin of the base. We set x := − M t y = − M − y to be the preimage of the origin of the platform and r := (cid:104) x, x (cid:105) = (cid:104) y, y (cid:105) , where (cid:104)· , ·(cid:105) is theEuclidean scalar product. If we take coordinates m , . . . , m , x , x , x , y , y , y and r , togetherwith a homogenizing variable h , in P , then a direct isometry defines a point in projective spacesatisfying h (cid:54) = 0 and M M t = M t M = h · id R , adj( M ) = hM t ,M t y + hx = 0 , M x + hy = 0 , (cid:104) x, x (cid:105) = (cid:104) y, y (cid:105) = rh, (3)where adj( M ) is the adjugate matrix. Recall that A · adj( A ) = adj( A ) · A = det( A ) · id R forany A ∈ R × , therefore the above equations imply det( M ) = h . The equations (3) define avariety X of dimension 6 and degree 40 in P , whose real points satisfying h (cid:54) = 0 are in one toone correspondence with the elements of SE . We call it the group variety ; its projective space P containing X is called group space .Mathematically, a leg is a triple ( a, b, d ), where a ∈ R is a point of the base, b ∈ R is a pointof the platform, and d ∈ R is a positive number, the length of the leg. We define the leg variety Y as the cone over the Segre variety Σ , ∼ = P × P in the projective space ˇ P ; recall that the Segrevariety is a subvariety of a projective space of dimension 15 and degree (cid:0) (cid:1) = 20, hence Y hasdimension 7 and degree 20. The values of projective coordinates of a leg ( a, b, d ) are u := 1, a i , b j and z ij := a i b j for i = 1 , ,
3, and the corrected leg length l := (cid:104) a, a (cid:105) + (cid:104) b, b (cid:105) − d . The projectivespace ˇ P containing Y is called leg space .The reason for this very specific choice of coordinates is the following. The algebraic condition (cid:104) M a + y − b, M a + y − b (cid:105) = d is bilinear in these coordinates: lh + ur − (cid:88) i =1 a i x i − (cid:88) j =1 b j y j − (cid:88) i,j =1 z ij m ij = 0 . (4)Hence it defines a duality between group space and leg space. Every point in group space, inparticular every group element, corresponds to a hyperplane in leg space; every point in leg space,in particular every leg, corresponds to a hyperplane in group space. More generally, to every k -plane in group space there is a dual (15 − k )-plane in leg space, for k = 0 , . . . , X in deg( X ) = 40 points(real or complex). Hence a generic hexapod has 40 configurations, possibly complex.Now, we choose 5 generic legs. They span a generic 4-plane in leg space, dual to a generic11-plane in group space, which intersects X in a curve C of degree 40: the configuration curve ofa generic pentapod. We can compute its genus. We first compute the Hilbert series of X from agenerating set of its ideal: H ( X )( t ) = t +18 t +10 t + t (1 − t ) . Because C is a codimension 5 subvarietyof X defined by 5 linear forms, we may compute the Hilbert series of C from the Hilbert series of X : H C ( t ) = H X ( t )(1 − t ) = 1 + 10 t + 18 t + 10 t + t (1 − t ) = 1 + 12 t + 41 t + 80 t + 120 t + . . . . This implies that C is a curve of genus 41, and its embedding in P is half canonical.18 he Bricard-Borel Infinity-Pod. Here is the infinity-pod that has won the Prix Vallaint toBorel [3] and Bricard [5]. We intersect X with the 3-space defined by r + βh − αm = m − m = m + m = m − h = x + y = m = m = m = m = x = x = y = y = 0 , where α, β ∈ R are real parameters such that α (cid:54) = 0. The result is a quartic curve defined by theequations m + m − h = x − αm h + βh = 0and by the linear equations above. It parametrizes a motion C contained in the two-dimensionalstabilizer of the third axes L , generated by rotations around L and translations in the directionof L . The dual 12-plane in leg space is defined by z − z = z + z − αu = a − b = l − z − βu = 0 . A leg ( a, b, d ) in the intersection with Y if and only if a b − a b = a b + a b − α = a − b = a + a + b + b − d − β = 0 . For any point ( a , a , a ) in the base such that ( a , a ) (cid:54) = (0 , b , b , b )in the platform and a length such that the motion C keeps the distance of base and platform pointequal to d . To get the platform point corresponding to a given base point ( a , a , a ), we invertits projection ( a , a ) on the circle with radius (cid:112) | α | and keep the third coordinate; if α <
0, thenwe also have to rotate the projection by an angle of π .In the degenerate case α = β = 0, one of the equations of the quartic curve is a perfect square,and the reduce equations m + m − h = x define a conic in a 2-space. In leg space, we haveone less linear equation: z − z = z + z = l − z = 0, or equivalently a b − a b = a b + a b = a + a + b + b − d = 0 . Here we get a four-dimensional set of possible legs with two components, namely the set of legswhere the platform point or the base point lies on the z -axis. The motion is just a revolutionaround the z -axis. Planar Multipods.
We consider now the linear subspace L p ⊂ ˇ P of dimension 9 in the legspace defined by the equations a = b = z = z = z = z = z = 0 . Its intersection Y p with the leg variety consists of all legs such that the two anchor points lie on afixed plane. The variety Y p is the Segre variety Σ , ∼ = P × P ; let us call its elements informally planar legs . The degree of Y p is (cid:0) (cid:1) = 6.A multipod such that all its base points are coplanar and all its platform points are coplanaris called a planar multipod (see Figure 13). To obtain the configuration of a planar multipod,one has to intersect the dual space of the linear span of all legs with the group variety X . Thelinear span of the legs is contained in L p , hence the dual space of the linear span contains thedual space L ⊥ p . This linear space does not intersect the group variety, otherwise we would have adisplacement that preserves the length of all legs in Y p , which is impossible. What we can say isthat the projection P (cid:57)(cid:57)(cid:75) P with center L ⊥ p projects the group variety to a subvariety X p ∈ P of dimension 6 and degree 20 by a map that is generically 2:1. Hence the configuration of a planarmultipod come in pairs: for every configuration, there is a conjugated configuration. It can beobtained by an outer automorphism of SE , namely the conjugation by the reflection with respectto the plane containing the anchor points.It is surprisingly easy to construct paradoxically moving planar hexapods. Here is the reason.19 a a a a a b b b b b b Figure 13: A planar hexapod. For any configuration, there is also a conjugated configuration thatcan be obtained by reflection on the green base plane.
Theorem 5.1 (Duporcq) . Let y , . . . , y ∈ Y p be five generic planar legs. Then there exists aplanar leg y ∈ Y p such that the configuration space of the pentapod defined by ( y , y , y , y , y ) isequal to the configuration space of the hexapod defined by ( y , y , y , y , y , y ) .Proof. Let V ⊂ ˇ P be the linear span of y , . . . , y . Its dimension is 4. The dimension of Y p is 5.Both V and Y p are contained in L p ∼ = P , hence the intersection Y p ∩ V is finite. Its cardinality isequal to the degree of Y p , which is 6. We know already 5 points; we choose y to be the 6-th.For both, the pentapod and the hexapod, the configuration set of the pentapod is the inter-section of the group variety X with the dual space V ⊥ . The linear condition imposed by the 6-legdoes not impose an independent condition because it lies in the linear span of the other five. Line-symmetric Multipods.
Another class of paradoxically moving hexapods is the class ofline symmetric hexapods. They can be obtained as special cases of line symmetric moving graphs(see Section 4). The graph consists of two octahedra G , G together with six edges each joiningone point of G to one point of G , so that these six edges provide a graph symmetry between G and G . The automorphism τ of the whole graph G maps each vertex v of G to the vertex in G connected with the unique vertex in G that is not connected with v (see Figure 14). This graphautomorphism does not fix any vertex or any edge. We fix a line L of symmetry and embed G sothat the half turn around L maps each vertex v to the image of τ ( v ), generically with respect tothis condition. By the count in Section 4, the configurations are solutions of an algebraic systemin 16 unknowns and 15 equations, implying mobility.It pays off to analyze the situation again by group-leg duality, following an analysis from [3].Let L i ⊂ P be the linear subspace in group space defined by the linear equations M = M t and x = y ; it intersects X in the subset X i of all displacement of order 2 or 1. Note that the order 2elements in SE are exactly the rotations around lines by an angle of π . These are six equations,hence dim( L i ) = 10. The dual subspace L ⊥ i in leg space has dimension 5 and is defined by theequations l = u = z i i = a i + b j = z ij = 0 for i, j = 1 , . . . , i (cid:54) = j . We have a situation thatmirror the planar hexapod case: the subspace L ⊥ i does not intersect the leg variety, otherwise therewould be a leg which does not change length in all involutions. But the projection ˇ P (cid:57)(cid:57)(cid:75) ˇ P with center L ⊥ i projects the leg variety Y to a subvariety Y i ∈ ˇ P of dimension 7 and degree 10by a map that is generically 2:1. Hence the legs of a multipod with involutive displacements comein pairs: if ( a, b, d ) is a leg, then ( b, a, d ) is also a leg. This can also be shown directly: if σ ∈ SE has order 2, then || σ ( a ) − b || = || σ ( a ) − σ ( b ) || = || σ ( b ) − a || . Group-leg duality induces a duality between the projective subspace L i of dimension 10 thatcontains X i and the projective image space ˇ P that contains Y i . Let us call the elements in Y i twin pairs of legs ; each such pair of legs is constituted by a leg ( a, b, d ) and by its conjugated20
234 5 64561 2 3
Figure 14: A graph consisting of two octahedra and 6 additional edges, with a graph automorphismof order 2 that does not fix and vertex or any edge. The automorphism is shown by vertex orbits:conjugated vertices have equal labels. By symmetric counting of variables and equations, a genericline symmetric embedding does move. In this motion, the two octahedra are rigid, and we obtaina moving hexapod.leg ( b, a, d ). Generically, three twin pairs in Y p correspond to three hypersurfaces in L i . Sincedim( X i ) = 4, the intersection of these three hypersurfaces and X p is a curve. So, we have explainedagain the paradoxical mobility.But there is more. We have not just constructed a moving hexapod, we have even constructed,at the same time, a moving icosapod! Here is the precise statement. Theorem 5.2.
Let p , p , p be three generic twin pairs of legs. Let C ⊂ X s be the configurationcurve of the hexapod defined by all six legs. Then there exist seven additional twin pairs, maybecomplex, such that C is the set of all order 2 displacements compatible with all 20 legs.Proof. The three twin pairs span a generic 2-plane in V ⊂ ˇ P . The subvariety Y i ⊂ ˇ P hasdimension 7, hence V and Y intersect in deg( Y i ) = 10 points. Three of them correspond to p , p , p , and the remaining seven are the additional pairs we require. The linear span of all 10points is equal to the linear span of p , p , p , namely V , hence the conditions for displacementsdo not change.In [14], it is shown that there exist examples where all twenty legs are real. The proof is basedon a result on quartic spectahedra in [9, 32]. In enumerative algebraic geometry, for instance for the problem of counting rational curves ona projective variety, compactifications of moduli spaces are known as a powerful tool. Here,we compactify the algebraic varieties in which the configuration spaces are naturally embedded:products of subgroups of SE in the case of linkages with revolute joints, SE itself in the case ofmultipods, and products of the plane in the case of moving graphs.21 .1 Moving Graphs The main theorem in [19] states that for a given graph, the existence of a flexible labeling isequivalent to the existence of a NAC coloring. The construction of a flexible linkage from a givenNAC-coloring was already explained in Section 2. For a construction proving the other implication,we need a compactification.Let (
V, E, λ ) be a graph with an edge assigment. We would like to projectivize in order tocompactify; for this purpose, it is convenient to slightly change the notion of a configurationslightly. A homogeneous configuration is an assignment of vertices by points in R such that forany two edges e = ( i, j ), f = ( k, l ), the equality λ e || p k − p l || = λ f || p i − p j || holds. For each vertex i ∈ V with assigned point p k , we write p k = ( x k , y k ) and z k := x k + i y k , w k := x k − i y k . In other words, the complex numbers z , . . . , z | V | represent the vertices in theGaussian plane of complex numbers. In order to normalize, we require p = (0 , p defines a point in P | V |− × P | V |− as follows: its first com-ponent has projective coordinates ( z : · · · : z | V | ), and its second component has coordinates( w : · · · : w | V | ). The equality above reads λ e ( z k − z l )( w k − w l ) − λ f ( z i − z j )( w i − w j ) (5)in these projective coordinates. This is a bihomogeneous equation of bidegree (1 , P | V |− × P | V |− , the configuration variety of( V, E, λ ). Equivalent homogeneous configurations define the same point in the configuration va-riety: since we fixed p = (0 , z -coordinates by a complex nonzero constant andall w -coordinates by a different complex nonzero constant, hence does not change the points in P | V |− .A point α ∈ P | V |− × P | V |− corresponds to a homogeneous configuration if and only if it fulfillstwo extra conditions. First, the conjugate has to coincide with the flip of the first and secondcomponent; if this condition fails, then some of the corresponding points in the plane have nonrealcoordinates. Second, for some edge e = ( i, j ), we have ( z i − z j )( w i − w j ) (cid:54) = 0. By (5), the choiceof the edge has no influence on the validity of this extra condition.The boundary of the configuration set is defined as the set of points in the configuration varietythat fail to satisfy the two extra conditions. In particular, for some edge e = ( i, j ), or equivalentlyfor all edges, we have ( z i − z j )( w i − w j ) = 0. For each point β in the boundary, we define a coloringof the edges of the graph in the following way: the edge ( i, j ) is colored red if z i − z j vanishes at β , and blue otherwise. Lemma 6.1.
For any point β in the boundary of the configuration variety, the coloring definedby it is a NAC-coloring.Proof. Assume, indirectly, that all edges are red. Then the first projection of β to P | V |− has onlyzero coordinates, which is impossible.Assume, indirectly, that all edges are blue. For any edge ( i, j ), we have ( z i − z j )( w i − w j ) = 0and z i − z j (cid:54) = 0. It follows that the second projection of β to P | V |− has only zero coordinates,which is impossible.Assume, indirectly, that ( i , . . . , i k , i ) is cycle such that ( i r , i r +1 ) is red for all r = 1 , . . . , k − i k , i ) is blue. Then z i = · · · = z i k and z i k (cid:54) = z i , which is impossible.Assume, indirectly, that ( i , . . . , i k , i ) is cycle such that ( i r , i r +1 ) is blue for all r = 1 , . . . , k − i k , i ) is red. Then w i = · · · = w i k , hence w i k = w i . In addition, we also have z i k = z i as ( i k , i ) is red. Therefore the form ( z i − z i k )( w i − w i k ) vanishes with order m ≥ β . Theorder of this form is the same for every edge, and because ( i r , i r +1 ) is blue, the forms z r − z r +1 have order zero for r = 1 , . . . , k −
1. Hence the order of the forms w r − w r +1 is at least m , for all r . Then the form w i − w i k vanishes with order at least m , and this is a contradiction.22 heorem 6.2. A ( V, E ) has a flexible labeling λ if and only if it has a NAC-coloring.Proof. If (
V, E, λ ) is flexible, then its configuration set is a projective variety K of positive degreein P | V |− × P | V |− . For any edge ( i, j ) ∈ E , the form ( z i − z j )( w i − w j ) has to vanish somewhere in K . Therefore, K meets the boundary. By Lemma 6.1, it follows that ( V, E ) has a NAC-coloring.Conversely, assume that we have a NAC-coloring of the edges. Then we make the graph movingby a construction already given in Section 1: the red edges always keep their direction and moveby translations only, while the blue edges rotate with uniform speed.For example, the graph in Figure 15 has no NAC-coloring and therefore never moves for anylabeling λ .Figure 15: A graph that does not have a NAC coloring. Consequently, the graph is rigid for everypossible labeling of its edges.A weakness of Theorem 6.2 is that its constructive part – the construction of flexible labelings– produces only a particular type of motions that we my call “uniform speed motions”. Also,these motions sometimes map different non-adjacent vertices to the same point in the plane. Forexample, in the case of the complete bipartite graph K , , all uniform speed motions map at leasttwo pairs of vertices to the same point in the plane, and the moving graph looks like a movingparallelogram. Deciding if a given graph has labeling with a generically injective motion is muchharder than deciding the existence of a flexible labeling; see [20]. The complete classification of mobile 4R loops was given by Delassus (see Section 3). The completeclassification of mobile 5R loops was given in [25] with the help of computer algebra. For 6R loops,the classification is still open; the difficult part is to come up with necessary conditions for mobility.In this subsection, we explain a method for deriving necessary criteria, which applies to n R loopsfor n = 4 , , d , . . . , d n (normal dis-tances), s , . . . , s n (offsets), and w , . . . , w n (cotangents of half angles) the invariant Denavit-Hartenberg parameters. For r = 0 , . . . , n −
1, the dual quaternion g r := (1 − s r (cid:15) i )( w r − k )(1 − d i k )is the displacement that transforms the internal coordinate system of link r to the internal coordi-nate system of link r + 1 (modulo n ), if the configuration parameter is zero. The closure equationis an equation in the variables t , . . . , t n − , which denote the cotangents of the half configurationangles: the dual quaternion x ( t , . . . , x n − ) := ( t − i ) g ( t − i ) g · · · ( t n − − i ) g n − is a multiple of 1, hence 7 of its 8 coefficients are zero. The variables t , . . . , t n − may also assumethe value ∞ ; in this case, the corresponding factor ( t r − i ) is replaced by the scalar 1, or is simplyomitted. In this section, we will avoid this technicality.We focus on solutions on the boundary, but this time we do not consider t r = ∞ as boundary.Instead, we define the boundary of ( P ) n as the set of n -tuples ( t , . . . , t n − ) such that t r + 1 = 0for at least one r . Indeed, if we remove the boundary, then we get a group variety isomorphicto (SO ) n , with an isomorphism respecting real structures. The statement that t r + 1 = 0 for23t least one r is equivalent to the statement N ( x ( t , . . . , t n )) = 0, by the multiplicativity of thenorm. Boundary solutions can never be real, at least one of the variable must be equal to ± i. Note:
Throughout this paper, we use i for the first quaternion unit in H , i for the imaginaryunit in C , and i for a running integer. In this sections, both i and i will appear, sometimes in thesame expression; but we will try to avoid using i for an integer.Unfortunately, the closure equation often has many solutions that are not of interest. But wecan obtain more equations by cyclic permutation of its factors, or by using quaternion conjugationto bring some factors to the other side, as in λ ( t − i ) g = νg n − ( t n − + i ) g n − · · · g ( t + i )for some scalars λ, ν that are not both equal to zero. This condition can be expressed by polynomialequations, namely the 2-minors of the 2 × t − i ) g and of g n − ( t n − + i ) g n − · · · g ( t + i ). After having added all these reformulation of the closureequations to our system of equations, we look for solutions on the boundary. These are called bonds .It is an easy exercise to prove that at least two of t , . . . , t n − must be ± i. Hint: use aformulation of the closure equation with factors on both sides, and then take the norm on bothsides. There are many examples with exactly two of t , . . . , t n − being ± i. If, say, t +1 = t k +1 = 0for some k < n , and t i + 1 (cid:54) = 0 for i (cid:54) = 1 , k , then we say that the first joint and the k -th joint are entangled in the respective bond. We can then prove the following equations:( t − i ) g ( t − i ) g · · · ( t k − i ) = 0 , ( t k − i ) g k ( t k +1 − i ) g k +1 · · · ( t − i ) g n ( t − i ) = 0 . (6)If the number of coordinates t r with t r + 1 = 0 is bigger than two, then Equation 6 also holdsform some k , up to cyclic permutation by [22, Lemma 2 and Theorem 3].Equation 6 together with t + 1 = t k + 1 = 0 is quite restrictive and often has implications onthe invariant parameters that are hard-coded in g , . . . , g n − . The case k = 2 is easy to analyze:assume (i − i ) g (i − i ) = 0 . Then it follows that w = d = 0; geometrically this means that the first two rotation axes areequal, except that they have opposite orientation in the closure equation. We may exclude thisdegenerate case, and then we always have k > n , this also excludes k = 0).If k = 3, then we get the equation(i − i ) g ( t − i ) g (i − i ) = 0 , up to orientation of the first and/or third axis. This is a system of inhomogeneous linear equationsfor t . It has a solution in three cases: either the three axes are parallel, or the three axes areconcurrent, or the equations s = 0 , d sin( α ) = d sin( α )are true. This should be compared with Bennett’s condition for a 4R loop to be mobile in Section 3:it is exactly the condition that has to be fulfilled for three axis that is true if and only if thereexists a 4th line that supplements the three lines to a mobile 4R loop. The “Benett condition forthree lines” mysteriously appears in Dietmaier’s collection [12] of known families of 5R loops and6R loops. The bond equation explains why: in a mobile linkage, bonds have to be present, andfor each bond there must be two non-neighboring joints entangled in a bond. In a 5R loop, wethen have k = 3 up to a cyclic permutation. In a 6R loop, we have either k = 3 – entanglementof diagonal joints –, or k = 4, entanglement of opposite joints. Many known families have somebond that entangles diagonal joints.The analysis of the case k = 4 is more involved; however, it is necessary in order to explainmobility of 6R linkages that have no three consecutive axis fulfilling the Bennett condition for24hree lines. Assume that n = 6, and we have a bond (cid:126)t that entangles the first and the fourth joint.Without loss of generality, we may assume t = t = i. Then we obtain the equations(i − i ) g ( t − i ) g ( t − i ) g (i − i ) = 0 , (i − i ) g ( t − i ) g ( t − i ) g (i − i ) = 0 . (7)Excluding some degenerate cases (4 parallel lines, or 4 lines meeting in a point), the first equationallows two solutions for ( t , t ), while the second equation allows two solutions for ( t , t ). Thesepartial solutions are not independent. They have to satisfy another reformulation of the closureequation: λ (i − i ) g ( t − i ) g ( t − i ) g = νg ( t + i ) g ( t + i ) g (i + i ) , for some complex numbers λ, ν that are not both equal to zero. By resultants, we can eliminatethe variables t , t , t , t and obtain an equivalent formulation without these variables: the twoquadratic polynomials Q +1 ( x ) = (cid:18) x + b c − b c − s (cid:19) +i2 ( b s + b s + s b c + s b c ) − b b c − s s c s + s − b + b − b − b c ,Q +4 ( x ) = (cid:18) x + b c − b c − s (cid:19) +i2 ( b s + b s + s b c + s b c ) − b b c − s s c s + s − b + b − b − b c C [ x ] have a common zero; here, c i := cos( α i ) and b i := d i sin( α i ) for i = 1 , . . . ,
6. For the detailsthis elimination of variables, we refer to [29].If one of the coordinates of t and t , or both, are equal to − i, then there are two quadraticpolynomials that are similarily defined, having a common zero. In total, the number of bondsentangling the first joint and fourth joint is even (because bonds always appear in complex conju-gate pairs) and at most 8. It is equal to 8 if and only if the two polynomials are equal in each ofthe four pairs of quadratic polynomials.Suppose that we have the maximal number of 8 bonds entangling opposite axes, for all threepairs of opposite axes. This assumption leads to a system of algebraic equations in the invariantDenavit/Hartenberg parameters (18 variables). Using computer algebra, we can compute thesolution set (see [29]). It turns out that there are two components F and F , of dimension 6 and7, respectively. Both are families of mobile 6R loops that have not been known before bonds wereused in kinematics. But the family F (the one of dimension 6) has a 5-dimensional subfamilywhich is classical: Bricard’s orthogonal 6R loops, characterized by the vanishing of c , . . . , c (i.e., all angles are right angles) and s , . . . , s (i.e., all offsets are zero), and the single equation b − b + b − b + b − b = 0. The two varieties that play a role in the analysis of multipods, namely the group variety X ∈ P and the leg variety Y ∈ ˇ P , both come with a natural definition of a boundary: the boundary of X is defined by the linear equation h = 0 and the boundary of Y is defined by the linear equation u = 0, with the variable setting as in Section 5. The group variety is more interesting, because theconfiguration set of a mobile multipod – defined as the intersection of X with hyperplanes dual25o the legs – will always intersect the boundary h = 0. The leg set of a mobile multipod, on theother hand, might be disjoint from the boundary u = 0.Let us have a closer look at the boundary B := X ∩ H , where H is the hyperplane H : h = 0.We refer to [16] for the calculation; here we report on only the facts we will use later. First, B is avariety of dimension 5 and degree 20. The variety X – which has degree 20 – and the hyperplane H intersect tangentially along B , with intersection multiplicity 2. The boundary B has a naturaldecomposition into five locally closed subsets, which we denote by Z i , Z b , Z s , Z c , and Z v . Thestratum Z i has dimension 5 and consists of all points in B which are smooth points of X such thatat least one of the m ij -coordinates is not zero. The stratum Z i has dimension 4 and consists ofall points in B which are singular points of X such that at least one of the m ij -coordinates is notzero. The stratum Z s has dimension 3 and consists of all remaining boundary points such thatone of the coordinates x , x , x is not zero and one of the coordinates y , y , y is not zero. Thestratum Z c has dimension 2, and here one of the previous three triples of coordinates has valuesall zero. Finally, the stratum Z v consists of a single point: all coordinates except r are zero. It isthe only point on B defined over the reals, all other boundary points are complex.For a multipod given by a set of legs, we have a set of hyperplanes in P dual to the legs. Wenow define the set of bonds of the multipod as the set of intersection points of all these hyperplaneswith the boundary B . The main point of the analysis of the boundary is that the presence ofbonds in particular strata has geometric implications for the geometry of the legs. Let us showthis for the stratum Z s . Here, the projections x := ( x : x : x ) and y := ( y : y : y ) bothexist, because there is at least one in both triples of coordinates that does not vanish. From theequations, it is easy to derive that both x and y have to lie on the absolute conic x + x + x = 0,which clearly has no real points. Theorem 6.3.
Let { ( a l , b l , d l ) | l ∈ L } be the leg set of a multipod, where L is an index setparametrizing the legs. Assume that this multipod has a bond in Z s . Then there exist orthogonalprojections p a : R → R and p b : R → R and a similarity transformation s : R → R such that s ( p ( a l )) = p ( b l ) for all l ∈ L .Proof. The variety X has an automorphism group of dimension 12, by left and right multi-plication of group elements. The statement of the theorem is invariant under these automor-phism. We use suitable automorphisms to transform the bond in Z s to a point with coordinates( x , x , x ) = (1 , i , y , y , y ) = ( λ, λ i ,
0) and all remaining coordinates zero; it is importantthat this transformation can be achieved by real transformations (see [16] for the calculation).The corresponding hyperplane in leg space has equation a + a i + λb + λb i = 0 . For all l ∈ L , the leg ( a l , b l , d l ) must lie on this hyperplane. The real part and the imaginary partof this equation must both be zero: a l, + λb l, = a l, + λb l, = 0. Therefore the claim follows.The stratum Z s ⊂ B is also called the similarity stratum , and the any bond in this stratum iscalled a similarity bond . The theorem above states, in more informal language, that the presence ofa similarity bond implies the existence of two similar projections of base and platform. If a linkagehas an infinite number of similarity bonds, then it can be shown that for all projections of theplatform points, there is a similar projection of the base points. This implies that the platformpoints and the base points are related by a similarity transformation of R . This geometricobservation plays a crucial role in the classification of multipods of mobility 2 in [31].There is the analoguous statement for the stratum Z i ; also the proof is analoguous. Theorem 6.4.
Assume that the multipod above has a bond in Z i . Then there exist orthogonalprojections p a : R → R and p b : R → R and an inversion i : R → R such that i ( p ( a l )) = p ( b l ) for all l ∈ L . Recall the Bricard-Borel multipod with infinitely many legs, described i Section 5: all its legs( a, b, d ) satisfy the condition a b − a b = a b + a b − α = 026or some fixed α ∈ R , α (cid:54) = 0. As we already saw, this is an inversion relation between theprojections of base and platform.If a multipod has a bond in Z b , then there are two lines G a , G b ⊂ R such that for any leg( a, b, d ) in the dual hyperplane in leg space, either a lies in G a or b lies in G b . The presence of abond in Z b implies the existence of two lines and a partition of the set of legs into two subsets,with the first subset having collinear anchor points in the base and the second subset havingcollinear anchor points in the platform. Let us called such a configuration a combined collineation .The existence of such a combined collineation already implies mobility for a suitable choice of leglengths. To see this, we start with a configuration such that the lines G a and G b coincide – theleg lengths have to chosen so that such a configuration exists. Then we can rotate the platformaround this line (similar as the “double banana” Figure 4).The stratum Z c has two irreducible components. For one of these components, the projection( x : x : x ) is defined. The geometric implication is stronger than the implication from a bond in Z b : all anchor points in the base have to be collinear. For the second component, all anchor pointsin the platform have to be collinear. If one of these two conditions is fulfilled, then a rotationmotion is possible from any starting position.The hyperplane corresponding to the one point in Z v is the hyperplane at infinity. Hence thereis no multipod with a finite leg that has a bond in Z v .In summary, any boundary point implies some geometric condition on the two configurationsof platform points and of base points. Hence the compactification gives necessary conditions formobility: if a multipod is mobile, then it must have some bonds, therefore one of the abovegeometric conditions hold.Many mobile multipods have more than just one pair of complex conjugate bonds, since thenumber of bonds is related to the degree of the configuration curve embedded in P . The corre-lation between the degree of the mobility curve of a hexapod and the number of special geometricevents – similar projections, inverse projections, or combined collineations – motives the questionon the maximal number of such events. Here are the answers. Theorem 6.5.
Assume that the six-tuple of points in the base and the six-tuple of points in theplatform are not similar, and that neither the base nor the platform consist of coplanar points.a) The number of combined collineations is at most 16. If every anchor point appears in atmost one leg, for both base and platform, then the maximal number of combined collineations is 4.b) The number of projections related by a similarity transformations is at most 2. The maxi-mum is reached if and only if the two six-tuples are affine equivalent.c) The number of projections related by an inversion is at most 7.
The proofs of (a) and (b) are left as exercises. For the proof of (c), we refer to [17].It is conjectured in [17] that for a generic choice of six points in R , there exists a second six-tuple of points, such that the maximal number of 7 projections related by an inversion is reached;such a six-tuple would then be unique up to similarity. The conjecture continues to state that thereis a unique scaling and choice of leg length such that the so constructed hexapod is mobile, with amobility curve of maximal degree 28. For a numeric random choice, the conjecture can be testedby a construction taking about 300 seconds using computer algebra. Using this construction, theconjecture has been verified for 50 random choices. Theoretically, it is still theoretically possiblethat these 50 random choices were picked on some unknown subvariety with non-generic behavior,but it is quite unlikely. References [1] J. E. Baker. The single screw reciprocal to the general plane-symmetric six-screw linkage.
J.Geom. Graph. , 1:5–12, 1997.[2] G. T. Bennett. The skew isogramm-mechanism.
Proc. London Math. Soc. , 13(2nd Series):151–173, 1913–1914. 273] E. Borel. M´emoire sur les d´eplacements `a trajectoires sph´eriques.
M´emoire pr´esente´es pardivers savants `a l’Acad´emie des Sciences de l’Institut National de France , 33(1):1–128, 1908.[4] R. Bricard. M´emoire sur la th´eorie de l’octa`edre articul´e.
Journal de math´ematiques pures etappliqu´ees e s´erie , 3:113–148, 1897.[5] R. Bricard. M´emoire sur les d´eplacements `a trajectoires sph´eriques. Journal de ´Ecole Poly-technique(2) , 11:1–96, 1906.[6] R. Bricard.
Le¸cons de cin´ematique . Gauthier-Villars, 1927.[7] K. Brunnthaler, H.-P. Schr¨ocker, and M. Husty. A new method for the synthesis of Ben-nett mechanisms. In
Proceedings of CK 2005, International Workshop on ComputationalKinematics , Cassino, 2005.[8] J. Capco, M. Gallet, G. Grasegger, C. Koutschan, N. Lubbes, and J. Schicho. The numberof realizations of a Laman graph.
SIAM J. Appl. Alg. Geom. , 2:94–125, 2018.[9] A. Degtyarev and I. Itenberg. On real determinantal quartics. In
Proceedings of the G¨okovaGeometry-Topology Conference 2010 , pages 110–128. Int. Press, Somerville, MA, 2011.[10] E. Delassus. The closed and deformable linkage chains with four bars.
Bull. Sci. Math. ,46:283–304, 1922.[11] J. Denavit and R. S. Hartenberg. A kinematic notation for lower-pair mechanisms based onmatrices.
Trans. A.S.M.E. , 22:215–221, 1955.[12] P. Dietmaier.
Einfach ¨ubergeschlossene Mechanismen mit Drehgelenken . Habilitation thesis,Graz University of Technology, 1995.[13] A. C. Dixon. On certain deformable frameworks.
Messenger , 29(2):1–21, 1899.[14] M. Gallet, Nawratil G, J. Schicho, and J. Selig. Mobile icosapods.
Adv. Appl. Math. , 88:1–25,2017.[15] M. Gallet, C. Koutschan, Z. Li, G. Regensburger, J. Schicho, and N. Villamizar. Planarlinkages following a prescribed motion.
Math. Comp. , 86:473–506, 2017.[16] M. Gallet, G. Nawratil, and J. Schicho. Bond theory for pentapods and hexapods.
J. Geom-etry , 2014.[17] M. Gallet, G. Nawratil, and J. Schicho. Liaison linkages.
J. Symb. Comp. , 79:65–98, 2017.[18] B. Gordon and T. S. Motzkin. On the zeros of polynomials over division rings.
Trans. Amer.Math. Soc. , 116:218–226, 1965.[19] G. Grasegger, J. Legersky, and J. Schicho. Graphs with flexible labelings.
Discr. Comp.Geom. , 62:461–480, 2019.[20] G. Grasegger, J. Legersky, and J. Schicho. Graphs with flexible labelings allowing injectverealizations.
Discr. Math. , pages 1–14, 2020. to appear.[21] G. Heged¨us, J. Schicho, and H.-P. Schr¨ocker. Factorization of rational curves in the Studyquadric and revolute linkages.
Mech. Mach. Theory , 69(1):142–152, 2013.[22] G. Heged¨us, J. Schicho, and H.-P. Schr¨ocker. The theory of bonds: A new method for theanalysis of linkages.
Mechanism and Machine Theory , 70(0):407–424, 2013.[23] B. Jackson and T. Jord´an. Rigid components in molecular graphs.
Algorithmica , 48(4):399–412, 2007. 2824] C. Jialong and M. Sitharam. Maxwell-independence: a new rank estimate for the 3-dimensional generic rigidity matroid.
J. Combin. Theory Ser. B , 105:26–43, 2014.[25] A. Karger. Classification of 5R closed kinematic chains with self mobility.
Mech. Mach. Th. ,pages 213–222, 1998.[26] N. Katoh and S. i. Tanigawa. A proof of the molecular conjecture.
Discrete Comput. Geom. ,45(4):647–700, 2011.[27] A. Kempe. On a general method of describing plane curves of the nth degree by linkwork.
Proc. LMS , S1-7(1):213, 1876.[28] G. Laman. On graphs and rigidity of plane skeletal structures.
Journal of EngineeringMathematics , 4:331–340, 1970.[29] Z. Li and J. Schicho. A technique for deriving equational conditions on the denavit-hartenbergparameters of a 6r linkage that are necessary for movability.
Mech. Mach. Theory , 94:1–8,2015.[30] Z. Li, J. Schicho, and H.-P. Schr¨ocker. Kempe’s universality theorem for rational space curves.
Found. Comp. Math. , 18:509–536, 2018.[31] G. Nawratil and J. Schicho. Pentapods with mobility 2.
J. Mechanisms Robotics , 7, 2015.[32] J. C. Ottem, K. Ranestad, B. Sturmfels, and C. Vinzant. Quartic spectrahedra.
Math.Program. , 151(2, Ser. B):585–612, 2015.[33] H. Pollaczek-Geiringer. ¨Uber die Gliederung ebener Fachwerke.
Zeitschrift f¨ur AngewandteMathematik und Mechanik (ZAMM) , 7:58–72, 1927.[34] B. Schulze. Symmetry as a sufficient condition for a finite flex.
SIAM J. Discrete Math. ,24(4):1291–1312, 2010.[35] J. Selig.
Geometric Fundamentals of Robotics . Monographs in Computer Science. Springer,2 edition, 2005.[36] A. J. Sommese and C. W. Wampler. Numerical algebraic geometry and algebraic kinematics.
Acta Numer. , 20:469–567, 2011.[37] E. Study.
Geometrie der Dynamen . Teubner, Leipzig, 1903.[38] D. Walter and M. L. Husty. On a nine-bar linkage, its possible configurations and condi-tions for paradoxical mobility. In12th World Congress on Mechanism and Machine Science,IFToMM 2007