Branch reconﬁguration of Bricard linkages based on toroids intersections: Plane-symmetric case.

This paper for the ﬁrst time investigates a family of plane-symmetric Bricard linkages studying two generated toroids. By means of their intersection, a set of special Bricard linkages with various branches of reconﬁguration are designed. An analysis of the intersection of these two toroids reveals the presence of coincident conical singularities which lead to the design of plane-symmetric linkages that evolve to spherical 4R linkages. By examining the tangents to the curves of intersection at the conical singularities it is found that the linkage can be reconﬁgured between the two possible branches of spherical 4R motion without disassembling it and without requiring the usual special conﬁguration connecting the branches. Thestudy of tangent intersections between concentric singular toroids also reveals the presence of isolated points in the intersection which suggests that some linkages satisfying the Bricard plane-symmetry conditions are actually structures with zero ﬁnite degrees of freedom but with higher instantaneous mobility. This paper is the second part of a paper submitted in parallel by the authors in which the method is applied to the line-symmetric case.


Introduction
Among the reported overconstrained linkages, the Bricard linkages (along with the famous Bennett 4R linkage) are the most studied due to their very special geometry that allow the mobility of these 6R loops.A total of six * Address all correspondence to this author.cases were discovered by Bricard [1,2]: the line-symmetric case, the plane-symmetric case, the trihedral case, the linesymmetric octahedral case, the plane-symmetric octahedral case and the doubly collapsible octahedral case.The geometry of these loops was thoroughly studied [3] [4] [5] [2] in order to explain the mobility of these linkages.However, some other properties of the Bricard linkages are still being studied, including optimization, application, combination of loops in large structures and reconfigurability, which has not been systematically analized and whose intuitive presence on the linkages has not been revealed.This paper deals with these issues.
A reconfigurable linkage [6] has a configuration space that includes at least two space components that are connected through singular configurations.Thus, the linkage can work in different motion branches without being disassembled [7].If these connected components of the configuration space are of different dimensions, then the linkage is able to change its degrees of freedom or local mobility and it is said to be a kinematotropic linkage [8,9].At the constraint singularity with change of joint functionality or link annexation resulting in configuration space change, a different kind of reconfigurable linkage emerges as the metamorphic linkages [10,11,12,13].Recent advances in reconfigurable linkages include: a closed chain with 14 states that constitute 28 configurations [14], a reconfigurable platform whose hybrid legs include a 4R diamond [15], a parallel manipulator with 15 motion branch [16], type synthesis of kinematotropic platforms [17], various reconfigurable kinematic JMR-  López-Custodio 1 chains with mobility one [18], a family of single-loop reconfigurable linkages with an infinity of motion branches [19].Some advances in the study of reconfigurable linkages include: the analysis of the connections between components of the configuration space by means of algebraic geometry [20], the study of reconfigurability by means of reciprocity of screws [21], the use of morphing systemization to analyze ways of reconfiguration of linkages [22], the application of higher order kinematic analysis to prove the local mobility of kinematotropic linkages [23,24,25] and the application of dual quaternions in the analysis of reconfigurability [26,16].
In regard to Bricard linkages that lead to different motion branches, the following contributions have already been presented: the analysis of reconfiguration of a plane-and line-symmetric Bricard linkage by means of geometric constraints and screw-system variations [18], spatial triangle formed joint and variable axis joint are used to obtain a linesymmetric linkage that can behave as a Bennett linkage [27], the analysis of a plane-and line-symmetric Bricard linkage with different motion branches in order to avoid singularities [28] and a line-symmetric Bricard linkage evolved from a metamorphic 8R linkage [29].In these studies, a theory of the reconfigurability of Bricard loops that can help the design of these linkages has not been studied thoroughly.However, the authors searched for a method that could be used for this study and the method of generated surfaces appeared as a very promising alternative.
The technique of generated surfaces, based on kinematic dyads joined by spherical pairs or any possible reduction [30,31,32], was formerly applied to prove the mobility of some overconstrained linkages [33], design of linkages with dwell motion [34], analysis of non-overconstrained linkages [35], synthesis of parallel platforms [36] and software graphics [37].The method was recently applied to reconfigurable linkages [38,19,39], taking advantage of the rich knowledge on tangent intersection of surfaces.
In this paper, the method is for the first time applied to the branch reconfiguration of Bricard plane-symmetric linkages, following the previous paper in which the authors study the line-symmetric case.The plane-symmetric case is characterized by the following DH parameters [5]: ( where the positive direction of the z i axes1 is given by the screw direction shown in Fig. 1.In this case each member of the linkage is symmetric to another member through a plane π (Fig. 1).Therefore, axes S 2 and S 6 (S 3 and S 5 ) intersect in a point lying on π and axes S 1 and S 4 also lie on π.Hence, a line containing the points of intersection of pairs of axes {S 2 , S 6 } and {S 5 , S 3 } lies on π and, therefore also intersects S 1 and S 4 .Such line is the central axis of the linear complex [40,41,3,42] which the six axes belong to.The pitch of such linear complex is zero since the central axis always intersects the six axes.The plane-symmetric Bricard linkage is analyzed in this paper by means of the intersection of two generated toroids, building a complete theory of the reconfigurability of these loops.The design is made by manipulating the construction parameters of two concentric singular toroids.An interesting result is the discovery of spherical 4R linkages evolved from the plane-symmetric linkages that are always able to reconfigure between their two branches without being disassembled and without passing through the special configuration that connects the branches in common 4R linkages.To the knowledge of the authors, this is the first time that a linkage with such reconfiguration between disjoint spherical 4R branches is presented and studied.However, Bricard 6R linkages that also work as spherical 4R linkages were first presented in [43].
This paper is organized as follows: The toroids, its generators and its singular forms are revisited in Section 2.Then, in Section 3, it is find out that some examples of Bricard plane-symmetric linkages can be explained and designed as the intersection of concentric singular toroids.This intersection is analyzed in Section 4. In Section 5 any possibility of tangent intersection is explored.In Section 5 the two branches of spherical 4R motion are studied in order to figure out how to reconfigure the linkage from Bricard branches to spherical 4R branches.Finally, two examples are presented in Section 7.This paper is a continuation of the paper submitted by the authors on the line-symmetric case, [44], some theory, notation, and basic concepts used here are introduced in the aforementioned.
2 Singular toroids generated by kinematic RR chains.
In the second section in the paper submitted by the authors on the line-symmetric case, [44], a discussion on general toroids generated by RR dyads was presented.Now, the particular case in which the radius of the secondary and base circles are the same, l = r, and there is no secondary offset, JMR-17-1188 López-Custodio s = 0, is discussed.Such a toroid, T r,r,γ,0 , can be generated by the following parameterization: and T l,r,γ,s = im(σ(T2 )).In a similar way, the implicit form φ ∈ R[x, y, z] is reduced to: For this case, it is easy to prove that: Therefore it can be concluded that (u, π) ∈ T 2 is a conical singularity of T r,r,γ,0 .Hence, we call T r,r,γ,0 a singular toroid 2 .
In this case the singularity maps to the point σ(u, π) = 0 = O.
Figure 2 shows T r,r,γ,0 with the singularity coincident with the origin.B, the intersection of the toroid and any plane containing the Z axis, becomes an 8-shapped curve that is symmetric with respect to the intersection of the XY plane and the plane containing B. The self-crossing of B occurs at O. 3 The concentric singular toroids generated by the plane-symmetric Bricard linkage.
In the plane-symmetric case of Bricard linkage any adjacent pair of revolute joints with skew axes generates a toroid.For a general plane-symmetric linkage, like the one shown in Fig. 1, let a 2,3 = 0. Then the point of intersection of S 2 and S 3 describes a toroid with respect to the fixed link between axes S 6 and S 5 , this is generated by joints with axes S 6 and S 1 .Furthermore, the same point describes another toroid with respect to the fixed link, this time generated by joints with axes S 4 and S 5 .Since the same point describes two toroids, such point is confined to move in the intersection of these toroids.
From the restrictions for plane symmetry in Eq. ( 1) , observe that if a 2,3 = 0 then a 5,6 = 0.This implies that both toroids are concentric, as shown in Fig. 3a.Furthermore, since a 6,1 = a 1,2 and a 3,4 = a 4,5 , then r = l for both toroids.Finally, since d 1 = d 4 = 0 both toroids have secondary offset s = 0. From these observations it can be concluded that both generated toroids are singular, with the singularity coinciding with the intersection of fixed axes S 6 and S 5 .Summarizing, in addition to the restrictions on the DH parameters of the plane-symmetric linkage in Eq. ( 1), the following conditions are required to analyze and design these linkages using the intersection of two toroids: Fig. 3b shows an example of plane-symmetric Bricard linkage that generates two singular toroids.For the sake of identifying the construction parameters of each toroid, the joints have been renamed with respect to Fig. 1: S A1 = S 6 , S A2 = S 1 , S A3 = S 2 , S B3 = S 3 , S B2 = S 4 and S B1 = S 5 .In such case, a singular toroid T i r i ,r i ,γ i ,0 := T r i ,r i ,γ i ,0 ( ki , O) is generated by the joints with axes S i1 and S i2 and is referred to coordinate systems i, i = A, B. The point that describes the intersection C := T A r A ,r A ,γ A ,0 ∩ T B r B ,r B ,γ B ,0 is the intersection of axes S A3 and S B3 and is called E. The point where S A1 and S B1 intersect is called O.
For each toroid T i r i ,r i ,γ i ,0 := {E i (q i ) | q i ∈ T 2 }, i = A, B, the joint variables vector is given by the variables of the parameterization in Eq. (2), so that q i = (u i , v i ) ∈ T 2 .Once the link D joins axes S A3 and S B3 , E A (q A ) = E B (q B ) = E(q), where q := (u A , v A , q A3 , q B3 , v B , u B ) ∈ V ⊂ T 6 , where V is the configuration space of the linkage whose elements have to fulfill the closure equation of the loop.Observe that, due to symmetry, q A3 and q B3 are in linear correspondence with u A and u B , respectively.Hence, finding the intersection C completely describes the behavior of the linkage.In fact, finding a parameterization of C in terms of any of the four variables of the toroids would be equivalent to solve the position analysis of the linkage.
C may be composed of several components such that the corresponding component of V , and the linkage has 1 D.O.F. when assembled in this mode, this leads to the typical overconstrained behavior of the linkage.On the other hand, if C i is an isolated point and dim(V i ) = 0.In such component of V , the linkage can be assembled as a structure.
An important phenomenon occurs when two conical singularities coincide.We call this coincidence the double singularity.The arrangement of singular toroids for planesymmetric linkages includes a double singularity since both toroids are concentric.In the general method of generated surfaces the two generators are connected by a spherical pair, it was proved [38] that for such linkages when E is in the double singularity the linkage has 2 D.O.F.since variables u A and u B can take any value without restriction while v A and v B remain constant.However, for the problem of planesymmetric Bricard linkages the spherical pair has been reduced to a pair of coincident revolute joints.When E reaches the double singularity it coincides with O, therefore the axes S A1 , S B1 , S A3 and S B3 are intersecting in the double singularity while the joints with axes S A2 and S B2 remain idle.u A and u B are now dependent one from another since the linkage becomes a spherical 4R linkage.This property will be thoroughly studied in section 8.
Refer to figure 4 which shows the common perpendiculars diagram of the plane-symmetric linkage obtained from the intersection of two toroids.In any configuration the common perpendiculars between the adjacent joint axes conform two isosceles triangles, OP A E and OP B E, these triangles share the same base OE, the lines from the mid-point of EO to P A and P B are perpendicular to OE and therefore, a plane π containing these two lines is perpendicular to the planes that contain the two triangles.The axes S A2 and S B2 contain P B and P A , respectively, and are perpendicular to the planes including the triangles OP A E and OP B E, respectively.Hence, both axes belong to π and therefore they either intersect or are parallel.This ensures that the symmetry condition is always present while E moves through all the components of C .

Concentric singular toroid-toroid intersection.
To analyze the intersection of toroids in the planesymmetric case, C = T A r A ,r A ,γ A ,0 ∩ T B r B ,r B ,γ B ,0 , let the relationship between coordinate systems A and B be given by A B T = T(R(θ, ĵ), 0), so that the toroids are concentric and the axis of B is obtained by rotating the axis of A θ radians about the The parameterizations of both surfaces referred to coordinate system A are:

JMR-17-1188
López-Custodio and the implicit forms are given by: A direct way to find C is to solve However, in this case it turns out to be more complicated.An alternative technique, taken from [45], is applied instead: Since, for any of both implicit forms referred to coordinate system A, (x, y, z) = 0 is a scalar equation with two variables from which the restrictions u A (v A ) or v A (u A ) can be obtained.This restriction fully defines C since it can be replaced in A σ A to obtain the whole parameterization of the intersection, for example using the restriction An immediate first possibility is observed: v A = π.This solution leads to the double singularity in which E(q) = O, ∀q ∈ V 1 , where V 1 is the spherical 4R component of V related to the double singularity.Since making 0 the first factor in Equation 7 would compromise the construction parameters of the toroids, the only remaining possibility is solving the third factor.This factor is solved to obtain the restriction u A (v A ), two solutions are found which are not presented here due to reasons of space since these are quite long expressions.In a similar manner, two solutions for the restriction v A (u A ) are obtained.Therefore, C may feature a maximum of two curves.Expressions for a parameterization of these components can be computed as explained in the previous paragraph, however, due to the length of the terms involved in the restriction u A (v A ), these are not presented here.
If in the concentric toroid-toroid intersection ∃ i, j 0 the linkage is reconfigurable with at least 2 motion branches, which are connected through at least one configuration q i j ∈ V i ∩ V j .It can be proved [38] that for the 1-dimensional components of V , the toroids are tangent to each other at E(q i j ).The intersection is non-transverse in E(q i j ).Therefore, ∇φ A (x P , y P , z P ) × ∇φ B (x P , y P , z P ) = 0, where (x P , y P , z P ) = E(q i j ).The points in V that map to points of tangency may be bifurcation configurations of the linkage.These points in V may represent the intersection of two components of V , or may be the self-crossing of the same component.The surfaces are also tangent to each other when they touch in one point, which would lead to an isolated point in V .In addition, if a continuum of points of tangency is found, the surfaces are touching in a curve that is a component of C .To find the points where the intersection may become non-transverse, the real points (x, y, z) ∈ R 3 that make ∇φ A (x, y, z) × ∇φ B (x, y, z) = 0 and also satisfy φ A (x, y, z) = φ B (x, y, z) = 0 are explored.Two points in the Y axis are found, however, they imply r A = r B , so that the points are (0, r A , 0) = (0, r B , 0) and (0, −r A , 0) = (0, −r B , 0).This case leads to a linkage that is both line-and plane-symmetric.This example was analyzed before in [28] and the linesymmetric case was investigated by the authors of this paper in another work yet to be published.The other solutions that do not degenerate the toroids imply y = 0. Therefore, any point of tangency must lie in the X A Z A plane if the linkage is not the plane-and line-symmetric case.The solutions in the X A Z A plane are large expressions that involve not only JMR-  López-Custodio the construction parameters of the toroids, but also the angle θ.Considering the singular curves B A and B B obtained by 5a), it is found that the curves become tangent to each other if: (8) Each possibility leads to two solutions, therefore there are in total 4 values of θ that make the surfaces tanget to each other in the X A Y A plane.Note that the argument of the square root is always positive, however, to obtain real values of θ, it is necessary that (r 2 This makes toroid B looking more flattened than toroid A.
6 Isolated points of tangency and Bricard structures.
The nature of the intersection when the two concentric singular toroids are tangent to each other in the X A Z A plane is now investigated.For this aim, consider the following proposition: Proposition 6.1:The intersection of two concentric singular toroids with different radius contains only two isolated points if the toroids are tangent to each other at some point.
Proof: Let the two singular toroids to be intersected be T A r A ,r A ,γ A ,0 and T B r B ,r B ,γ B ,0 , where r A = r B , thus the only possibility for tangent intersection is that the surfaces are tangent to each other in the X A Z A plane.Replacing the values of θ from Eq. ( 8) in the parameterizations of the surfaces and trying to find C would lead to quite complicated expressions.A simpler way to proceed is to analyze the normal curvatures of the toroids in one of the points where the surfaces are tangent.The normal curvatures of both surfaces must be the same in the direction that is tangent to the intersection curve.If there is no intersection curve and the surfaces are only touching in such point, the curvatures are always different for both surfaces in any direction.Since normal curvature is invariant to frame transformations both toroids can be analyzed in their own coordinate systems.According to Euler's formula, the normal curvature is given by: κ(ψ) = κ 1 cos 2 ψ + κ 2 sin 2 ψ, where κ 1 and κ 2 are the curvatures in the principal directions ê1 and ê2 and ψ is the angle that defines the direction of the normal curvature with respect to one of the principal directions.
The singular toroids are surfaces of revolution with B being rotated about the Z axis.It is known (see for example [46]) that in surfaces of revolution the principal curvatures are the tangents to the meridian and parallel crossing the point in analysis.Hence, for the arrangement shown in Fig. 5b ê1 = ĵ and ê2 = t, where t is the mutual tangent vector to B A and B B at P 1 .If the intersection includes a curve crossing P 1 , it should be possible to find an angle ψ ∈ T, such that: where κ iA and κ iB , i = 1, 2, are the principal curvatures at P 1 of toroids A and B, respectively.From Fig. 5b Note that κ 2 j , j = A, B are the curvatures of plane curves B j , while κ 1 j are the curvatures of plane curves D j obtained by intersecting the toroids with the plane that contains P 1 and is spanned by ê1 = ĵ and n.If, w.l.o.g.r A > r B as shown in Fig. 5, it is clear that κ 2B > κ 2A and κ 2B − κ 2A > 0. Hence, in order to have a real solution of Eq. ( 9), it is necessary that κ 1A − κ 1B > 0. These curvatures can be computed using the following expression [47]: where, Hess : R[x, y, z] → M 3×3 (R) is the Hessian matrix of the given implicit form.Upon calculations it is concluded that: where |r P 1 |,is the magnitude of the position vector of P 1 , which is the same value for both toroids and is invariant to frame transformations.|r P 1 | is calculated using the value of θ in Eq. ( 8), leading to: Replacing this value in Eq. ( 10) and carrying out simplifications it can be concluded that, κ 1A − κ 1B has the same sign as r B − r A .Therefore, if r A > r B (as first supposed for this proof), κ 1A − κ 1B < 0, tan 2 ψ < 0 ⇒ ψ / ∈ T and there is no real solution for Eq. ( 9).Hence, both toroids are touching each other in P 1 and P 2 but these are isolated points in Two important conclusions can be drawn from the previous proposition: First, a Bricard linkage fulfilling the plane symmetry conditions can be a 0-DOF structure which can be assembled in two different configurations.However, if the linkage is assembled in E(q) = O the same linkage has 1 DOF and works as a spherical 4R linkage.In such case, V is composed of 3 regions: 2 isolated points and a 1-dimensional curve in T 6 .And second, there is no way to reconfigure these JMR-  López-Custodio linkages directly from one curve to another at least the linkage is also line-symmetric.
Consider the linkage assembled in an isolated point E(q 1 ) = P 1 in figure 8, some interesting results regarding the reciprocal system of the screw system of the linkage are now obtained: First, the plane of symmetry is perpendicu-lar to and bisects the segment OE, which lies on the X A Z A plane, thus the plane of symmetry is perpendicular to the X A Z A plane.As a consequence of this, S A3 and S B3 lie on the plane X A Z A , since their symmetric members, S A1 and S B1 lie on X A Z A which is perpendicular to the plane of symmetry.Then, the axis of the special linear complex is the intersection of the plane of symmetry and the X A Z A plane as expected from Section 2. Knowing the value of θ, it is possible to calculate both u A and u B for any of the two configurations in which the linkage can be assembled without using the double singularity and making it a spherical 4R linkage.With u A and u B , the following Plücker coordinates are obtained for the screws A2 and B2, respectively: Previously, it was proved that S A2 and S B2 always intersect.If the screws are defined by S i = (ŝ i ; m i ), i = A2, B2, such intersection point is given by P = (ŝ A2 • m B2 ) −1 m A2 × m B2 .However, it turns out that for this example ( A m A2 × A m B2 ) • A j = 0, which means that P lies on the X A Z A plane.This implies that a pencil of lines in the X A Z A plane can be JMR-  López-Custodio drawn with center in P and each line in the pencil will be reciprocal to all the axes of the linkage.Hence, the reciprocal system is a 2-system and the first order mobility of the linkage is 2. A velocity analysis applied to this structure would reveal that the linkage has 2 degrees of freedom since the tangent space of V in such configuration is a 2-dimensional space.A tangent cone [48,49,50,51,52,23] analysis was made for this configuration.While the first order approximation is a 2-dimensional vector space, as expected, the second order approximation is 0-dimensional, proving that the linkage is actually a structure.It is then said that the structure has second order rigidity [53].
7 Motion branch reconfiguration through the double singularity.
Reconfiguration of motion branches can be achieved using the double singularity of the concentric toroids if there are curves crossing it.Due to the symmetry of the intersection, and since there are no points of tangency (excluding the known exceptions), it can be concluded that the intersection will have any of the following forms: 1. Two regular disjoint curves and the double singularity point 2. One singular 8-shaped curve with its self-crossing coincident with the double singularity, 3. Two singular 8-shaped curves that share the same selfcrossing point which is coincident with the double singularity.
The first case is generated by non-reconfigurable linkages since there is no way to migrate from one curve to another or to visit the double singularity.The second case is reconfigurable with two motion branches: a Bricard 6R operation mode and a spherical 4R linkage mode.The third case is the most interesting since the configuration space includes two Bricard branches which can be visited by the linkage without disassembling it and, in addition, the linkage can undergo spherical 4R motion branches.
In figure 8, note that if in the X A Z A plane B A ∩ B B = {O} the intersection of the toroids includes two singular curves as in the third case of intersection.Since the toroids are symmetric with the X A Z A plane, if B B crosses B A in four points, the intersection curve never includes O, as in the first case of intersection.If B B crosses B A in two points, the intersection curve crosses O and then intersects the X A Z A plane in the two points where B B crosses B A , the intersection of toroids is then a sole singular curve, as in the second case of intersection.Now imagine that the two points where the intersection crosses the X A Z A move through B A approaching O, since the intersection is symmetric with the X A Z A plane, the curve starts to sharpen in such points until they reach O and the intersection becomes two singular curves.Refer to figure 8a, it is easy to prove that the tangent to B i , i = A, B makes an angle γ * i with the x i axis.Then the conditions that make B B intersect B A only in O are the following: In the remaining part of the paper we focus exclusively in linkages whose generated toroids fulfill conditions 12, since these are the most complicated cases.
8 Bricard branches as a link between crank-rocker spherical 4R branches.
like in the third case of intersection, then C 1 ∩C 2 = {O}.it has to be considered that, even though the two curves intersect in one point, if , which means that the linkage cannot reconfigure from V 1 to V 2 directly.This is a consequence of the double-singularity O.In any regular point in a surface all curves intersecting the point do it with the same values of (u, v) ∈ U, as σ is a bijection from U to S \ sing(S ).But since in the conic singularity of the singular toroids ∂σ/∂u = 0 there are an infinity of pairs (u, v) that map to O and the only way to escape from the singularity is moving in the direction of the isoparametric curve of v, since ∂σ/∂v = 0 is such point.These isoparametric curves are the secondary circles, their tangent vectors in the singularity generate a cone that is tanget to the toroid in the singularity.Any two curves on the toroid crossing the singularity with non-parallel tangent vectors at O will have different values (u(t), v(t)) at O, since they reached the point in different secondary circles.

JMR-17-1188
López-Custodio The previous paragraph implies that in general the linkage cannot move from V 1 to V 2 since E reaches O in different configurations.In fact, in the self-crossing of each singular curve in C , the linkage is unable to chose between the two segments in the neighborhood of O, E smoothly passes the double singularity and V 1 and V 2 are free of singularities even though they are related to singular curves C 1 and C 2 .Despite V 1 and V 2 are disjoint, they are connected through the spherical 4R motion branch related to the double singularity.For the evolved spherical 4R linkage two opposite links have the same twist angle, θ, while the other two links have twist angles 2|γ A | and 2|γ B |.
Suppose E lies on C 1 and approaches O, once the linkage starts working in the spherical 4R branch axes S A2 and S B2 can move until the secondary circles are both tangent to C 2 and E can scape from the double singularity allowing the linkage to enter the V 2 branch.Since there are two singular curves crossing O, there are in total four different directions in which E can move to scape from O. Two of these will reconfigure a spherical 4R branch into the same planesymmetric Bricard branch related to C 1 , while the other two will reconfigure to the branch related to C 2 .In each of these configurations the evolved spherical 4R linkage most be in a plane-symmetric configuration since such configuration also belongs to a Bricard branch.Fig. 9 shows a plane-symmetric linkage in a spherical 4R branch with E = O, the linkage is about to scape to V 1 since the secondary circles C 2A and C 2B are both tangent to C 1 at O. Fig. 9.A reconfigurable Bricard plane-symmetric linkage in its spherical 4R operation mode, with its two secondary circles tangent to each other and to curve C 1 and the linkage is about to escape to V 1 . .However, it is known that the configuration space of the spherical 4R linkages may include two branches which may or may not intersect.It is possible that the two configurations that allow to scape to V 1 (E scaping from O to C 1 ) belong to the same branch of the 4R linkage, while the other two configurations to scape to V 2 (E scaping from O to C 2 ) belong to the other branch of the 4R linkage.In such case, in order to reconfigure the linkage from V 1 to V 2 it is necessary to disassemble it if the two spherical 4R branches are disjoint.Therefore, the following paragraphs investigate the two spherical 4R branches and the four scape configurations in order to establish the restrictions that ensure that the linkage can reconfigure through all of its branches.We begin by analyzing the rotability of the evolved spherical 4R linkages since branch identification is different depending on the rotability of the links.However, we restrict this analysis to the cases in which the toroid generators are built using |γ| = γ * Fig. 10.A plane-symmetric configuration of a spherical 4R linkage with two opposite links of the same angle.
. Proposition 6.1:The spherical 4R linkages obtained as a behavior of Bricard plane-symmetric linkages that generate two concentric toroids intersecting in two curves are either crank-rocker or change-point.
Proof: Let the twist angles of the spherical four-bar linkage evolved from the Bricard plane-symmetric linkage be α A1,A3 = 2|γ A |, α A3,B3 = α B1,A1 = θ and α B3,B1 = 2|γ B | (Fig. 10).From the second condition in Eq. ( 12): According to Gupta and Ma [54], if the sum of twist angles is less that 2π, the full rotability criterion is quite similar to that for the planar case: if α min + α max < α p + α q , then at least one of the links is fully rotatable.
From the second condition in Eq. ( 12), . By contradiction consider that none of the links is fully rotatable.Then, if α max = α A3,B3 = α B1,A1 = θ, from the criterion it follows: which is a contradiction.In a similar way, now consider α p = α q = α A3,B3 = α B1,A1 = θ: which contradicts the second condition in Eq. ( 12).Hence, it is proved that at least one of the links is fully rotatable.This link is the one whose twist angle is α min .In Fig. 10 it can be seen that the twist angle for the coupler and fixed links is θ, which is proved to be different to α min .Therefore, the smallest twist angle corresponds to either the input or output links.Hence, all the linkages are crank-rocker or, if It is known [55] that in crank-rocker (or rocker-crank) 4R linkages both branches are disjoint.Therefore, from Proposition 6.1 it is concluded that the only way to have a special configuration joining the two branches is the spherical equivalent of a parallelogram linkage.In such a very special case the criterion for branch change is simply the parallel-or anti-parallelism of the links.For crank-rockers (or rocker-cranks), the following two propositions allow the identification of branch change.
Proposition 6.2:A 4R linkage with two opposite links of the same twist angle can reach four plane-symmetric configurations, two of them belong to the same branch while the other two belong to the other branch.
Proof: Refer to Fig. 10, which shows the spherical 4R linkage with two opposite links of the same twist angle, θ, the angle of the other two links are 2γ A and 2γ B .The linkage is shown in a plane-symmetric configuration.ŝA1 , ŝA3 , ŝB3 and ŝB1 are the unit vectors parallel to the axes of the revolute joints.A coordinate system X 0 Y 0 Z 0 is placed fixed to the symmetry plane π, so that π coincides with the plane X 0 Z 0 and X 0 bisects the angle between ŝA1 and ŝA3 .
From Proposition 6.2 it can be seen that the two different Bricard branches may reconfigure to spherical 4R modes in different branches, making impossible to move from one Bricard branch to the other.Each of the four plane-symmetric configurations presented in Proposition 6.2 is a bifurcation configuration between Bricard branches and spherical 4R branches.Therefore, a vector v j tangent to the curve of intersection at O can be calculated for each of these configurations.For the sake of simplicity we call these vectors escape directions Proposition 6.3 Given a plane-symmetric Bricard linkage generated from the intersection of two concentric singular toroids with the axis of one rotated about the Y axis from the other, the escape directions lying on the same side of the plane XZ correspond to configurations lying in the same spherical 4R branch.
Proof: From the geometry of the plane-symmetric linkages obtained from the intersection of two concentric singular toroids it can be proved that the escape directions v j are parallel to 4 (ŝ B3 × ŝB1 ) × (ŝ A3 × ŝA1 )|| ĵ0 , where ĵ0 is the unit vector in the direction of Y 0 in Fig. 10.Each of the four configurations obtained in Proposition 6.2 lead to a escape direction v j .We are interested in obtaining such vectors in the coordinate system A, which is fixed, while coordinate system 0 moves from one configuration to another.Therefore, for coordinate system A, vectors A ŝA1 and A ŝB1 are fixed, while there are four sets of vectors A ŝ j A3 and A ŝ j B3 .The escape directions are calculated by finding the bases 0 î j A , 0 ĵ j A , 0 k j A : The following four escape directions are found, each related to each of the symmetric configurations found in Propo-sition 6.2: It can be seen that A v1 and A v2 ( A v3 and A v4 ) are symmetric with respect to the X A Z A plane as the only difference between them is the sign of the Y component.The sign of the Y components of A v1 and A v3 ( A v2 and A v4 ) is the same, namely −sign(cos γ B ) (sign(cos γ B )), hence they lie in the same side of the X A Z A plane.In addition, all these vectors lie in the same side of the X A Y A plane as their Z components are the same, namely −sign(sin γ B ). From Proposition 6.2 it is known that configurations 1 and 3 (2 and 4) lie in the same branch, therefore it can be concluded that the escape directions lying in the same side of the X A Z A plane belong to the same branch of the spherical 4R linkage.Fig. 11 shows three possible cases of intersection composed by two curves: r A > r B , r A = r B and r A < r B .In each case the tangent vector to the curves at O, the double singularity point, are shown.These tangent vectors are the same that were calculated in Proposition 6.3, in which it was proven that if these 4 vectors have Z A component of the same sign, then: when the vectors lie in the same side of the X A Z A plane the configurations of the spherical 4R linkage lie in the same branch.From Fig. 11 it can be seen that for the case r A > r B the vectors lying on the same side of the X A Z A plane are tangent to the same curve, and since these configurations belong to the same spherical 4R branch reconfiguration to the other curve is impossible and the two spherical 4R branches cannot be reached without disassembling the linkage.For r A = r B and r A < r B the two vectors lying on the same side of the X A Z A are tangent to different curves, this means that it is possible to reconfigure the disjoint spherical 4R branches without disassembling the linkage as both branches are connected through a Bricard branch related to each of the two curves.
It can be proved 5 that for the change-point evolved spherical 4R linkages the escape directions that are symmetric with respect to the plane X A Z A belong to the same branch.Therefore, from Fig. 11, if r A > r B the same Bricard branch can be reconfigured to parallel or anti-parallel branches of change-point spherical 4R linkage.Fig. 12 shows the reconfiguration diagrams for two cases of reconfigurable plane-symmetric linkage obtained from the toroids intersection.The diagram in Fig. 12a is related to a case with two crank-rocker spherical 4R branches, it is known that these branches are disjoint, but it is possible to reach both branches without disassembling the linkage since they are communicated through two different Bricard plane-symmetric branches.Figure 14 shows the linkage in several configurations in each motion branch.Observe that the two configurations belonging to V 1 (V 2 ) for which E = O are different as expected.None of these four symmetric configurations coincide since V 1 and V 2 are disjoint.However, each of these is singular in V , allowing the reconfiguration to V O1 and V O2 , the spherical 4R branches, for which E(q) = O, ∀q ∈ V O1 ∪V O2 .The reconfiguration between branches is presented in the diagram in Fig. 12a, which shows how the two branches of the evolved spherical 4R linkage are connected through planesymmetric Bricard branches allowing reconfigurability without disassembling.

Conclusions
The plane-symmetric case of Bricard loops was analyzed using the intersection of two concentric singular toroids, allowing the design of reconfigurable linkages with several motion branches which can be either planesymmetric 6R branches or spherical 4R branches.The conditions for having two singular curves in the intersection set were presented.Each of these curves is related to a planesymmetric 6R branch of motion.The phenomenon of double singularity leads to kinematotropy when the two surface generators are joined by a spherical pair or a reduction of this to a pair of revolute joints each being parallel to the axis of rotation of surfaces of revolution.However, in the case of overconstrained plane-symmetric linkages it was found that such double singularity leads to a spherical 4R branch. .The study of the escape directions, which are the tangents to the intersection curves at the double singularity revealed the existence of linkages whose evolved crank-rocker spherical 4R linkage can work in its two branches without disassembling it.To the knowledge of the authors, this is the first time that a linkage with this property is presented.These interesting results, along with those for the line-symmetric case, which the authors are presenting in a different paper, shed light on whether it is possible to design more overconstrained linkages that can be reconfigured between different branches using the method of generated surfaces.The paper in which the authors apply the method to the line-symmetric case, [44], is published in parallel with this paper.

Fig. 5 .
Fig. 5. Two singular toroids that are tangent to each other in the X A Z A plane.

Fig. 6 .
Fig. 6.A Bricard plane-symmetric linkage with finite mobility zero in its two different assembly modes.

Fig. 7 .
Fig. 7.A Bricard plane-symmetric linkage with finite mobility zero in the assembly mode with E(q 1 ) = P 1 .

Fig. 8 .
Fig. 8.A case with two singular curves in the intersection: a) B A and B B curves, b) surfaces and intersection.

Fig. 11 .Fig. 12 .
Fig. 11.Tangent vectors to the intersection curves at the double singularity, for the three possible cases: r A > r B , r A = r B and r A < r B

Fig. 13 .
Fig. 13.A reconfigurable Bricard plane-symmetric linkage that allows the reconfiguration between two spherical 4R branches through two Bricard branches.The intersection of the concentric singular toroids is composed of two singular curves.

Fig. 14 .
Fig. 14.Several configurations of the plane-symmetric Bricard linkage when the intersection of concentric toroids contains two singular curves.