We demonstrate analytically that it is possible to construct a developable mechanism on a cone that has rigid motion. We solve for the paths of rigid motion and analyze the properties of this motion. In particular, we provide an analytical method for predicting the behavior of the mechanism with respect to the conical surface. Moreover, we observe that the conical developable mechanisms specified in this article have motion paths that necessarily contain bifurcation points, which lead to an unbounded array of motion paths in the parameterization plane.
The basic question that we begin to address in this article is, “Given a developable surface, what types of mechanisms can be constructed on the surface which conform to the surface and have rigid motion?” Rigid motion for such a mechanism means motion without deforming the surface or the links.
Developable surfaces are of interest in design because they can be obtained by bending a flat surface , without stretching, tearing, or creasing. In particular, a developable surface is represented mathematically as the image of a smooth path isometry defined on a flat surface. For developable surfaces, the Gaussian curvature, or the product of the two principle curvatures, is necessarily zero [2–4]. Basic families of developable surfaces include planar, cylindrical, conical, and tangent surfaces . Developable surfaces can be represented as the union of a one parameter family of lines in , called ruling lines. The existence of ruling lines allows the possibility of creating the mechanisms described in this article by introducing hinges along the ruling lines of the surface.
Engineers can take advantage of the lower production costs and complexity associated with using developable surfaces in their designs. Developable surfaces designed utilizing flexible materials can be manufactured in a flat state and later transformed into their desired curved forms. In addition, developable surfaces can be often manufactured without the heat treatment required for the production of other types of surfaces . Some applications of developable surfaces include steel ship hulls, cartography, architecture, aerostructures, and texture mapping in computer graphics [5–9].
Because developable surfaces are commonly used in design, it is of interest to discover innovative ways to create functionality on these surfaces. One way to increase functionality is to create mechanisms that conform to the surface and are able to achieve motion off of the static surface.
Developable mechanisms are mechanisms that “conform to developable surfaces when both are modeled with zero thickness” . This zero-thickness surface is called the developable mechanism’s reference surface . The links of a developable mechanism should not be required to deform for the mechanism to have motion. This can be achieved by aligning hinge lines with the reference surface’s ruling lines . In at least one position, the conformed position, the mechanism’s links must conform to the developable reference surface. This requires the rigid links to be shaped to the surface when in their conformed position . Cylindrical developable mechanisms have been discussed in Ref.  and have inspired the creation of surgical devices . Since the beginning of this study, additional work has been done on conical developable mechanisms, which are presented in Ref. .
The motivation of this article is the demonstration of the mathematical modeling and analysis of developable mechanisms that can be constructed using kirigami techniques, similar to designs for planar surfaces. Kirigami is a variation of origami that includes cutting in addition to folding . Kirigami has inspired the creation of lamina emergent mechanisms, or mechanisms that can be fabricated in a plane, and then emerge from the surface . Although mechanisms on cones are the subject of this article, the planar counterpart can provide an analogy that is helpful for understanding some of the work that follows. The lamina emergent mechanism shown in Fig. 1 is a developable mechanism that is constructed from a planar surface using kirigami and consists of panels linked at hinge joints. As the hinge lines are all parallel, this mechanism will have planar motion. Figure 1(a) shows the mechanism in its as-fabricated state, conforming to a plane. The planar state also represents a bifurcation point (change-point), where the mechanism can change between two different paths depending how it begins to move from this point. These two paths are illustrated in Figs. 1(b) and 1(c). The existence of bifurcation points occurs if the mechanism is a special-case Grashof mechanism . For planar four-bar mechanisms, this depends on link lengths. In the planar case presented in Fig. 1, the opposite links are the same length, making it planar parallelogram linkage, which is a special-case Grashof mechanism. For spherical mechanisms, the Grashof condition is based on link angles .
In this article, we construct an analogous mechanism cut out of a cone as shown in Fig. 2. This mechanism will be referred to as a conical developable mechanism because it is constructed from a cone. This particular conical developable mechanism is constructed to be a spherical parallelogram linkage, where the link angles of opposite links are equal. This means it is a special-case Grashof mechanism and, if a rigid motion exits, it will necessarily have bifurcation points (change points) . We will proceed by providing a model of the motion of the mechanism and then prove analytically that a rigid motion does exist. We will then provide a detailed description of its motion, with special attention to the initial motion from its conformed position at bifurcation points in the motion path.
2 Construction and Setup
In this section, we detail the construction of the conical developable mechanism and begin to setup the mathematical model utilized in determining the rigid motion of the panels on the mechanism.
2.1 The Mechanism.
Let be a cone centered on the positive z-axis, with its cone point at the origin, and having cone angle ϕ, i.e. ϕ is the angle between the cone axis and the cone surface. The conical developable mechanism is constructed by cutting out a section of the cone and folding along hinge lines to form three panels (links) with the remainder of the cone forming a panel (a ground link), as shown in Fig. 2. The panel P0 is the main body of the cone. The panels P1 and P2 are joined to the body of the cone along hinge lines, which we will call H1 and H2, respectively. Panel P3 emerges out of the cone and is connected to panels P1 and P2 along hinge lines, which we will call H3 and H4, respectively.
Viewing the main body of the cone P0 as fixed, the panels P1 and P2 will rotate rigidly about their respective hinge lines H1 and H2. The angle from which the panel P1 rotates about hinge line H1 is denoted α1, with α1 = 0 corresponding to P1 being in the conformed position (i.e., being flush with the body of the cone). Similarly, the angle from which the panel P2 rotates about the hinge line H2 is denoted as α2, with α2 = 0 corresponding to P2 being in the conformed position. We define the positive direction of the angle to correspond to an initial outward movement. Thus, αi is the angle between the normal vectors to the panel P0 and the rotated panel Pi at any point of the intersection of P0 and Pi. We desire to find a relationship between α1 and α2, so that we can ensure that the rigid motion of panels P1 and P2 will admit a rigid motion for P3, so that P3 remains joined to P1 and P2 along the hinge lines H3 and H4.
The panel P0, and therefore the hinge lines H1 and H2, are held fixed. Because the position of panel P1 in depends on α1, we refer to the image of P1 in resulting from a rotation of α1 as P1[α1]. The image of P2 resulting from a rotation of α2 is denoted as P2[α2]. Similarly, the image of the hinge lines H3 and H4 in , with respect to the angles on which they depend, will be denoted H3[α1] and H4[α2], respectively. The dependence of a possible position for P3 on α1 and α2 is what is in question.
We can now clearly see that the conical developable mechanism (termed for the type of surface from which it is constructed) behaves kinematically like a spherical mechanism (termed for the existence of a point of concurrence of the hinge lines). A spherical mechanism consists of bars linked at hinge joints whose hinge lines all intersect at a point. The conical reference surface has ruling lines that meet at the cone point (or apex of the cone). A necessary, although not sufficient, condition for a conical developable mechanism to have rigid motion is that the hinges are constructed along straight lines. Thus, the hinges must be constructed along ruling lines. Since P0 is fixed, any rigid motion of panels P1 and P2 maintain that all four hinge lines meet at the cone point throughout the motion. Thus, the mechanism must be a spherical mechanism centered about the cone point (see also Ref. ). Methods traditionally used for analyzing spherical mechanisms  could also be used in the analysis that follows, but the mathematical approach used here is helpful for the particular analysis and the resulting theorems.
2.2 Defining Points.
2.3 Motion via Linear Transformation.
The motion of a panel about a hinge line Hi, for i = 1, 2, can be described through a series of linear transformations. The composition of the following transformations will provide the transformations that describe the motion of panel Pi, so that we can determine Pi[αi]:
- The first transformation moves hinge line Hi to the xz-plane by a clockwise rotation by an angle ω about the z-axis:(3)
- Next, we move the image of hinge line Hi to the z-axis by a clockwise rotation of the cone angle ϕ about the y-axis:(4)
- This next rotation about the z-axis by an angle αi is the key transformation. Having applied the transformations A0(ω) and A1(ϕ), the image of the hinge line Hi now lies on the z-axis. Hence, the rotation of panel Pi about the hinge line Hi at this step is realized by(5)
- The transformation A1(−ϕ) reverses the action of A1(ϕ):(6)
- Finally, the transformation A0(−ω) reverses the action of A0(ω):(7)
3 Rigid Motion
For the conical developable mechanism to have rigid motion, panels P1 and P2 must move by a rotation about their hinge lines H1 and H2, respectively. Our goal is to find an open interval U of the real line containing 0 and a function , so that the rigid motion of panels P1 and P2 given by P1[α1] and P2[f(α1)] admits a rigid motion for P3 as well.
Supposing that such a function f exists, a necessary condition for a rigid motion on panel P3 is that the distance between points a[α1] and b[f(α1)] remains constant as α1 varies. In fact, as we shall see from the rigidity theorem in the next section, this condition is both necessary and sufficient.
Note that D(α1, α2) is dependent on the design parameters ϕ, δ, ξ, and z2. However, it is sufficient for our analysis to set z2 = 1. This is the case because although z2 modifies the magnitude of the function D, it does not affect the D(0, 0)-level set, which determines possible motion paths. In other words, the movement of two mechanisms with the same design parameters, except the zi values (i = 1, 2, 3), are exactly the same.
3.1 Existence of a Rigid Motion.
z2 ≠ 0 because we chose it to be positive.
Since 0 < ξ < 2π, we have .
Since 0 < ϕ < π/2, we have sin ϕ ≠ 0 and tan ϕ ≠ 0.
- Showing thatrequires some work. We will prove by contradiction. Supposing that equality holds and applying the angle addition formula for cosine gives usRearranging terms gives usApplying the half angle identity for tangent, we are left withThis means that for some integer n, orBut we have chosen ξ and δ, so that 0 < ξ + δ < 2π. So this is a contradiction. Thus,
3.2 Solving for the Motion Path.
We illustrate graphs of the D(α1, α2) for several variations of the design parameters δ, ξ, and ψ in Fig. 4. Note that D(0, 0) is the functional value of D when the conical developable mechanism is in its conformed position. The curves indicated within the graphs are the D(0, 0)-level curves (i.e., the set of points for which D(α1, α2) = D(0, 0)) and are obtained by numerically solving the differential equation Eq. (13) for α2(t) and then plotting the collection of points (t, α2(t)).
As illustrated in Fig. 5, there are multiple possible paths that are connected to the origin. In Sec. 3.3, we verify that these parameter functions are sufficient to define a rigid motion. It is clear that the relationship between the parameter functions α1 and α2 is necessary. An animation of how panels move on these paths is given online.1
3.3 The Rigid Transformation.
In this section, we define the rigid motion that acts on the developable conical four-bar mechanism. The rigid motion is piecewise defined as follows:
T(t) defines a rigid motion.
To see that T(t) is well defined, first note that by construction, T1*(t) and T2*(t) are the identity on hinge lines H1 and H2, respectively. Thus, the mapping T(t) is well defined on the points of P0 intersecting P1 or P2. Next, we need to verify that T3*(t) is consistent with T1*(t) and T2*(t) on hinge lines H3 and H4, respectively. Note that:
- If x ∈ H3, then x = kaa. Thus, T*3(t)x = kaT*1(t)a = T*1(t)(kaa) = T*1(t)x. Hence,
- If x ∈ H2, then x = kbb. Thus, T*3(t)x = kbT*2(t)b = T*2(t)(kbb) = T*2(t)x. Hence,
Thus, we have the desired result. Therefore, T(t) is well defined.
Note that the aforementioned argument does not depend on being a circular cone, nor that a and b have the same z-coordinate. It is only required that T1*(t) and T2*(t) are orthogonal transforms and that the distance between T1*(t)a and T2*(t)b is constant throughout the motion. Thus, we can summarize these results by the following theorem.
SupposeCis a generalized cone inwith cone point at0and a conical developable mechanism is constructed onC, similarly as in Fig. 2, with hinge linesH1, H2, H3, andH4passing through the origin. Letaandbbe points distinct from the origin on the hinge linesH3andH4, respectively. If there are linear transformation pathsT1*(t) andT2*(t) acting on panelsP1andP2, respectively, so that the distance betweenT1*(t)aandT2*(t)bare constant astvaries, then the motion defined by (14) is a rigid motion.
4 Observational Analysis
These results can be helpful in analyzing the behaviors of conical developable mechanisms that are particularly relevant for the use of these mechanisms in future applications. Identifying the location of bifurcation points is important, so that the motion can be adequately known and controlled. Determining whether a mechanism’s motion is exclusively inside or outside of the reference cone is valuable for understanding which geometry is appropriate for use on solid surfaces (such as a rocket nose cone) to ensure that the mechanism’s motion does not penetrate the surface. These concepts are discussed in this section.
Considering Fig. 5, note that in all cases there is a class of upward slanting curves, which represent motion in which α1 and α2 are increasing at nearly the same rate. We will refer to these curves as the -curves. The other curves we will refer to as the -curves. Note that the origin is contained in a -curve in each case.
The points where two motion curves intersect are called bifurcation points, and correspond to the change points of the mechanism. A bifurcation point represents a point in the motion in which there is more than one possible continuation of the motion, other than reversing the motion. For planar lamina emergent mechanisms, there must be a bifurcation point corresponding to when the mechanism is in its conformed position . However, for this conical developable mechanism, the conformed position corresponds to the origin in the α1α2-plane (i.e., (α1, α2) = (0, 0)), which is not a bifurcation point as illustrated in each case in Fig. 5. Indeed, bifurcation points occur at positions where the hinge lines lie in a single plane (see Ref. ). When δ ≠ ξ, in the conformed position, no three of the axes are coplanar. When δ = ξ, in the conformed position, the axes for H2 and H3 coincide (consider Fig. 3 when δ = ξ), but the four axes together are not coplanar. Further note that when δ = ξ, if l is the axis containing H2 and H3 in the conformed position, then P2 and P3, moving together, can rotate freely about l while holding l fixed, and hence holding P1 fixed. Likewise, when the axes of H1 and H4 coincide, then P1 and P3, moving together, can rotate freely about the axis containing H1 and H4 while holding P2 fixed.
Consider again Fig. 5. When δ ≠ ξ, the bifurcation points arise only from the intersection of -curves with -curves. However, when δ = ξ, bifurcation points may also arise from the intersection of -curves. We will refer to a bifurcation point that is the intersection of an -curve with a -curve as an ordinary bifurcation point and a bifurcation point that is the intersection of two -curves as an extraordinary bifurcation point. In particular, an extraordinary bifurcation point is the intersection of a horizontal and a vertical line in the motion path. Along a vertical line in the motion path, P1 is fixed as P2 moves freely. Along a horizontal line in the motion path, P2 is fixed as P1 moves freely.
To understand the transition of the shapes of the -curves as δ changes size in comparison to ξ, i.e., the transitions from Figs. 5(a) through 5(c), note that at δ = ξ, the -curves have a stair-step pattern that, when pieced together differently, can be represented as a set of vertical and horizontal lines. Thus, the region of space near an extraordinary bifurcation point is divided into four quadrants. When δ decreases away from ξ, the -curves break into two continuous curves: one in the first quadrant and one in the third quadrant. Likewise, when δ increases away from ξ, the -curves break into two continuous curves: one in the second quadrant and one in the fourth quadrant.
4.1 Initial Motion.
The compact nature of the conical developable mechanism is achieved when the mechanism is in its conformed position. As such, it is important to consider the initial motion as the mechanism moves from the conformed position. Greenwood described three behaviors (intramobility, extramobility, and transmobility) that characterize the motion of developable mechanisms as they move from their conformed position. For regular cylindrical  and conical  developable mechanisms, these behaviors can be predicted using graphical methods. We also note that as a change point mechanism, there are two possible configurations, open and crossed, and that the conformed position represents a crossed configuration (see Ref. ).
We provide an analytical perspective for regular conical developable mechanisms. Note that if all panels start in the conformed position, the initial motion must be defined by a path that moves along a -curve. The initial direction of the -curve depends on the relative sizes of δ and ξ as follows:
If δ < ξ, we observe that when α1 is initially increasing, then α2 is initially decreasing (see Fig. 5(a)). Hence, if panel P1 is initially moving outward, then panel P2 must be initially moving inward, and vice versa. Greenwood et al.  refer to this type of behavior as transmobile (also see Ref. .)
If δ = ξ, recall that this is the case where H2 and H3 are colinear in the conformed position. Thus, panel P1 must initially be kept fixed, while panel P2 moves in either direction (see Fig. 5(b)).
If δ > ξ, we observe that both panels initially move in the same direction, but P1 moves at a slower rate than P2 (see Fig. 5(c)). This behavior is called intramobile if the motion is toward the interior of the surface and extramobile if the motion is toward the exterior of the surface .
4.2 Bifurcation Points.
The characteristics of the possible continued motions at a bifurcation point also depends on the relative sizes of δ and ξ. The existence of the bifurcation points lead to unbounded motion paths in the α1α2-plane.
If δ < ξ, at a bifurcation point, we observe that it is possible to move panels P1 and P2 in the same direction by continuing the motion along an -curve or in a different direction by continuing the motion along a -curve. Each -curve intersects each -curve at precisely one point and all bifurcation points connect to the origin. They form an array that is periodic in two directions.
If δ = ξ, we observe that at an ordinary bifurcation point there is a choice to keep one panel, P1 or P2, fixed while moving the other or to keep both panels in motion at nearly the same rate. At an extraordinary bifurcation point, only one panel can be put in motion while fixing the other, but either panel can be selected to be put in motion. In this case, all bifurcation points are connected to the origin. Both the set of ordinary bifurcation points and the set of extraordinary bifurcation points each form an array that is periodic in two direction.
If δ > ξ, we observe that both panels P1 and P2 must continue to move in the same direction. However, there are two possible rates at which this occurs. In this case, all bifurcation points are ordinary. There is a one-to-one correspondence between the -curves and -curves that intersect. The set of bifurcation points connected to the origin is periodic in one direction. There are an infinite number of parallel sets.
Figure 6 illustrates positions of the conical developable mechanism corresponding to various points in the motion path when δ < ξ. Recall that when δ < ξ, we observe transmobile behavior. In particular, starting from the conformed position, the panels P1 and P2 must move in opposite directions, inward and outward, relative to the surface of the cone. Only in the case that δ > ξ is there a rigid motion that allows both of the panels to move in the same direction initially.
Note that, by design, our conical developable mechanism is a spherical parallelogram linkage, where the link angles of opposite links are equal. For modifications of our design that are not parallelogram linkages, see Ref. . For these more generally designed mechanisms, the existence and types of bifurcation points in the motion will depend on whether they are Grashof mechanisms.
In this article, we have demonstrated that conical developable mechanisms, as designed here, have rigid motion. We have also demonstrated how to analytically determine the motion and have provided a general description of the motion. The relationship between variables δ and ξ determines the motion of the mechanism with respect to the conical reference surface and predicts the behaviors (intramobile, extramobile, and transmobile) the mechanism can exhibit. The relative sizes of δ and ξ also determine the variety of bifurcation points that arise in the motion path. We have described how these bifurcation points arise and the behavior of the mechanism around the various types of bifurcation points. Furthermore, we proved that a conical four-bar mechanism constructed on a generalized cone has rigid motion provided that a motion can be found that preserves the distance between any two distinct points, one on each of the hinge lines H3 and H4.
National Science Foundation (NSF Grant No. 1663345.)
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The authors attest that all data for this study are included in the paper.