Thick Rigidly Foldable Origami Mechanisms Based on Synchronized Offset Rolling Contact Elements

Origami-inspired, origami-adapted, and origami-applied mechanisms have been used to realize a wide variety of deployable forms, ranging from space-based solar arrays [1] to arterial stents [2] to backpacks [3]. While the applications are quite diverse, most folding-based engineering designs begin with a concept that is fundamentally a zero-thickness mathematical model and then transform it so that it can be realized with nonzero thickness materials.

An origami mechanism—often represented by folded paper and often prototyped in paper—consists of facets, usually flexible and of negligible thickness, connected by folds, where the intended motion occurs. When translated into an engineering application, the facets are replaced by panels, which are commonly rigid and whose thickness is non-negligible. The folds that connect facets are replaced by hinges, which replicate the folding motion of the underlying origami mechanism, typically pure revolution motion about an axis that is at the position of the fold.

When the panels are rigid, the underlying origami mechanism must be rigidly foldable, meaning that facets of the fold pattern remain planar throughout the range of motion; all motion takes place along the hinges. While many zero-thickness origami mechanisms are theoretically rigidly foldable, e.g., the Miura-ori [4] and Yoshimura pattern [5], the introduction of non-negligible thickness into the model commonly breaks the kinematic behavior associated with the zero-thickness mechanisms.

A fundamental problem with thick rigidly foldable mechanisms is that if one simply “thickens” the panels of a zero-thickness mechanism, one almost immediately runs into problems with mechanical interference between pairs of panels; indeed, for mechanisms that fold flat (highly desirable for deployable structures), in the flat folded state, all of the panels would be coplanar (and thus, may be mutually interfering) in the stowed (flat-folded) state.

This issue has led to significant study of thick rigidly foldable mechanisms in recent years, and, in particular, of ways of adapting zero-thickness mechanisms so that they use nonzero-thickness panels but still exhibit the desirable motion of the zero-thickness form. There are now several distinct techniques for creating origami mechanisms from thick materials.

The simplest and most straightforward technique to avoid mechanical interference is to offset the fold hinges to top and bottom surfaces of the panels (placing valley folds on top, mountain folds on the bottom). This works perfectly well for linear chains of panels, but once one introduces one or more interior vertices (vertices entirely surrounded by panels) into the fold pattern, most such vertices (with a few exceptions) become bistable, existing only in the flat and fully folded state, but with no unstrained motion between them. To achieve kinematic motions, several improved approaches have been developed.

Hoberman demonstrated a version of the Miura-ori based on laterally offsetting hinges in a way that preserves kinematic behavior across the range of motion [6]. However, this technique was only applicable to the symmetric bird's-foot vertex used in the Miura-ori fold pattern. More recently, Chen et al. showed a technique that generalizes the concept of vertically displacing hinges relative to the unfolded state and preserving kinematic behavior by careful choice of sector angles and panel thicknesses at each vertex [7]. She enumerated the possible vertex configurations for degree-4, −5, and −6 vertices. A challenge for this approach is that only a limited set of vertices support such mechanisms: for degree-4 vertices, for example, only flat-foldable vertices are allowed. Furthermore, in all of these examples, the relative hinge offsets (and thus relative panel thicknesses) are fully specified and dependent upon the specific sector angles; one cannot choose desired offsets arbitrarily.

Trautz and Künstler showed sliding hinges (SH) that did generalize to various folding mechanisms [8]. The ability to slide panels laterally relative to one another can introduce sufficient degrees-of-freedom to accommodate offsetting the hinges to top and bottom surfaces. A challenge with this approach is that it introduces multiple degrees-of-freedom, and the amount of required sliding accumulates in complex ways that can still lead to mechanical interferences.

Hoberman also described a technique in which several of the hinges are split into two parallel or near-parallel hinges [9], which allows an entirely planar unfolded state and introduces offsets that allow panels to stack side-by-side in the folded form. This technique was generalized by Ku and Demaine, who described how to modify an arbitrary fold pattern to accommodate arbitrary offsets via such hinge-doubling [10]. A challenge of this approach is that because the number of folds is increased at each vertex, in general, in an intermediate state (between fully unfolded and fully flatly folded), there are extra degrees-of-freedom in the mechanism motion, and even with the extra degrees-of-freedom, there is not yet a guarantee of a continuous path in phase space from the unfolded to fully folded state for all possible arrangements of vertices.

Tachi showed that for arbitrary rigidly foldable zero-thickness origami mechanisms, one can create a thick rigidly foldable mechanism that preserves the kinematics of the zero-thickness model by embedding the zero-thickness model within the thick panels and locating the axes of the hinges at the positions of the folds of the zero-thickness model [11]. For some mechanisms, some of the hinges can be located on top and bottom surfaces of the unfolded panels, while others must tilt diagonally from top to bottom, and usually, material must be trimmed away near the hinges. A challenge with this approach is that there can remain mechanical interferences to be dealt with, especially for fold angles close to $±180 deg$⁠.

Edmonson et al. showed an approach that allows rigidly foldable motion and parallel stacking of panels in which the panels can be arbitrarily independently offset from one another [12], so long as the hinges (though not the panels) remain located at their positions in the zero-thickness model. Challenges of this approach are that many configurations require through-holes in nonadjacent panels to allow hinges to penetrate to the zero-thickness surface; the offsets can impart high stresses on hinges; and in the unfolded state, the fixed offsets give rise to a nonplanar surface.

Zirbel et al. showed a technique for realizing thick origami mechanisms by introducing finite-width compliant membrane hinges (MH) that allowed for both flexing and incorporation of offsets between adjacent panels [13]. A challenge of this approach is that the required hinge width is difficult to predict a priori, and the extra compliance of the membrane hinges can give rise to unpredictable degrees-of-freedom.

In this work, we describe a new concept for thick rigidly foldable mechanisms: the synchronized-offset rolling-contact element (SORCE), along with a corresponding technique for designing thick rigidly foldable origami mechanisms that combines advantages of many of these techniques and avoids many of the disadvantages. Beginning from a rigidly foldable zero-thickness model, one can design a mechanism that:

• accommodates panels of arbitrary thickness

• begins with a planar unfolded state

• ends with side-by-side parallel stacked panels in the fully folded state

• preserves the kinematic motion of the zero-thickness model, including single-DOF motion.

We accomplish these goals by using rolling contacts, albeit ones designed to incorporate specific offsets between the panels in the fully folded state. The panel offsets vary as the mechanism transitions between the unfolded and folded states, and the relative rates of offset are synchronized between adjacent panels by the shapes of the contacting surfaces. Rolling contacts are, of course, well known in mechanical engineering—they are the basis of bearing joints. Most rolling contacts have involved circular or spherical cross sections, though more specialized configurations have been developed. We will show that it is possible to design rolling contact surfaces that give the precise offsets needed to realize thick rigidly foldable origami patterns. Furthermore, for certain configurations, it is possible to implement the rolling contacts as CORE (compliant rolling-contact element) joints [14], giving rise to complex thick-panel origami mechanisms that are monolithic and fully compliant, with all flexing remaining in the elastic domain. In this paper, we present the basic concept; we develop equations that allow one to design SORCE joints for arbitrary vertices and networks of panels and vertices; and we present two realized examples for individual vertices. We end by discussing avenues for future development.

Consider an origami mechanism based on a zero-thickness model where we wish to replace the individual zero-thickness facets with non-negligible thickness panels in such a way that:

• The motion of the panels approximates the motion of the facets in the zero-thickness model as it flexes from the unfolded state to the fully folded state (which, for purposes of discussion, we will take to be flat-folded).

• During the motion, the panels are shifted from their zero-thickness positions so as to avoid mechanical interferences throughout the range of motion.

The offset panel technique of Edmonson et al. [12] accomplishes these goals by enabling the designer to independently specify fixed offsets for each panel in the flat-folded state, thereby ensuring that arbitrary thickness panels may be accommodated. Each panel is offset perpendicularly relative to its zero-thickness position. However, those offsets are fixed throughout the range of motion; so that in the unfolded state, they are also present. Thus, the panels are offset relative to one another in the unfolded state and do not lie in a common plane, which is often a desirable goal.

Ideally, in the unfolded state, all of the panels would be precisely aligned with their zero-thickness corresponding facets, with zero offset. In the flat-folded state, we want a nonzero offset, as illustrated in Fig. 1, which shows a representative degree-4 vertex in unfolded and flat-folded configurations.

Let us parameterize the desired motion on a variable $t∈[0,1]$⁠, where t = 0 is the unfolded state, t = 1 is the fully folded state. Variable t is an arbitrary parameter, but throughout this analysis we will treat it as if it were a time variable describing the temporal evolution of the folding motion.

If we had total control over the panel motion, then we would ideally like to assign each ith panel an offset function $zi(t):[0,1]→ℝ$ that varies continuously over the range of motion, as illustrated in Fig. 2. The offset function describes the perpendicular offset of the panel from the position of its zero-thickness facet at time t.

At the flat state, t = 0, we would have $zi(0)=0$⁠, no offset, for each i. At the fully folded state, t = 1, we would have each $zi(1)$ equal to some nonzero value that allows the panels to stack without interference. Each $zi(t)$ would vary continuously (perhaps smoothly, perhaps monotonically) between those two values across the range of folding motion.

The fold angles, too, can be parameterized; for the jth fold of the zero-thickness model, we describe its fold angle by some function $γj(t):[0,1]→[0,±π]$⁠, which parameterizes the fold angle on t. For flat-foldable vertices, we will have $γj(1)=±π$⁠; for non-flat-foldable vertices, the final values will be some other, but definite, angle.

Consider now a plane that is perpendicular to one of the axes of the zero-thickness model and to the two panels adjacent to that axis, such as the cut-plane used in Fig. 3. Since the panel offsets are pure translation relative to the zero-thickness facets, any point in either panel that lies within the plane in the unfolded state must stay within that plane throughout the full range of motion. In addition, the fold angle between the two adjacent panels is the same as the fold angle between the corresponding two zero-thickness facets, i.e., $γj(t)$⁠. Thus, the complete desired motion of the two panels relative to the zero-thickness model, and more importantly, relative to each other, can be fully described by three parametric functions: the fold angle $γj(t)$ between the panels and the two offset functions $zi(t)$ of the panels on either side of the joint. For a general joint and perpendicular plane, we will denote the offset function on the left by $zl(t)$ and the one on the right by $zr(t)$⁠, as illustrated in Fig. 3.

Note that $zi(t)$ is a signed value that gives the offset relative to the zero-thickness facet in the facet's local coordinate system. In the flat-folded form, half of the facets are flipped upside down, and so their local coordinate systems have a local z-axis that points downward. Thus, for example, if we define the local coordinate system so that in Fig. 3(a), up is positive, then in Fig. 3(c), the local z-axis for the left panel points up, while the local z-axis for the right panel points down. Thus, in Fig. 3(c), $zl(1)$ and $zr(1)$ are both negative.

The offsets $zl(t)$ and $zr(t)$ are the desired perpendicular offsets of the panels from their corresponding zero-thickness facets. If we are given a particular rotational mechanism, though the values of $zl(t)$ and $zr(t)$ will be determined by the mechanism itself. The position of the right panel relative to the left is some combination of rotation (described by $γ(t)$⁠) and translation; the translation can be described by an appropriate choice of $zl(t)$ and $zr(t)$⁠.

The simplest mechanism to analyze is a pure revolute joint, as illustrated in Fig. 4. In this case, the offsets are both identically zero for all t; both panels remain centered on the zero-thickness facets throughout the range of motion.

Clearly, then, pure revolute joints would not work for a mechanism where we want the offsets to take on nonzero values over the range of motion. Fortunately, there are other forms of joint that can provide perpendicular panel offset along with rotation, and we now examine one of them.

In a previous work, Cannon et al. introduced the compliant rolling-contact element mechanism [14], or CORE mechanism, which is based on the older notion of no-slip rolling contacts in which the nonslip condition is typically enforced by gear teeth. A circular rolling-contact joint has two circular contacts that roll against one another without slipping, and in such joints, the instantaneous center of rotation of the two surfaces varies with the motion of the joint. The kinematics of circular rolling contact joints have been analyzed by, e.g., Collins [15]; more recently, Cai noted that circular rolling contacts are suited to certain classes of folded plate structure [16]. Here, we examine whether a circular rolling-contact joint could provide the combination of rotation and relative offsets that would enable a thick rigid panel origami mechanism.

Figure 5 shows the configuration of two circular rolling contacts in an intermediate state, along with some additional construction lines.

where r is the radius of curvature of the circular edge of each panel.

Instead of circular cross section, we could pick various other cross-sectional shapes, e.g., ellipses, hyperbolas, etc., and see what form of offset functions $zi(t)$ we wind up with, but even that is unlikely to prove fruitful. Equation (2) shows that the offset function depends directly upon the parameterized angle $γ(t)$⁠. Since each sector angle is bounded by two different folds that will, in general, have two different angle parameterizations, we would be lucky indeed for both joints to give rise to the same offset function for their common panel.

Instead, what we would like to do is choose the offset function $zi(t)$ for each panel, then solve for the joint configuration that gives rise to the desired offset functions at all times t; in effect, we wish to find contact surfaces that synchronize all of the offset functions ${zi(t)}$ to our desired values.

We now assume an arbitrary cross section for both contacts. We assume that the left contact surface is parameterized by a vector-valued function $sl(t):[0,1]→ℝ2$ and the right contact surface is parameterized by $sr(t):[0,1]→ℝ2$⁠, where $sl(t)$ and $sr(t)$ are the points that are brought into contact at fold parameter t, as illustrated in Fig. 6. We fix the left contact to ground and locate the center of rotation of the zero-thickness fold at the origin (0, 0). For full generality, we assume that the initial point of contact between the two surfaces is at position $(s0,0)$ for some constant s₀, i.e., not necessarily at the origin.

Now we can enumerate the constraints that define the desired surface functions $sl(t)$ and $sr(t)$⁠.

The combination of Eq. (14) and the rolling condition, Eq. (11), constitute a set of first-order differential equations that define the two surfaces $sl(t)$ and $sr(t)$⁠.

Thus, for any arbitrary parameterization of the fold angle $γ(t)$ and the two offset functions $zl(t)$ and $zr(t)$⁠, we can—at least in principle—solve for the surface functions $sl(t)$ and $sr(t)$ that, when used in a rolling contact, induce the adjacent panels to undergo the desired offsets synchronously as they rotate through the desired range of motion.

Let us now work out an example.

The two opposite fold angles of equal sign, γ₂ and γ₄, are the major fold angles; the two opposite fold angles of opposite sign, γ₁ and γ₃, are the minor fold angles.

Figure 7 shows the four surfaces along with schematic representations of the panels on either side at four different values of the fold parameter t. For clarity, we label each panel with its corresponding sector angle α_i and each fold with its corresponding fold angle γ_i (see Fig. 2).

If we join both surfaces (darker lines) to their corresponding panels to form monolithic elements, the resulting panels create rolling contacts that synchronize the offset shifts (relative to the zero-thickness model) with the fold angles. Each rolling contact is a single-degree-of-freedom (1DOF) joint, like a pure revolute joint; the kinematics of the rolling-contact vertex will therefore match the kinematics of the zero-thickness model, giving a 1DOF mechanism for this degree-4 vertex.

Several features of interest (or, perhaps, concern) are visible in the figure:

• In two of the joints (γ₁ and γ₃), both contacting surfaces are strictly convex toward each other. That means that these joints could be realized as CORE joints.

• In the other two joints, however (γ₂ and γ₄), one of the surfaces has both convex and concave regions. From a design point of view, this is a weakness, as it complicates realization of a CORE joint architecture.

• In three of the joints (γ₁, γ₂, and γ₄), the contacting surfaces extend above or below the panel surfaces. These extensions arise directly from the required offsets, as we will presently see. In the case of γ₂ and γ₄, this means that one of the panels will need to incorporate clearance holes to allow the penetration of the contact surface of its mating panel.

• While it is not immediately obvious from the figure, for γ₁ and γ₃, toward the end of the range of motion (⁠$t→1$⁠), rotation takes place about a point, rather than along rolling surfaces. While the contact is still nonsliding, over a range of t-values, the motion approaches that of a pure revolute joint, which would also be problematic to implement as a CORE joint.

So, while we have a solution for rolling contacts that gives a 1DOF mechanism with the desired offset behavior, it does present some challenges for practical implementation. However, we chose the fold angle functions ${γi(t)}$ and the offset functions ${zi(t)}$ for this example rather arbitrarily. We have considerable freedom in our choice of functions. In Sec. 6, we will explore the freedom and constraints to choose these functions and show how to achieve better-behaved surface functions that still give the desired kinematic behavior.

We have considerable latitude in choosing the offset functions ${zi(t)}$⁠; their values at the beginning and the end of motion are specified by panel planarity (at the beginning of motion) and parallel stacking (at the end of motion), but we can choose how they get from one state to the other. However, there are several considerations that will affect the functions we might choose.

For many origami mechanisms, the fold angles will vary monotonically from 0 deg in the unfolded state to some nonzero value— $±180 deg$ for a flat-foldable pattern. However, there are technologically useful patterns in which some folds do not flex monotonically. An example is the central diagonal crease in a split-diagonal mountain/valley/mountain/valley (MVMV) twist, which increases from $0 deg$ up to a maximum value, then declines again to $0 deg$ as the mechanism continues to the flat-folded state.

A nonmonotonic fold angle function is problematic: not fatal, but significant, because of the one-to-one mapping between the fold angle $γ(t)$ and the position of the point of contact between adjacent panels. If one of the fold angles is nonmonotonic, then offset functions ${zl(t),zr(t)}$ for adjacent panels will need to be chosen so that the surface function positions ${sl(t),sr(t)}$ are equal for values of t that give the same fold angle.

So we can adjust the lateral offset of the point of contact at flatness by choosing appropriately the slopes of the offset functions relative to that of the fold angle function.

This is a bit more interesting; it tells us that the lateral offsets of the ends of the surfaces are equal and opposite. (This also follows from the fact that in the flat-folded state, the two panels are $180 deg$ flipped and pure z-translated from one another.)

It also tells us that the end of each surface extends vertically by the total amount of z-offset that is programmed for its panel, or, equivalently, at flat-folding that the point of contact between the surfaces lies in the zero-thickness plane. Since this point starts and ends in the zero-thickness plane, one might wonder if it always remains in that plane. It does not; for most choices of offset function, each point of contact will move in and out of that plane and then back in across the range of motion. This behavior can be seen in Fig. 7.

As noted above, for at least some sets of offset functions, one (or both) of the surfaces can contain both convex and concave surfaces. If both surfaces are convex toward each other, then this permits a straightforward implementation of a CORE joint, and all of the associated robustness and maintenance of relative alignment that ensues. It is useful, then, to have a means of gauging when or whether double-convex surfaces arise, and how that can be ensured in the design.

The convexity functions $cl(t)$ and $cr(t)$ are the surface function curvatures with the signs chosen so that a positive value for either surface corresponds to a surface that is convex toward its mating surface, independently of whether the fold is mountain or valley.

We now plot in Fig. 8 the convexity functions for both surfaces for all four joints for the surface functions defined in Eq. (28).

Both of the pathologies noted above are visible in the convexity plots: the divergence of curvatures near t = 1 and the negative curvature of one surface (corresponding to a concave surface).

Now that we have expressions for the convexities, though, we can relate these functions back to our choice of offset functions: the convexities $cl,i$ and $cr,i$ come from the surface functions $sl,i$ and $sr,i$ via Eq. (33), which depend on the fold angles and offset functions. The offset functions are pinned by their endpoints, but we are free to search for sets of functions that connect the various endpoints in such a way that the convexities are positive, finite, and generally well-behaved.

By then using variational calculus, we could find an optimal set of offset functions ${zi(t)}$ that meet their endpoint values and minimize this figure of merit.

Given the complexity inherent in Eq. (34), the variational analysis is going to be nontrivial. However, we want our functions to be relatively smooth, to minimize the worst-case curvature. This allows a strategy for finding a near-optimal solution numerically.

where $Bj(t)$ is the jth expansion polynomial. Choosing a degree-n expansion gives a total of 2n coefficients ${bi,j,bo,j}$ that can be optimized to minimize the figure of merit.

which correspond, respectively, to a bump (or divot) in the function at the beginning, middle, and end of the motion.

In Figs. 9 and 10, we plot the fold angle functions, offset functions, convexity functions, and show the surfaces at t = 0 for the four joints. Now all four joints have finite positive convexity toward each other, thus permits a CORE joint implementation.

We fabricated two thick-panel origami vertices to demonstrate implementation of the SORCE joints. The rolling contact surfaces and panels were 3D printed (Makerbot Replicator 2) as a single part for each sector of a vertex using polylactic acid (PLA) filament. The thickness of the panels for these two vertices was chosen to be 1.016 cm (0.4 in.). For the first vertex constructed, the rolling contact surface geometries from Sec. 7 were printed. A photo of these 3D printed joints is shown in Fig. 11. Three canvas bands were used to join each pair of rolling surfaces. These bands were arranged such that no net twisting moment would be introduced by any tension in the bands.

The first origami vertex constructed was the flat-foldable degree-4 vertex described in Sec. 5 with sector angles $(120 deg,80 deg,60 deg, and 100 deg)$⁠. The rolling joint contact pairs were assembled with the flexible bands and attached to their corresponding panels with an adhesive. Care was taken to ensure that the lateral offset s₀ was properly aligned relative to the zero-thickness pattern for each joint pair; because s₀ is nonzero for γ₂ and γ₄ (as can be seen in Fig. 10), the line of contact between the two surfaces is not collinear with the zero-thickness fold line. Nevertheless, the motion is kinematic and 1DOF (subject to minimal extension in the connecting bands). The progression of the motion of the vertex from a closed, folded position to an open, unfolded position is shown in Fig. 12.

This prototype exhibited a smooth folding and unfolding motion while using the rolling joints to create the panel offsets required in the folding.

The key concept of the rolling-contact configuration described above is that restricting the displacements of the panels from their zero-thickness facets to be purely perpendicular to the facets ensures that relative motion between adjacent panels lies within the plane perpendicular to the fold between the corresponding facets. Relative planarity of adjacent facet motion means that we can “soak up” the varying spacing between adjacent panels as the panels rotate and translate relative to each other using rolling contacts that have translational symmetry in the direction of the fold.

Defining the motions by scalar offset functions ${zi(t)}$ ensures relative planarity, but does not take in all possible allowed relative motions of the panels. By broadening the description of relative panel motions, we can access a broader range of single-DOF origami mechanisms based on rolling contacts.

In particular, we can introduce time-varying gaps and other translational motions between the facets of the zero-thickness reference model, and, as long as the relative motion between every pair of panels lies within a plane perpendicular to their corresponding fold axis, we can absorb the additional relative motion into the surface functions and still achieve single-DOF kinematic motion.

Going forward, we will refer to the plane perpendicular to the zero-thickness fold as the normal plane of the fold.

We now assume two zero-thickness facets F_i and F_j that both undergo some planar rigid-body motion, i.e., a combination of translation and/or rotation that all take place within the normal plane of the fold. To encompass full generality, we now allow both F_i and F_j to undergo their own independent planar motions relative to an arbitrary frame of reference, rather than (as we did in Sec. 4) working within the frame of reference of one of the zero-thickness facets.

Once again, we assume rolling contact surface functions. Define $si,j(t)$ as the position at t = 0 of the point connected to facet F_i that touches its corresponding point on facet F_j at time t. Similarly, we define $sj,i(t)$ as the position at t = 0 of the point connected to facet F_j that touches its corresponding point on facet F_i at time t. These surface functions are illustrated in Fig. 13.

which we can do because $Rj(t)$⁠, being a 2D rotation, is nonsingular and therefore invertible.

Thus, $[Rj−1·Ri]′$ is invertible if and only if the derivative of its rotation angle (analogous to the relative fold angle $γ(t)$ of Sec. 4) is nonzero. If the derivative of the rotation angle vanishes, then the matrix $[Rj−1·Ri]′$ is not just singular; it is the zero matrix.

Equations (50) and (51) give the desired surface functions for planar-motion rolling contacts for arbitrary planar rigid-body motions for either facet.

That assumes that the derivative matrix $[Rj−1·Ri]′$ (and its counterpart, $[Ri−1·Rj]′$⁠) do not vanish. If they do for some value t = t₀, then Eq. (47) cannot be solved by inverting the matrix. One of the two conditions must apply:

• The right side of Eq.(47)is nonzero. Then there is no solution to Eq. (47). We say that a set of trajectories for which the left side is zero and the right side is nonzero is ill-behaved at time t₀.

• The right side of Eq.(47)is zero. Then Eq. (47) is satisfied for any function $si,j(t0)$ (since both sides of the equation vanish).

If both sides of the equation vanish at a particular time t₀, Eq. (50) is undefined, but it is often still possible to find solutions by taking limits as one approaches t₀. (It is not uncommon to encounter this situation at the endpoints of the motion, depending on the trajectory functions.) If both derivatives vanish but the limit of Eq. (50) (and counterpart) exists as $t→t0$⁠, we can take $si,j(t0)$ to be the limit value; if the limit does not exist, we also say the motion is ill-behaved. Any set of trajectories that is nowhere ill-behaved is well-behaved.

We summarize our result as follows:

Theorem 1. (Planar Rolling-Contact Theorem) Given any collection of rigid objects undergoing independent arbitrary time-varying Euclidean motion in a plane, between every pair of objects, if the motion is well-behaved, there exists a pair of translationally symmetric surfaces on each pair of objects that form a rolling contact between the objects, where the axis of translational symmetry is normal to the plane.

Said surfaces are given by the formulas in this section.

which now displays the symmetry between the two surface functions directly.

The theory in this section is quite general; we now apply it to a specific mechanism of practical interest.

There are various ways we could describe the additional relative motion supported by the planar rolling contact model. A particularly useful way is the following. For a given fold of a zero-thickness mechanism, we split the mechanism along the fold and introduce a general relative motion vector $d(t)$⁠, which is the vector between the two halves of the split fold lying in the plane perpendicular to the fold, as illustrated in Fig. 14 (compare with Fig. 6(c)).

So, in addition to the panels being perpendicularly offset from their zero-thickness facets, we have introduced the additional offset $d(t)$ between the zero-thickness facets themselves.

Note that both $d$ and $d′$ appear in $sl$⁠, but only $d′$ appears in $sr$⁠.

We can apply this concept to any vertex or collection of vertices that can be split in such a way that over the full range of folding motion, the offset $d(t)$ stays perpendicular to every split fold. An example is the symmetric bird's-foot vertex shown in Fig. 15.

We split the vertex along folds γ₁ and γ₃, choosing the 3D offset vector $d(t)$ to be perpendicular to the plane containing both γ₁ and γ₃.

We define the scalar function $d(t)≡||d(t)||$⁠. Although the direction of $d(t)$ is prescribed by the symmetry of the pattern, we have considerable freedom in choosing d(t) in order to accommodate panel thicknesses.

The utility of splitting a vertex (or entire crease pattern) is that it gives us an additional degree in freedom in choosing the zero-thickness reference planes and surface functions. For example, we can choose the zero-thickness reference plane for γ₂ so that it lies between the two adjacent panels, and similarly for γ₄, then choose the scalar offset function d(t) so that panels α₂ and α₃ stack one atop the other in the fully folded form.

The elevation functions $zi(t)$ and offset function d(t) must be 0 at t = 0 (so that the panels are coplanar when unfolded) and their values at t = 1 are chosen so that they stack neatly in the folded form. In between, we have complete freedom in choosing their functional form; for this example, we have chosen to make them linear with t. Figure 16 shows the surface functions and panel positions for four different values of t across the range of motion.

This choice of parameters gives a set of surface function pairs that gives the desired motion—all of the layers stack neatly in the fully folded state and are kinematic single-DOF in between, but there is a problem. The surfaces for fold γ₃ have cusps in them and they overlap each other (at least in planar projection), which would make fabrication problematic.

However, we can choose different elevation and/or offset functions to give better behaved surfaces. We have considerable freedom in the choice of the ${zi(t)}$ and d(t); they need to hit certain values at their endpoints, but we can choose different functional forms to connect the endpoints, and different functions will give rise to different rolling-contact surfaces.

This gives the surface functions and motion illustrated in Fig. 17. Now all of the surfaces are smooth and convex toward one another.

But we can do even better than this. Instead of just picking various elevation and offset functions and seeing what the results are, we can actually choose specific rolling-contact configurations and then solve for the elevation and/or offset functions that give rise to them, as we now show.

Arguably, the most desirable configuration for a rolling-contact joint would be one with two matching circular surfaces, because the constant curvature results in constant stresses in a circular-CORE implementation and can be in static equilibrium at any position. Since, in the bird's-foot vertex, fold angles γ₂ and γ₄ are equal from symmetry, it would be desirable to make both of these joints have circular cross sections.

where r is half the thickness of the panel and the radius of the circular cross section.

Next, we would like to make γ₃a circular-CORE joint as well. The problem is that we can no longer choose its surrounding elevation functions $z2(t)$ and $z3(t)$⁠, since those are already fixed; instead, we can, at best, choose the panel offset function d(t) to give the relative motion of the two panels that induces circular-cross-section surfaces between them.