Capital letters represent non-dimensional (n.d.) values. The first subscript indicates the logic element: *a* for AND gate and *b* for bistable element. A second subscript indicates *a* for axial, *c* for coupling, and *t* for transverse. Variables are standard where L is length, K is stiffness, X and Y are displacements, F is force, and E is energy.

## Bistable element design

A bistable element and coupling linkage form a repeating unit, comprised of kinematics and energetics layers. (Fig. S1a) The kinematics layer determines the motion of the bearing, while the energetics layer fine tunes the stiffness values. The kinematics layers, which constrain motion, are the same for each bistable element in a chain ensuring equivalent kinematic performance. Coupling linkages, designed with 6 DOF compliance, connect bistable elements without over-constraint. Design details are elaborated in depth in the supplemental information section ‘Layered design of bistable elements’.

## Bistable element general model

The general model of a bistable element is parametrically defined by: lever arm characteristic length *l**b*, axial stiffness *k**ba*, y-axis displacement *y**b*, transverse stiffness *k**bt*, x-axis displacement *x**b*, and coupling stiffness *k**bc* (Fig. S1b). Degeneracy is required in the bistable states to ensure equivalent response to signal propagation of either value; symmetry is the simplest approach to ensuring degeneracy. Symmetry requires that the unstressed structure hold the orientation of the unstable equilibrium and be energized via preload *y**b* after fabrication. These conditions hold regardless of the means of compliance, structure scale, or bistable displacement orientation relative to the chain orientation since the coupling stiffness simply represents the compliance between relative node locations independent of the actual axis these motions. Thus, the general model represents a wide range of designs if equivalents for *l**b*, *k**ba*, *y**b*, *k**bt*, and *k**bc* are identified.

General model parameters are nondimensionalized (n.d.) to capture scale independent behavior, Eq. (1), where *E* is the n.d. energy, *F* is the n.d. force, *K* is the n.d. stiffness, and *X**b*, *Y**b*, *R**b*, are the n.d. displacements or lengths. Normalization parameters are generated through combinations of the axial stiffness and lever arm.

$$E= \frac{e}{\frac{1}{2}{k}_{ba}{l}_{b}^{2}}, F= \frac{f}{{k}_{ba}{l}_{b}}, K= \frac{k}{{k}_{ba}}, {X}_{b},{Y}_{b},{R}_{b}=\frac{{x}_{b,}{y}_{b},{r}_{b}}{{l}_{b}}$$

1

The bistable element potential energy, *E**b*, is a function of displacements *X**b*, *Y**b*, *R**b*, Eq. (2).

$${E}_{b}= {K}_{bt}{X}_{b}^{2}+{\left[{Y}_{b}-(2-4{R}_{b})(1-\sqrt{1-{X}_{b}^{2}})\right]}^{2}$$

2

The energy derivative is the force generated by the bistable element bearing on the node, Eq. (3).

$${F}_{bx}=\left[4{(1-2{R}_{b})}^{2}\left(1-\frac{1-\frac{{Y}_{b}}{2-4{R}_{b}}}{\sqrt{1-{X}_{b}^{2}}}\right)-{K}_{bt}\right]{X}_{b}$$

3

The force derivate is the bistable element bearing stiffness on the node, reduced to Eq. (4).

$${K}_{bx}=\left[4{(1-2{R}_{b})}^{2}\left(1-\frac{1-\frac{{Y}_{b}}{2-4{R}_{b}}}{{1-{X}_{b}^{2}}^{3/2}}\right)-{K}_{bt}\right]$$

4

A range of important parameters are defined from these expressions, including Eq. (5) the n.d. scale of the potential energy change from bistable equilibrium to unstable equilibrium, *Δ**EbMax*, Eq. (6) the n.d. minimum potential energy at bistable equilibrium, *E**bMin*, Eq. (7) the n.d. relative bistable element potential energy, *ΔE**b*, Eq. (8) the n.d. maximum restoring force exerted on the node by the bistable bearing, *F**bxMax*, Eq. (9) the n.d. nodal displacement associated with generating the maximum restoring force, *X**bFMax*, Eq. (10) the n.d. characteristic stiffness of the bistable profile, *K**bFmax*, and Eq. (11) the n.d. bistable equilibrium distance, *X**be*.

\(\varDelta {E}_{bMax}= \frac{{\left(2{Y}_{b}-\frac{{K}_{bt}}{1-2{R}_{b}}\right)}^{2}}{\left(4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}\right)}\) | (5) |

\({E}_{bMin}= {Y}_{b}^{2}-\varDelta {E}_{bMax}\) | (6) |

\(\varDelta {E}_{b}= {E}_{b}-{E}_{bMin}\) | (7) |

\({F}_{bxMax}={\left(1-2{R}_{b}\right)}^{2}{\left[{\left(4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}\right)}^{2/3}- {\left(4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}\right)}^{2/3}\right]}^{3/2}\) | (8) |

\({X}_{bFMax}=\sqrt{1-{\left(\frac{4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}}{4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}}\right)}^{2/3}}\) | (9) |

\({K}_{bFMax}=\frac{{F}_{bxMax}}{{X}_{bFMax}}\) | (10) |

\({X}_{be}=\sqrt{1-{\left(\frac{4-\frac{2{Y}_{b}}{\left(1-2{R}_{b}\right)}}{4-\frac{{K}_{bt}}{{\left(1-2{R}_{b}\right)}^{2}}}\right)}^{2}}\) | (11) |

### Propagation Regimes

The general model shows signal propagation only occurs in certain compliance regimes. In these regimes, a soliton can propagate along a chain of bistable elements (Fig. 1b). We define a decision hierarchy to simplify propagation regime discovery. First, choose the element energy and size scale via *k**ba* and *l**b* selection, which then determine explicit values for all following terms. Second, choose *Y**b* the scale of bistable energy storage, which determines the amount of bistable element axial compression, as well as the approximate angle of the lever arm at equilibrium, *θ**b*, via Eq. (12). Generally, *Y**b* ≈ 0.2 is an upper bound as this corresponds to ≈ 26° rotation, which is large scale motion for compliant bearings.

$${\theta }_{b}\approx {\text{cos}}^{-1}(1-Y/2)$$

12

Third, we chose the compliant bearing approach and set the effective cross-pivot rolling radius *R**b* to describe the scale of rolling, which occurs in the bearing during large range rotation (Supplemental Information section ‘Cross-pivot rolling radius’). Fourth, the transverse stiffness modulates the depth of the bistable potential energy well. Ideally, *K**bt* should approach 1 to keep all stiffnesses at the same scale and simplify fabrication, but bistability disappears when *K**bt* > *K**btMax*, defined in Eq. (13). A practical value of *K**bt* = *K**btMax*/2 balances the need for high bistable element stiffness to allow for manufacturing feasibility without suppressing bistability.

$${K}_{btMax}= 2Y\left(1-2{R}_{b}\right)$$

13

Fifth, the coupling stiffness is used to engineer the pulse properties, including the propagation speed, pulse size, and pulse energy. The pulse properties are governed by *r**kPulse* which describes the balance between the nondimensionalized coupling linkage stiffness, *K**bc*, with the nondimensionalized characteristic stiffness of the bistable element profile, *K**bFMax*. The coupling stiffness captures the ability of nodes to affect neighbors while the characteristic stiffness captures the ability of nodes to resist neighbors. *r**kPulse* also captures a lower limit to pulse propagation which occurs when the coupling stiffness is too low to pull the rising node over the peak restoring force. A simple force balance is created with the edge nodes approximated at bistable equilibria, even though the finite compliance of the chain always results in the node being closer than this. The rising node *n* is then controlled by the balance of the three dominant forces, i) the coupling stiffness dragging it up from the prior *n*-1 node at the positive equilibrium, ii) the bistable restoring force pulling it back down towards the initial negative equilibrium, and iii) the coupling stiffness of the following *n* + 1 node at the negative equilibrium, shown in Eq. (14).

$$\stackrel{n-1}{\underset{\left({x}_{be}+ {x}_{bFMax}\right){k}_{bc}}{⏞} }- \stackrel{n+1}{\underset{\left({x}_{be}- {x}_{bFMax}\right){k}_{bc}}{⏞} }- \stackrel{n}{\underset{{F}_{bxMax}}{⏞} }\ge 0$$

14

The simplified expression compares the coupling stiffness and characteristic profile stiffness in Eq. (15), producing a simple constraint *r**kPulse* > ½ to propagate the pulse.

$$2{k}_{bc}-{k}_{bFMax}\ge 0$$

15

## Pulse propagation

The minimum energy step is used to engineer the pulse chain for pulse propagation. The node ‘energy’ is calculated as the potential energy of a pulse in the unstable equilibrium centered on the node, assuming no energy decrement across the pulse. This allows the measurement to be a spot analysis of the pulse characteristics at the node. The pulse chain energy profile is engineered by calculating the energy and minimum energy step of the *n*’th nodes. The *n* + 1 nodal energy decrements from the *n* node by the minimum energy step, scaled by a factor around 1.5x to ensure fabrication variation does not halt pulse operation. The energy of each node is tuned by adjusting *k**a* for the bistable element, while holding the nondimensional parameters constant. This process tunes the energy for each node, resulting in bistable elements with the same kinematics but successively less potential energy. The process is repeated with a calculation of the minimum energy step for the *n* + 1 node being used to determine the energy of the *n* + 2 node and so on.

## Pulse propagation – steady state velocity

A pulse propagates along a bistable element chain at an approximately steady state velocity, determined by the energy flow from the pulse potential energy per nodal step into energy dissipation *e**d*, where the unknown is the time increment for the pulse to advance one node, *dt*. The velocity of the *i*’th node in the profile [*d**xp*]i/*dt*, and the distance each node travels [*d**xp*]i, are used to calculate the dissipated energy, Eq. (16).

$${e}_{d}={b}_{b}{\sum }_{i}\left(\frac{{\left[d{x}_{p}\right]}_{i}}{dt}{\left[d{x}_{p}\right]}_{i}\right)={e}_{step}$$

16

The known energy decrement between nodes, *e**step*, leads to the time increment between nodes, Eq. (17).

$$dt=\frac{{b}_{b}}{{e}_{step}}{\sum }_{i}{\left({\left[d{x}_{p}\right]}_{i}\right)}^{2}$$

17

## AND gate design

Figure S3a shows the detailed structure of the AND gate, composed of several linkages, which freely rotate relative to one another. The *l**a1*, *l**a2* linkage from the input ports to the first rotary joint is structured in a V to allow for the two input ports to expand away from one another along the Y-axis. This ensures the structure does not lock when it rotates, since the rotation forces *l**a1* to increase slightly when the input ports are guided by linear shuttles. Fig. S3b defines and labels the geometry. The gate inputs and output are connected to bistable elements, which properly constrain the AND gate shuttle motion. The bistable elements are connected on separate levels, however right-angle linkages could maintain an alternative planar design. Rigid linkages provide high stiffness to reject unwanted state changes via back-driving. Compressive compliance at the output interface retains the basic structure of bistable element nodes separated by compliant couplings to enable propagation. The AND gate kinematics are initially tuned based on the assumption that the two inputs reach the full equilibrium location, however these displacements are attenuated by finite compliance of the structure and coupling linkages. The central AND gate lever arms are modified beyond the static operation constraint to compensate for this effect and ensure full output propagation despite incomplete input state transitions.

A five-dimensional design space (three geometric and two stiffness parameters) represents the AND gate, where only a small regime enables pulse propagation. Kinematic analysis ensures the rotational nonlinearity produces AND operator performance in ideal conditions. The basic AND gate dimensional parameters are *k**ad*, *k**ac*, *l**a1*, *l**a2* and *l**a3*. An additional parameter captures the energy transfer behavior of the AND gate, *r**aE*, which describes the ratio of the pulse energy out of the gate, *e**aOut*, compared to the theoretical maximum energy output, *e**aOutMax*. Eq. (18) shows the nondimensionalized (n.d.) terms. The n.d. differential motion stiffness, *K**ad*, is normalized to the compressive stiffness, simplifying elasto-mechanic calculations. The n.d. compressive stiffness, *K**ac*, is normalized to the coupling linkage stiffness of the output port, *k**acp3*, as this output port stiffness determines scales the requirements on the compressive stiffness. All displacements in the AND gate, *X**a* = [*x**ap1*, *x**ap2*, *x**ap3*, *x**ap3o*, *x**ad*, *x**ac*, *x**as*] and linkage lengths *L**a* = [*l**a1*, *l**a2*, *l**a3*] are normalized by the equilibrium displacement of the bistable elements, *x**be*.

$${K}_{ad}=\frac{{k}_{ad}}{{k}_{ac}}, {K}_{ac}=\frac{{k}_{ac}}{{k}_{acp3}}, {X}_{a}, {L}_{a}=\frac{{x}_{a}, {l}_{a}}{{x}_{be}}, {r}_{aE}=\frac{{e}_{aOut}}{{e}_{aOutMax}}$$

18

The lever arm lengths, *l**a1*, *l**a2*, and *l**a3*, are set by the requirements to ensure AND gate nonlinear behavior as well as the design maximum values set for the two linkage angles, *θ**aaMax*, *θ**abMax*. The stiffness and energy terms, *K**ad*, *K**ac*, and *r**aE*, are set by the requirements of pulse propagation. The AND gate nonlinear response to the two inputs (*p1* and *p2*) is captured in the output port (*p3*) uncompressed displacement, *x**ap3o*.

The input motion *X**ap1*, *X**ap2*, is redefined as differential, *X**ad*, and shared, *X**as*, terms, shown in Eq. (19) and Eq. (20).

$${X}_{as}=\frac{{X}_{ap1}+{X}_{ap2}}{2}$$

19

$${X}_{ad}={X}_{ap2}-{X}_{ap1}$$

20

Eq. (21) determines the compression displacement *X**ac* by comparing the output motion, *X**ap3* to the uncompressed displacement, *X**ap3o*.

$${X}_{ac}={X}_{ap3}-{X}_{ap3o}$$

21

Eq. (22) demonstrates the uncompressed displacement, calculated geometrically, and determined by the differential and shared motion.

$${X}_{ap3o}={X}_{as}+{L}_{a2}\left(\sqrt{1-{\left(\frac{{X}_{ad}}{{L}_{a1}}\right)}^{2}}-1\right)+{L}_{a3}\left(\sqrt{1-{\left(\frac{{L}_{a2}{X}_{ad}}{{L}_{a1}{L}_{a3}}\right)}^{2}-1}\right)$$

22

The AND gate nonlinear behavior constrains the output port to the same location, regardless of the input port states such that the 000 and 010 states are equivalent. Three constraints are supplied to set the three AND gate linkage sizes when the nonlinear behavior constraint is combined with the design maximum values set for the two linkage angles, *θ**aaMax*, *θ**abMax*. The requirement on the horizontal linkage *L**a1* is simply constrained by the *θ**aa* rotation angle, shown in Eq. (23).

$${L}_{a1}= \frac{2}{\text{s}\text{i}\text{n}{\theta }_{aaMax}}$$

23

The requirements on *L**a2* Eq. (24) and *L**a3* Eq. (25) capture the nonlinear complexity.

$${L}_{a3}=\frac{\frac{1}{\sqrt{\frac{1}{{L}_{a1}^{2}\left(\sqrt{\frac{1}{4}-{L}_{a1}^{-2}}-\frac{1}{2}\right)+1}+{L}_{a1}^{2}-2}}+\frac{{\left[\text{cos}{\theta }_{abMax}\right]}^{-1}}{\frac{1}{{L}_{a1}\left(\sqrt{\frac{1}{4}-{L}_{a1}^{-2}}-\frac{1}{2}\right) }+{L}_{a1}}+\text{tan}{\theta }_{abMax}}{\text{tan}{\theta }_{abMax}[2+{L}_{a1}sin{ \theta }_{abMax}]}$$

24

$${L}_{a2}=\frac{1}{2} \sqrt{\frac{1-2{L}_{a3}}{{L}_{a1}^{2} \left( \sqrt{\frac{1}{4}-{L}_{a1}^{-2}-\frac{1}{2}}\right)+1}+{L}_{a3}^{2}}+\frac{1}{2}-\frac{{L}_{a3}}{2}$$

25

The next design steps define gate elastomechanics based on the two internal stiffnesses. The force of the AND gate on each port’s node, *f**a* = [*f**ap1*, *f**ap2*, *f**ap3*] is defined in each case as positive in the + x-axis direction and nondimensionalized to *F**a* = [*F**ap1*, *F**ap2*, *F**ap3*] by the characteristic gate force, *k**ac* *x**be*, as shown in Eq. (26).

$${F}_{a}=\frac{{f}_{a}}{{k}_{\alpha }{x}_{be}}$$

26

Differential stiffness between the input ports generates equal and opposite forces on the input gates, while compression stiffness generates a return force, modified by gate kinematics, captured by the term *α**aF* as shown in Eq. (27).

$${\alpha }_{aF}=\frac{{L}_{a2}{X}_{ad}}{{L}_{a1}^{2}}\left[{\left[{\left(\frac{{L}_{a3}}{{L}_{a2}}\right)}^{2}-{\left(\frac{{X}_{ad}}{{L}_{a1}}\right)}^{2}\right]}^{-\frac{1}{2}}+{\left[1-{\left(\frac{{X}_{ad}}{{L}_{a1}}\right)}^{2}\right]}^{-\frac{1}{2}}\right]$$

27

Eq. (28) is the nodal force at the first input port, *F**ap1*.

$${F}_{ap1}={K}_{ad}{X}_{ad}+\left(\frac{1}{2}+ {\alpha }_{aF}\right){X}_{\alpha }$$

28

Eq. (29) is the nodal force at the second input port, *F**ap2*.

$${F}_{ap2}= -{K}_{ad}{X}_{ad}+\left(\frac{1}{2}-{\alpha }_{aF}\right){X}_{ac}$$

29

Eq. (30) is the nodal force at the output port, *F**ap3*.

$${F}_{ap3}= -{X}_{ac}$$

30

Propagation through the AND gate requires the definition of several stiffness and energetics terms. The maximum possible differential stiffness, *k**adMax*, is defined by the total energy flow of the pulse into charging the differential mode accounting for the energy lost to dissipation during the traversal of the pulse between nodes, *e**aDis*, and the minimum value of the pulse energy at the inputs, *e**aIn*, as shown in Eq. (31).

$${k}_{adMax}=\frac{{e}_{aIn}-{e}_{aDis}}{2{x}_{be}^{2}}$$

31

The differential stiffness ratio captures the extent to which the pulse energy is routed to the differential mode as shown in Eq. (32). Figure 2f shows that high efficiency AND gates exist in the *r**kad* = 0.15 regime, where only 15% of the pulse energy is drained to the differential mode.

$${r}_{kad}=\frac{{k}_{ad}}{{k}_{adMax}}$$

32

The energy ratio, *r**aE*, is the ratio of the output energy over the available energy after accounting for losses, and it determines the pulse energy at the gate output, which is generally set around 0.8 to ensure propagation despite fabrication variation, and the theoretical maximum energy output of the gate, *e**aOutMax*, calculated as shown in Eq. (33). The theoretical maximum energy output considers energy dissipation and storage in the differential mode. This theoretical value provides an energetics-based system constraint, but as shown in Fig. 2f, it is often not the limiting factor.

$${e}_{aOut}={r}_{aE}\stackrel{{e}_{aOutMax}}{\underset{({e}_{aIn}-{e}_{aDis}-2{k}_{ad}{x}_{be}^{2} }{⏞} }$$

33

The overall gate efficiency *η**a* is the ratio of the gate energy output over input, and thus will always be a lower than *r**aE*. Kinematic modifications are necessary to ensure propagation, as the preceding analysis assumes all ports are at the bistable equilibrium locations. Tuning a parameter to overcompensate the gate output ensures the incomplete input transitions still produce sufficient output displacement to drive pulse propagation. In this work, *l**a1* is the preferred term for modification and is slightly reduced to amplify the gate response to input motions. All nodes in the profile are then displaced beyond the threshold *x**trans*/*x**be* ≈ 0.98 to add less energy to the profile than lost by advancing by a node. Propagation switches from energetically unfavorable to favorable once the output port passes through this threshold. The AND gate static force balance captures the unstable equilibrium when the output port reaches the transition displacement, accounting for the input ports being connected to the tail end of a pulse profile rather than locked at equilibrium displacements. Eq. (34) shows where functions *f**bxp1*(…), *f**bxp2*(…) are the nodal bistable element forces at the input 1 and 2 locations; *f**ap1*(…), *f**ap2*(…) and *f**ap3*(…) are functions calculating the nodal AND gate forces at the input 1, 2 and output; *k**cp1* and *k**cp1* are the coupling stiffnesses of the linkages to the input 1 and 2 ports; The coupling stiffness scaling terms account for the extended chain of nodes connected to the port node. We assume the output port node experiences no forces from the chain, since forces on the output port switch direction, passing through 0 during transition. The modified *l**a1* term is extracted from the system of equations, producing the value used in the fabricated AND gate to ensure pulse propagation.

\(given {x}_{p3}={x}_{trans}\) \({k}_{cp1}\left(-{x}_{be}-{x}_{p1}\right)(1-\frac{{k}_{cp1}}{2{k}_{cp1}+{k}_{bxp1}{ x}_{be}}+{f}_{bxp1} {x}_{p1}+{f}_{ap1}\left({x}_{p1}, {x}_{p2}, {x}_{p3}, {l}_{a1trans}\right)=0\) \({k}_{cp2}\left({x}_{be}-{x}_{p2}\right)(1-\frac{{k}_{cp2}}{2{k}_{cp2}+{k}_{bxp2}{ x}_{be}}+{f}_{bxp2} {x}_{p2}+{f}_{ap2}\left({x}_{p1}, {x}_{p2}, {x}_{p3}, {l}_{a1trans}\right)=0\) \({f}_{ap3 }\left({x}_{p1}, {x}_{p2}, {x}_{p3}, {l}_{a1trans}\right)=0\) \(find {x}_{p1}, {x}_{p2}, {l}_{a1trans}\to {l}_{a1trans}\) (34)

## Circuit synthesis

The bistable element chain Fig. 3a and AND gate chain Fig. 4a used the following parameters: *Y**b* = 0.1, *K**bt* = 0.065, *K**bc* = 0.06, initial *k**ba* = 37643 N/m, *l**b* = 35mm, *r**EstepAct* = 1.5, which produced a 1.96% energy drop per node. The AND gate was designed with maximum angular deviation for *θ**a*,*θ**b* = 10°, *r**kad* = 0.15 which produced *k**ad* = 154N/m, *k**ac* = 1876N/m, *l**a1* = 108.6mm, *l**a2* = 322.2mm, *l**a3* = 322.2mm.

## Fabrication

Our sequential printing method enables additive manufacturing of large and long overhanging features without support materials/structures using two-photon lithography (TPL). To print the model shown in Fig. S2, the overhanging geometries are divided into shorter substructures to reduce unwanted layer movement while printing and to increase the precision and quality of fabricated structures. The sub-structures are printed in an order ensuring the last printed feature is connected to a previously grounded/stable structure. We limited the maximum layer height of substructures to 12µm to avoid obscuring the TPL laser beam in the subsequent printing of features located on the lower layers.