Designing continuous equilibrium structures that counteract gravity in any orientation

This paper presents a framework that can transform reconfigurable structures into systems with continuous equilibrium. The method involves adding optimized springs that counteract gravity to achieve a system with a nearly flat potential energy curve. The resulting structures can move or reconfigure effortlessly through their kinematic paths and remain stable in all configurations. Remarkably, our framework can design systems that maintain continuous equilibrium during reorientation, so that a system maintains a nearly flat potential energy curve even when it is rotated with respect to a global reference frame. This ability to reorient while maintaining continuous equilibrium greatly enhances the versatility of deployable and reconfigurable structures by ensuring they remain efficient and stable for use in different scenarios. We apply our framework to several planar four-bar linkages and explore how spring placement, spring types, and system kinematics affect the optimized potential energy curves. Next, we show the generality of our method with more complex linkage systems that carry external masses and with a three-dimensional origami-inspired deployable structure. Finally, we adopt a traditional structural engineering approach to give insight on practical issues related to the stiffness, reduced actuation forces, and locking of continuous equilibrium systems. Physical prototypes support the computational results and demonstrate the effectiveness of our method. The framework introduced in this work enables the stable, and efficient actuation of reconfigurable structures under gravity, regardless of their global orientation. These principles have the potential to revolutionize the design of robotic limbs, retractable roofs, furniture, consumer products, vehicle systems, and more.

Watt's Linkage at ψ = 0 • Internal torsional springs were placed at up to four locations, labelled A, B, C, and D in Figure 1. Table S1 contains the optimized rest angles α j , stiffnesses k j , and Σ|∆PE T | values for all internal torsional spring location combinations for the Watt's Linkage at ψ = 0 • . For the combinations of BCD and ABCD, the results are nearly equivalent. The variation between the two cases is likely due to the arbitrary initial optimization point, convergence threshold, and the bounds placed on the design parameters. With small changes to the optimization parameters, it could be shown that these two combinations lead to equivalent results. Changes to the optimization parameters would slightly influence the final variables and Σ|∆PE T |, but the overall trends and conclusions presented in the results would be preserved.
Watt's Linkage at orientations ψ = 0 • to 90 • When considering orientations ψ = 0 • to 90 • , we explored three cases: four internal torsional springs, one external torsional spring, and both four internal and one external torsional spring. Spring parameters were found by minimizing the mean(Σ|∆PE T |). For the case with no springs, the mean(Σ|∆PE G |) = 1.382 N-m. Table S2 contains the spring properties and mean(Σ|∆PE T |) values for all three cases. The results can be improved if we optimize for a smaller range of ψ. Table S3 gives the spring properties and mean(Σ|∆PE T |) values for the Watt's linkage, considering ranges 0 • to 30 • , 60 • , and 90 • .
Scissor Mechanism at ψ = 0 • The first case we considered for the Scissor Mechanism at ψ = 0 • is with up to four internal torsional springs at locations A, B, C, and D (Figure 3(A)). Table S4 contains the optimized rest angles α j , stiffnesses k j , and Σ|∆PE T | values for all internal torsional location combinations for the Scissor Mechanism at ψ = 0 • . Figure S1 illustrates the potential energy contributions of components for each location combination.
In addition to four internal torsional springs, we considered adding two internal extensional springs and one external extensional spring to the Scissor Mechanism (Figure 3(B-C)). Table S5 contains the spring parameters and Σ|∆PE T | values for each of the cases. The ex-    tensional springs are labelled as spring 1, which spans one unit of the Scissor Mechanism (stiffness k 1 , rest length L 1 ), spring 2, which spans the entire structure (stiffness k 2 , rest length L 2 ), and external spring 3, which is connected to an external anchor point (stiffness k 3 , rest length L 3 ).
Scissor Mechanism at orientations ψ = 0 • to 90 • For the Scissor Mechanism at orientations ψ = 0 • to 90 • , we explored seven cases including internal, external, torsional, and extensional springs. Table S6 contains the spring properties and mean(Σ|∆PE T |) for all cases.
The shape of the gravity curve changes significantly as the Scissor Mechanism is reoriented ( Figure S2). As a result, the internal springs are less effective than the external springs at minimizing the mean(Σ|∆PE T |), because their potential energy contributions do not change with ψ. Furthermore, as shown in Figure 3 and Table S2, the optimal result for the mean(Σ|∆PE T |) can be substantially improved if all springs are used together.  Figure S1: Potential energy breakdowns for all possible location combinations of internal torsional springs on the Scissor Mechanism, optimized for ψ = 0 • . Due to symmetry in the system geometry, the total potential energy curve is nearly equivalent for all cases.  Figure S2: The potential energy due to gravity changes as the Scissor Mechanism is reoriented. As a result, the external torsional spring is more effective than the internal torsional springs at reducing the mean(Σ|∆PE T |). For the case with both internal and external torsional springs, the stiffness of each internal spring goes to zero and the system reverts to the case with one external torsional spring.

Double Rocker linkage
The Double Rocker linkage is a four-bar linkage with unequal bar lengths. The lengths of the input link, floating link, output link, and ground link are 0.25 m, 0.1 m, 0.3, and 0.2 m, respectively. The Double Rocker linkage has four angles with kinematic paths that are not symmetric nor linearly related ( Figure S4). When fitted, the orders of the potential energy contributions are 2nd, 3rd, 4th, and 3rd order for springs A, B, C, and D, respectively. This variety of higher order terms in PE T gives the system more freedom to offset PE G and leads to a more effective minimization of Σ|∆PE T |.
Adding four internal torsional springs to the Double Rocker linkage reduces the Σ|∆PE T | by over 99%, from 0.372 N-m to 0.003 N-m. Table S7 contains the optimized rest angles α j , stiffnesses k j , and Σ|∆PE T | values for all location combinations for the Double Rocker linkage. Figure S3 illustrates the potential energy contributions of components for each location combination.
Similarly to the Scissor Mechanism, we consider adding extensional springs to the Double Rocker linkage ( Figure S4(C-D)). For ψ = 0 • , internal or external extensional springs reduce the mean(Σ|∆PE T |) by 99.5% and 92%, respectively.
We can also explore which type of springs are most effective when the Double Rocker linkage is reoriented between ψ = 0 • to 90 • . For the system with only torsional springs, the combination of internal and external torsional springs reduces the fluctuation in potential energy the most ( Figure S4(E)). Due to the lack of symmetry in the kinematics, the internal torsional springs have an effect, unlike the Scissor Mechanism. The case with external torsional and external extensional springs reduces the mean(Σ|∆PE T |) nearly as much as the case with all springs ( Figure S4(E-F)); note the log scale and bar graph in Figure S4(F)). Table S9 contains the spring parameters for all spring cases of the Double Rocker linkage for orientations ψ = 0 • to 90 • .

Curve Fitting of Spring Kinematic Relationships
We use the MATLAB function fit to determine the order of the spring angle curves θ A , θ B , θ C , θ D when they are plotted against the kinematic angle φ. We consider polynomial fits with orders 1 (linear), 2 (quadratic), 3 (cubic), and 4 (quartic). We use the coefficient of determination (R 2 value) to determine the polynomial that best fits the kinematic curve. The maximum possible R 2 value is 1.
The angle kinematics for the Watt's linkage are plotted in Figure 1. We define θ A as equal to φ, so a linear fit provides an R 2 value of 1. The angles θ B , θ C , and θ D require fourth-order fits to reach an R 2 value of 1. The kinematics of the Scissor Mechanism are plotted in Figure 3. Due to symmetry in the system geometry, there are two sets of symmetric angles which are linearly related to φ and to each other. As a result, all of the angle curves have an R 2 value of 1 for a linear  fit. As discussed in the main text, because the kinematics of all springs are linearly related, the number and placement of the springs does not have a significant effect on the optimized potential energy behavior.
The lengths of the internal extensional springs (l 1 and l 2 ) are not linearly related to φ or to each other. They can be fitted using polynomial and sinusoidal functions. The length of extensional spring 1 is directly related to φ: l 1 = L sin φ, where L is the member length, so a one-term sinusoidal fit results in an R 2 value of 1. The length of extensional spring 2 requires a three-term sinusoidal fit for an R 2 value of 1.  The kinematics of the Double Rocker linkage are plotted in Figure S4. The angle θ A is linearly related to φ: θ A = π − φ, so the R 2 value for the linear fit is equal to 1. The angles θ B and θ C have third-order (cubic) fits with respect to φ, and θ D has a fourth-order (quartic) fit.

Physical Prototypes and Testing of the Watt's Linkage
We fabricated three versions of the Watt's linkage: one with no springs, one with four internal torsional springs, and one with four internal and one external torsional spring. The members of the linkages were fabricated using acrylic sheets with thickness = 2.7 mm (0.106"), length = 0.3048 m (12") and width = 0.0381 m (1.5"). The sheets were glued together to create members with a total thickness of 0.0081 m (0.329"). Additional acrylic pieces were used to attach the springs. Members were connected using bolts and nuts.
The spring parameters for the model with four internal torsional springs are presented in Table S16. The spring parameters for the model with four internal torsional springs and one external torsional spring are presented in Table S17. The tables provide both the calculated values (found using optimization) and the actual parameters of the springs used in fabrication. For the springs at locations B, C, and D, two springs of equal stiffness were used to create a composite spring with the total stiffness needed ( Figure S5(B)). Despite using springs with some deviation from the calculated parameters, our results show that the system with springs exhibits continuous equilibrium properties, where the system remains stable in different configurations and in different orientations ( Figure S5 We used a force gauge (load cell) to measure the reconfiguration force of the Watt's linkage along the kinematic path (Figure 2(G) in the main text). The load cell was attached to the Watt's linkage at location B and a vertical force (pulling or pushing) was applied. The force for the system with springs is nearly centered around 0 N, with higher forces developing at both ends of the kinematic path (positive force for pulling up, negative force for pushing down), where the linkage deviates from a straight vertical path. Overall, the forces for the system with springs are lower than the system without springs. Figure S5: Physical prototype of the Watt's linkage. (A) Members of the Watt's linkage were cut from acrylic sheets. (B) Two springs were used at locations B, C, and D to achieve the required spring stiffness. (C) For the system that can be re-oriented, an external torsional spring was installed with one end connected to the Watt's linkage at location A and one end connected to a horizontal bar (shown in black). An internal spring was not used at location A, because it does not add a significant influence on the overall system behavior (see Figure 2(B)). (D) With internal and external torsional springs, the Watt's linkage can be reconfigured at ψ = 0 • , 45 • , and 90 • without collapsing.

Physical Prototypes and Testing of the Scissor Mechanism
We fabricated two models of the Scissor Mechanism: one without springs and one with four internal torsional springs. The members of the linkages were fabricated using acrylic sheets with thickness = 2.7 mm (0.106"), length = 0.3048 m (12") and width = 0.0381 m (1.5") ( Figure S6(A)). The sheets were glued together to create members with a total thickness of 0.0162 m (0.638"). Additional acrylic pieces were used to attach the springs. Members were connected using bolts and nuts. A low-friction frame was built to support the linkages. The optimized stiffness of all of the springs is 0.09 N-m/rad, and we use springs with a stiffness of 0.1 N-m/rad in the physical model. The springs at locations A and C have a rest angle (optimized and used values) of 180 • ( Figure S6(B)) and two have a rest angle of 0 • (locations B and D, Figure S6(C)).
The model without springs collapses due to gravity (Movie S7). The model with optimized springs can be reconfigured easily and is stable at any position along its kinematic path (Movie S8).
We measured the force needed to reconfigure the Scissor Mechanism using a load cell ( Figure S6(D)). The forces required to reconfigure the model with springs are lower than those required to reconfigure the model without springs (Figure S6(E)). We can also calculate the analytical solution for the horizontal force required to prevent the Scissor Mechanism from collapsing: F x = Fg 2 tan φ (solid line in Figure S6(E)). The force gauge data matches this solution well for 45 • ≤ φ ≤ 90 • . For φ < 45 • , the force gauge method is not able to capture the increase in F x , which tends towards infinity as φ −→ 0. In the physical testing of the system with no springs, we have to reorient the load cell upward so that we can move the system, and this reorientation reduces our recorded force for φ < 45 • . A load cell (force gauge) was used to measure forces required for reconfiguration.(E) Force measurements for the system with springs and without springs. The system with springs requires lower forces to reconfigure than the system with no springs. An analytical solution for the horizontal force required to maintain equilibrium matches the load cell data well for φ > 45 • (solid line).

Additional Design Cases Scissor Lift
The Scissor Lift is modeled using bar lengths of 1 m, uniform mass distribution of 10 kg/m, and an external load of 200 kg applied at the center of the fifth scissor unit ( Figure S7). The optimized spring properties of the internal extensional springs placed at each unit of the structure (shown in light blue in Figure 4 Like the Scissor Mechanism, the scissor lift can be reoriented. For specific orientations, adding an external torsional spring and internal extensional springs successfully flatten the PE T curve, as was shown for the case of ψ = 90 • in Figure 4(A). The curves for a system that is are optimized for ψ = 45 • to ψ = 90 • are shown in Figure S7. When considering a range of orientations 45 • < ψ < 90 • , the Σ|∆PE T | is significantly lower than for the case with no springs. The mean(Σ|∆PE T |) for the case with no springs is 9640 N-m, and the mean(Σ|∆PE T |) with springs is 336.2 N-m; a 96.5% reduction. The spring properties are as follows: k 1 = 5043.5 N/m, L 1 = 0.235 m, k 2 = 639.8 N/m,

Knee exoskeleton
We model a knee exoskeleton as two 45 cm members connected to a "foot" which is anchored to the ground ( Figure S8(A)). A four-bar linkage (member length = 15 cm) with four internal torsional springs is placed at the knee joint. An internal extensional spring is connected to location B on the linkage and to the heel joint of the structure. The loads applied to the structure are the self weight of the members (2.6 kg per bar) and an external mass M = 30 kg. The internal torsional springs have kinematics that are symmetric and linearly related ( Figure S8(B)) while the internal extensional spring has a sinusoidal relationship with φ.  Figure S8: (A) The knee is modeled as two members connected to a "foot" which is anchored to the ground. A symmetric four-bar linkage is placed at the knee. (B) The internal torsional springs are linearly related to φ, while the internal extensional spring has a sinusoidal relationship with φ. (C) The magnitude of the potential energy due to gravity changes slightly for different orientations, but the overall shape of the curve remains constant. For orientations 70 • ≤ ψ ≤ 105 • , internal springs flatten the potential energy curve.

Origami Arch
The origami arch is a three-dimensional structure based on a variation of the Miura-ori unit cell (47). The arch deploys from a flat sheet. The deployment is defined by the kinematic angle φ; we consider a range of 100 • < φ < 175 • . The panel dimensions are shown in Figure S9(A). We optimized properties of three internal torsional springs and two internal extensional springs per unit cell. The optimized spring parameters are α A = 140.1 • , k A = 0.3572 N-m/rad, α B = 123.9 • , k B = 0.6526 N-m/rad, α C = 81.0 • , k C = 1.443 Nm/rad, L 1 = 0.136 m, k 1 = 12.4 N/m. The Σ|∆PE T | is reduced from 43.0 N-m to 1.69 N-m. The kinematics of the fold angles are not symmetric or linearly related to each other, and the length of the extensional spring has a quadratic relationship with φ ( Figure S9(B)). These factors result in the internal springs effectively minimizing the Σ|∆PE T |. While we have not optimized this system for reorientation, the origami arch could be rotated about a given axis and optimized for a range of orientations. When considering a range of orientations, we Figure S9: (A) The Miura-ori arch structure is made of an array of unit cells that are a variation of the Miura-ori pattern. Three internal torsional springs are added to the system at θ A , θ B , and θ C , and two extensional springs are included in each cell. (B) The angles of the Miura-ori arch are not symmetric or linearly related. The length of the extensional spring is also not linearly related to the angles with respect to the kinematics defined by φ. expect that external springs would be needed.

Stiffness Matrix Formulation
To use the stiffness method to analyze the Watt's linkage with prestressed springs, we constructed a stiffness matrix [K] with additional rotational degrees of freedom (DOFs). The formulation of the stiffness matrix is an adaptation of the formulation for flexible connections presented in McGuire et al [56]. Figure S10(A) shows the DOFs for one member with a flexible connection at the i-node. For the Watt's linkage, we define four members, each modeled as a frame element with a spring at one end. The local stiffness matrix for each member is 7x7. We use a de-coupled approach so that moments can be applied at the added rotational degrees of freedom (DOFs 1, 5, 12, and 13 in Figure S10(B)), representing the pre-stress that is developed in the springs. Figure S10(B) has a spring at its i-node with stiffness k A . The local stiffness matrix for member 1 is then:

Member 1 in
where E is the Young's Modulus, A is the member cross-sectional area, I is the moment of inertia, and L is the member length. For a member with a spring at its j-node, such as member 4, the additional DOF is included at that node. The local stiffness matrix for member 4 is then: DOF : 9 10 11 17 18 19 13 These local stiffness matrices can then be assembled into the global stiffness matrix [K] and used to solve for the nodal displacements and rotations {δ} = [K] −1 {F}, where {F} is a vector of applied loads. The external loads applied to the Watt's linkage are the gravity forces, which act downward at the center of mass of each of the bars, and the spring moments at each of the four spring locations. The gravity forces are equal to −mg, where m is the mass of a bar and g = 9.81 m/s 2 is the acceleration due to gravity. The moment at spring j is equal to k j (θ j − α j ), where k j and α j are found using optimization. The applied moment changes as the linkage moves through its the kinematic path, reflecting the spring moving toward or away from its rest angle. The properties of the members are assumed to be A = 0.0254 m 2 , I = 1 12 (0.0254)(0.0254 3 ) m 4 , and E = 200 GPa unless otherwise noted. The floating link is split into two members to give a DOF at the center of the linkage, so there are four members total. Members 1 and 4 have a length of 0.3 m, and members 2 and 3 have a length of 0.15 m.

Structural Stiffness of Watt's Linkage
To evaluate the stiffness of the Watt's linkage, we applied unit loads P = 1.0 to the centroid of the floating link, along with the gravity load and spring moments. Using the stiffness Figures S11 (A) and (B) show the structural stiffness of the Watt's linkage in the horizontal and vertical directions, respectively. With the addition of springs, the linkage has high stiffness for a load perpendicular to its kinematic path. For a load parallel to its kinematic path, the Watt's linkage has low stiffness, and thus is easily reconfigured. The Watt's linkage without torsional springs has no structural stiffness and collapses under gravity without locking. The springs are critical to the stiffness of the structure, more than the members of the linkage.

Residual Displacements and Actuation
Even with the addition of torsional springs with optimized parameters, the structure may experience some residual displacements in configurations where the gravity and spring forces do not directly offset. At ψ = 0 • , the horizontal displacement is nearly zero throughout the entire kinematic path; the Watt's linkage is known to approximate vertical straightline motion ( Figure S12(A)). The vertical displacements are reflected in the rotation angle ξ ( Figure S12(B)), which reflects the difference between the desired configuration angle φ and the equilibrium state of the structure due to the effects of gravity and springs. Adding springs reduces ξ significantly from the case with no springs (gray line in Figure S12(B)). The magnitude of the actuation moment M A is comparable for the cases with and without springs ( Figure S12(C)), but the actuation energy E A = M A * ξ is nearly zero throughout the entire kinematic path when springs are added ( Figure S12(D)).
We conduct the same study of the residual displacements, rotation angle, actuation mo-

Locking
Locking mechanisms can be used to make a reconfigurable linkage more suitable for loadbearing applications. Fixing all of the rotational DOFs of the Watt's linkage results in an increase in stiffness in both the horizontal and vertical directions ( Figure S13(A) and (B)). When all rotational DOFs are locked, the horizontal stiffness of the Watt's linkage depends on both the member cross-sectional area (A) and the moment of inertia (I) ( Figure  S13(C)). This is because when the structure is locked, the load results in axial and bending deformations. The vertical stiffness, however, only depends on the moment of inertia when the Watt's linkage is fully locked ( Figure S13(D)). The structure experiences bending as a result of the vertical load and there is no axial dependence. The most effective combination of rotational DOFs to lock varies along the kinematic path. For all configurations, locking all rotational DOFs (at locations A, B, C, and D) leads to a system with the highest stiffness in the vertical direction. For horizontal stiffness, however, locking at location combinations BC, ABC, or BCD lead to the same stiffness as combination ABCD for all configurations ( Figure S14). At the orientation ψ = 45 • ,  The stiffness of the Watt's linkage can be increased by locking one or more rotational DOFs, at locations A, B, C, and D. The stiffness also changes as the structure reconfigures along its kinematic path. The highest horizontal stiffness can be obtained by locking rotations at all locations (ABCD). The largest vertical stiffness, however, can be achieved by locking location combinations BC, ABC, ACD, or ABCD.