Noncircular rolling contact joints enable programmed behavior in robotic linkages

Significance

We describe an optimization routine to design rolling contact joints with customized kinematics. This compact and robust mechanism can be “programmed” with desired end-effector paths and variable mechanical transmission ratios while maintaining flexibility in off-axis directions. This hybrid system of rigid links and flexible joints results in mechanisms that are robust to impacts, while maintaining precise controllability and high load-bearing capacities. The programmability of the joints allow them to closely match desired paths such as the biomechanics of humans or animals, potentially improving assistive devices and bioinspired robots. The flexible optimization routine can be modified to generate multijoint kinematic chains coupled to a single actuation input, as well as passive joints coupled with elastic elements to create nonlinear springs.

Abstract

Rolling contact joints (RCJs) guide motion in robotic linkages, including manipulators, surgical devices, prosthetics, and more. In this work, we present a generalized optimization method to tailor the kinematic properties of RCJs by simultaneously optimizing both noncircular surface geometries and internal actuation pulley shapes. Our approach accommodates multiple joint types, including passively coupled systems with programmable spring stiffness as well as actuated single or multilink mechanisms. We explicitly incorporate common and practical manufacturing constraints into our optimization framework, such as size and convexity constraints. To demonstrate this approach, we optimize an RCJ designed to replicate the trajectory of a human knee, achieving a 99.6% reduction in alignment error compared to revolute joints and a 99.3% error reduction compared to circular RCJs. Additionally, we show that optimized RCJs increase the load-carrying capacity of a two-finger gripper by more than 3.5 times compared to a comparable circular-jointed design, showcasing how joint optimization can enhance robotic performance.

From the variable mechanical advantage that allows frogs to store elastic energy in their legs before leaping into the air (1), to robots that achieve efficient locomotion by adjusting force output across their gait (2), tailored kinematics are used in both natural and artificial systems to enhance performance. In humans, for example, Deshpande et al. observed that the force-transmission ratio in our hands increases when fingers are flexed (3). This effect amplifies palmar force and potentially enhances grasping and manipulation abilities. Across humans and animals, Catavitello et al. found that kinematic synergies are used to conserve energy during locomotion (4). Such understandings from biology indicate the potential for tailored kinematics to improve engineered systems.

A traditional approach to achieving specific kinematic profiles is the use of four-bar linkages. These mechanisms enable a variety of applications, from functional devices such as knee prostheses designed to replicate human joint motions (5), to artistic creations like Theo Jansen’s Strandbeests, which harness wind power to traverse beaches (6). However, conventional four-bar linkages typically rely on rigid links connected via bearings or gears, which introduce unwanted complexity, friction, backlash, and size constraints. As an alternative, rolling contact joints (RCJs) show promise to overcome these limitations through their simple construction and programmable kinematics, namely mechanical transmission ratio and end-effector path.

RCJs consist of cam-like contact surfaces constrained with wire “ligaments” to roll over each other without sliding. By leveraging rolling contact, designers can achieve lower friction, reduced backlash, and tunable stiffness profiles compared to systems comprised of bearings and gears (7–10). RCJs are easily fabricated with a variety of processes including three-dimensional (3D) printing (7), molding (11), and traditional machining (12). These advantages have led designers to favor rolling contact joints in applications that include dexterous manipulation and grasping (13–18), prosthetics and assistive devices (19–24), and surgical robotics (25–30). Furthermore, using elastic ligaments to constrain the rolling surfaces instead of inextensible wires enables out-of-plane flexibility, creating a hybrid system of rigid links connected by flexible joints (14, 18, 31, 32). Such flexibility can be desirable in assistive and rehabilitative devices which must be tuned to the stiffness properties of the human body, as well as applications like grasping, where rigid links efficiently transfer force and enable Jacobian-based control, while flexibility at the joints allows the system to conform to unknown objects (33, 34).

While the simplest RCJs use circular arcs as rolling surfaces, it is possible to tailor the trajectory of a joint by designing noncircular contact profiles. Previous work includes custom surfaces that follow the natural motion of human joints for implants (23), specially shaped joints in folded origami mechanisms (35), rotational springs with programmable stiffness (36), and magnetic joint-angle sensors that are compatible with noncircular RCJs (37). These are examples of passive RCJs and components. In contrast, actuated RCJs can be created using wire “tendons” anchored at a single point or wrapped around internal pulleys. Several efforts have optimized aspects of these configurations: Zhang et al. utilized noncircular pulleys at a motor to actuate rolling contact joints at different rates in a robotic hand (38); Lee et al. optimized circular surfaces and fixed tendon attachment points to optimize the mechanical transmission characteristics of a surgical device (27); Boisclair et al. optimized the sizes of circular pulleys and circular surfaces for grasping tasks (39); Lee et al. generated noncircular surfaces with circular pulleys for use in a prosthetic hand that can be 3D printed in a single step (24); Liang et al. optimized noncircular surfaces for use with a linear actuator to mimic the motion of the human hand (40). These studies demonstrate the vast design freedom that RCJs offer. However, the optimization efforts are largely limited to varying a small set of geometric parameters, instead of considering the full range of surfaces and pulleys, and often chose to hand-tune the optimization output instead of implementing manufacturing constraints directly. Additionally, to our knowledge, there has been no attempt to simultaneously optimize over both surface and pulley geometry to fully explore this design space. Addressing these challenges is critical to unlocking the full capabilities of RCJs and, more broadly, advancing the development of high-performance mechanisms that integrate tailored kinematics.

We present an optimization routine that is designed to optimize all aspects of the rolling contact joint, including both noncircular pulleys and rolling surfaces, given a desired end-effector trajectory. Our approach is scalable to a wide range of RCJ configurations, encompassing both actuated and passive joints, as well as coupled multijoint systems. Common design constraints are incorporated, such as size limits and tendon tension requirements. This treatment of RCJ optimization enables a thorough exploration of the design space while respecting practical considerations. As proof of concept, we showcase various passive joints including bistable and multistable joints along with nonlinear springs. We also fabricate an RCJ designed to follow the trajectory of a human knee, and find that a noncircular RCJ reduces misalignment error by 99.6% when compared to a revolute joint, and 99.3% when compared to circular RCJs. Finally, we demonstrate an actuated and coupled two-finger gripper with optimized RCJs that can pick up objects more than 3.5 times heavier than its circular-joint counterpart, illustrating the impact of a fully optimized design.

Mechanism Description.

Fig. 1A illustrates a schematic of a single RCJ, driven by a noncircular input pulley. A wire wraps around an idler bearing, routing the tendon to avoid collisions with the wall of the RCJ, then attaches to a second noncircular pulley on the distal end of the link. This wire couples rotations of the input pulley with those of the distal pulley, thus actuating the RCJ. An example physical prototype is shown in Fig. 1B, assembled using 3D-printed components. “S”-shaped ligaments on either side of the RCJ enforce rolling contact without slip. With inextensible ligaments, the two surfaces are constrained to pure rolling and exhibit no off-axis movement. By tuning the elasticity of these ligaments, one can introduce and customize off-axis compliance. Movie S1 displays these properties in example circular RCJs.

Fig. 1.

Examples of rolling contact joints. (A) An illustration of an RCJ including relevant optimization variables, noncircular pulleys and surfaces, and an idler bearing to route the tendon. (B) A picture of an example 3D printed RCJ with flexible ligaments. Note that the proximal noncircular pulley was omitted from this picture. (C–F) Illustrations of example RCJs optimized to follow different end effector paths. The width of the comb-plot denotes the mechanical transmission ratio of the end-effector at that point, with a thicker comb indicating higher force for a given input torque. (G) A comparison of the end-effector paths of the four example RCJs, overlaid on the same axes to highlight differences in position between each joint. The curves are color-coded to match the joint they belong to in (C–F). (H) A comparison of the end-effector transmission ratios of the four example RCJs, normalized to the maximum transmission ratio of each joint and overlaid on the same axes to highlight differences in transmission ratio. The curves are color-coded to match the joint they belong to in (C–F).

Optimization.

Our optimization routine aims to minimize the Euclidean distance between the optimized end-effector trajectory $d_{\mathit{opt}}$ and a specified goal trajectory $d_{\mathit{goal}}$. To achieve this, we discretize the motion into $N_{\mathit{pts}}$ time steps, with each step $t_k\in [0,1]$ and rotate the input pulley at a constant speed $\omega$, so ${\theta }_P=\omega t_k$. Because the input speed is held constant, variations in the end-effector velocity $v_{\mathit{opt}}$ directly reflect changes in the mechanism’s mechanical transmission ratio. Formally, if $F_{\mathit{opt}}$ is the force exerted at the end-effector, and $\tau$ is the torque from the input motor, then conservation of power implies:

$$\begin{array}{c}\hfill \frac{F_{\mathit{opt}}}{\tau }=\frac{\omega }{v_{\mathit{opt}}}\end{array}$$ [1]

Hence, the mechanical transmission ratio at any point on the optimized trajectory is inversely proportional to the end-effector speed. By providing goal trajectories $d_{\mathit{goal}}$ with specific speed profiles (i.e., derivatives), it is possible to control the mechanical transmission ratio of the mechanism and meet task-specific requirements. For example, a gripper might require higher force output near its closed position, a robotic leg may require higher speed at the end of its stride, or an assistive device may benefit from a noncircular path that maintains a constant mechanical transmission ratio. Fig. 1C–F illustrates examples of RCJs optimized for different goal trajectories and visualizes each joint’s corresponding path and force profile. Fig. 1 G and H compares these four example designs on the same set of axes.

To achieve these results, we optimize multiple decision variables at each time step $t_k$. These include $S_{P,k}$ and $S_{D,k}$, which are the coordinates of the contact point on both the proximal and distal surfaces, respectively; ${\alpha }_{P,k}$ and ${\alpha }_{D,k}$, the angle between the tendon and the idler bearing on both sides; and ${\theta }_{D,k}$, the rotation of the distal link relative to the proximal link. Additionally, we define fixed parameters $L_0$ and $L_1$, which set the locations of the input pulley and end effector relative to the idler bearing and the distal pulley respectively (Fig. 1A). Let $\mathbf{R}(\varphi )$ be the standard $2\times 2$ rotation matrix:

$$\begin{array}{c}\hfill \mathbf{R}(\varphi )=\left[\begin{array}{cc}cos(\varphi )& -sin(\varphi )\\ sin(\varphi )& cos(\varphi )\end{array}\right]\end{array}$$

Then given these optimization variables, the optimized end-effector position relative to the idler bearing at each time step is

$$\begin{array}{c}\hfill d_{op{t}_k}=S_{P,k}-\mathbf{R}({\theta }_{D,k})\left(S_{D,k}-\left[\begin{array}{c}{L}_1\\ 0\end{array}\right]\right)\end{array}$$

We minimize the squared distance between the optimized path and the given goal path at every timestep:

$$\begin{array}{c}\hfill min{\displaystyle \sum _{k=1}^{N_{\mathit{pts}}}}\frac{1}{N_{\mathit{pts}}}||d_{\text{opt},k}-d_{\text{goal},k}{||}^2\end{array}$$ [2]

A series of constraints ensures the physical realism of the mechanism and ensures manufacturability through constraint equations. First, we enforce the no-slip rolling condition by requiring that the surfaces remain tangent and that the arc-lengths rolled on each surface are equal. Using the two-dimensional cross product and the standard Euclidean norm:

$$\begin{array}{ccc}& (S_{P,k}-S_{P,k-1})\times \left(\mathbf{R}({\theta }_{D,k})(S_{D,k}-S_{D,k-1})\right)=0\hfill & \hfill \forall k>1\end{array}$$ [3]

$$\begin{array}{ccc}& \Vert S_{P,k}-S_{P,k-1}{\Vert }^2=\Vert S_{D,k}-S_{D,k-1}{\Vert }^2\hfill & \hfill \forall k>1\end{array}$$ [4]

Next, we incorporate the radius $R$ of the idler pulley as an input parameter to calculate the tendon’s moment arm around the input pulley, $r_{P,k}$, as well as around the contact point of the rolling surfaces, $r_{D,k}$. These moment arms are used to enforce the rotational coupling between the two noncircular pulleys that results from the inextensible wire connecting them (41):

$$\begin{array}{c}\hfill r_{P,k}=-L_{0}sin({\alpha }_{P,k})-R\end{array}$$

$$\begin{array}{c}\hfill r_{D,k}=\left[\begin{array}{cc}-sin({\alpha }_{D,k})& cos({\alpha }_{D,k})\end{array}\right]S_{P,k}+R\end{array}$$

$$\begin{array}{ccc}& r_{P,k}({\theta }_{P,k}-{\theta }_{P,k-1})+r_{D,k}({\theta }_{D,k}-{\theta }_{D,k-1})=0\hfill & \hfill \forall k>1\end{array}$$ [5]

Several constraints are implemented to govern the allowable range of moment arms, mechanism size, and surface geometry. First, we set bounds on the moment arms $r_{P,k}$ and $r_{D,k}$ to prevent excessive tendon tension:

$$g_{1,\max }\ge r_k\ge g_{1,\min }$$ [6]

Since $r_{P,k}$ and $r_{D,k}$ are signed moment arms, this constraint also ensures that wire passes on one side of the contact point. In combination with Eq. 5, both the sign of the moment arm and the sign of joint rotation are fixed, ensuring that the actuation wire remains in tension when actuated.

Additionally, size constraints are imposed on both the proximal and distal surfaces:

$$\begin{array}{c}\hfill g_{2,max}\ge ||S_k{||}^2\ge g_{2,min}\end{array}$$ [7]

To ensure the ligaments remain taut and the rolling contact surfaces do not lose contact, we require the surfaces to be convex. However, this constraint may be relaxed if a different no-slip enforcement mechanism (e.g., gear teeth) is used, similar to the joint mechanism used in the Pisa/IIT SoftHand (15). For both $S_P$ and $S_D$:

$$\begin{array}{ccc}& g_{3,max}\ge \left(S_{k-2}-S_{k-1}\right)\times \left(S_k-S_{k-2}\right)\ge g_{3,min}\hfill & \hfill \forall k>2\end{array}$$ [8]

Adjusting the bounds of this constraint can enforce minimum or maximum surface curvatures. Convexity must also hold for the pulley shapes, but because we do not optimize directly over pulley points, it must be enforced with angular constraints. First, we write the equation representing the line of action of the tendon in a frame local to each pulley:

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-sin({\gamma }_k)& cos({\gamma }_k)\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+c_k=0\end{array}$$ [9]

$$\begin{array}{c}\hfill {\gamma }_k={\alpha }_k-{\theta }_k\end{array}$$

$$\begin{array}{c}\hfill c_{P,k}=L_{0}sin({\alpha }_{P,k})+R\end{array}$$

$$\begin{array}{c}\hfill c_{D,k}=\left[\begin{array}{cc}-sin({\alpha }_{D,k})& cos({\alpha }_{D,k})\end{array}\right]\left(S_{P,k}-\mathbf{R}({\theta }_{D,k})S_{D,k}\right)+R_D\end{array}$$

Here, ${\gamma }_k$ is the rotation of the wire line with respect to each pulley, and $c_k$ is a translational offset. With the lines written in the correct frame, we ensure convexity by forcing ${\gamma }_k$ to be monotonic:

$$\begin{array}{ccc}& \mathrm{\Delta }{\gamma }_k={\gamma }_{k-1}-{\gamma }_k\ge 0\hfill & \hfill \forall k>1\end{array}$$ [10]

Successive intersection points of wire lines define the shape of the pulley. We constrain these intersection points to prevent the pulley tangent from spontaneously switching directions, which would cause cusps in the pulley shape and require a “negative” wire length:

$$\begin{array}{cc}\hfill g_{5max}\ge & \left(\frac{c_{k-2}-c_{k-1}cos(\mathrm{\Delta }{\gamma }_{k-1})}{sin(\mathrm{\Delta }{\gamma }_{k-1})}\right.\hfill \\ \hfill & \left.+\frac{c_k-c_{k-1}cos(\mathrm{\Delta }{\gamma }_k)}{sin(\mathrm{\Delta }{\gamma }_k)}\right)\ge g_{5min}\hfill & \hfill \forall k>2\end{array}$$ [11]

This constraint is discussed in more detail in SI Appendix, section S1. We also implement pulley size constraints using the Euclidean norm of the intersection points:

$$\begin{array}{cc}& g_{6max}\ge \left(\frac{2{(c_{k-1})}^2-4cos(\mathrm{\Delta }{\gamma }_k)c_{k-1}{c}_k+2{(c_k)}^2}{1-cos(2\mathrm{\Delta }{\gamma }_k)}\right)\hfill \\ \hfill & \ge g_{6min}\forall k>1\hfill \end{array}$$ [12]

Finally, to prevent degenerate solutions, we enforce a minimum spacing between neighboring contact points:

$$\begin{array}{ccc}& ||S_{P,k}-S_{P,k-1}{||}^2\ge g_{7min}\hfill & \hfill \forall k>1\end{array}$$ [13]

The constraints and objective function are inherently nonlinear. We implement the optimization in CasADI, an open-source interface that facilitates formulating and solving optimization problems (42). We then solve the resulting problems using IPOPT, an open-source large-scale nonlinear optimizer based on an interior-point algorithm (43). Additional details on how these constraints were derived can be found in SI Appendix,section S1.

In summary, we provide IPOPT with a goal path $d_{\mathit{goal}}$, parameters $\omega$, $L_0$, $L$, $R$, and the relevant minimum and maximum bounds in each of the constraint equations. SI Appendix, Tables S2–S7 give values for the parameters used throughout this work. We initialize the solver with a known feasible solution; specifically, the set of decision variables that would yield circular surfaces and circular pulleys (SI Appendix, section S2). IPOPT then attempts to minimize Eq. 2, subject to the constraints given by Eqs. 3–13, producing an optimized set of rolling contact joint designs that satisfy both kinematic and practical requirements. To reconstruct the shapes of the optimized pulleys, the intersection points of successive pulley lines (Eq. 9) are calculated and transformed to a global frame (SI Appendix,section S3).

Results

Simulated Experiments.

The end-effector trajectories achievable with a noncircular RCJ depend on the size, convexity, and moment arm constraints used in the optimization. If no constraints are imposed, any differentiable end-effector path can, in theory, be embedded in a rolling contact mechanism. In practice, however, convexity and size constraints substantially narrow the design space. To explore the range of feasible paths, we randomly generate goal trajectories and record the resulting optimized end-effector paths in Fig. 2A. The “default” path, or path that you achieve with circular surfaces is plotted in dark blue. We then introduce deviations from this default path by adding a random B-spline, whose control points are bounded by $[-5,5]$ mm. This range was chosen so that the added path deviations are on the same order of magnitude as the scale of the mechanism that the optimization is initialized with. A scaling factor $\alpha$ is applied to the B-spline, varying from 0 to 8, which adjusts the magnitude of deviation relative to the default circular path. In this study, the default circular surfaces are initialized with a radius of 10 mm and a maximum radius of 40 mm, meaning that at high $\alpha$ values, deviations can reach up to the maximum surface radius.

Fig. 2.

Physical and simulated experiments. (A) A plot of randomly generated RCJ paths that the solver successfully optimized. (B) A plot of the ratio of output end-effector speed to input speed for the paths displayed in the previous panel. The speed ratio is inversely proportional to the mechanical transmission ratio. (C) A picture of the testing setup used to measure force and position of the RCJ devices. (D) Schematics of the four tested RCJs, along with the predicted and measured force at the end effector for each, plotted against angle. These joints are identical to the joints presented in Fig. 1. (E) Predicted and measured position of the end-effector for the four tested RCJs, color-coded to match the corresponding joint in (D) and in Fig. 1.

To characterize the changes in mechanical transmission ratio, we define the “effective radius” $r_e$ of each mechanism as the ratio of input torque to output force $\tau =r_e{F}_{\mathit{opt}}$ at each time step. Since this value can be scaled arbitrarily by changing the sizes of the pulleys, we normalize it based on the average effective radius of the default circular RCJ. The normalized effective radius is then displayed for all trials in Fig. 2B, visualizing the mechanical transmission ratios that can be achieved with this mechanism.

When the desired path deviates too far from the default path, the solver either exits the feasible region and fails to converge (exceeding iteration limits), or converges to a local minimum whose objective value indicates a poor match with the goal path. For single-joint mechanisms, the solver achieved a high success rate: of the 400 trials in Fig. 2 E and F, 388 converged to a local minimum. The average solver time was 0.71 s. Beyond $\alpha =4$, however, the objective value began to increase above 0.1 cm², revealing that the solutions diverged from the target trajectory by an average of 3.16 mm at each decision point (SI Appendix, Fig. S4). At $\alpha =8$, the objective function had an average value of 0.623 cm². The optimized joints achieved variation in end-effector speed ranging from near 0 to up to 4 times the average speed of the default RCJ initialized with circular surfaces and pulleys. Movie S2 displays example optimized RCJs along with the associated goal path and mechanical transmission ratio, chosen randomly at increasing $\alpha$ values.

Physical Experiments.

We experimentally validate four representative RCJs by measuring their force output and end-effector position over their workspace (Fig. 2C). The four joints selected for testing are shown in Fig. 2D. The first joint is designed to mimic the circular path of a bearing with a constant mechanical transmission ratio and circular path. The second joint is designed with the (slightly noncircular) path that results from circular rolling surfaces, but with a mechanical transmission ratio that is much higher in one portion of the workspace. Next, we design a joint that follows a noncircular trajectory, but maintains a near-constant transmission ratio. Finally, we test a joint with a noncircular path and nonconstant mechanical transmission ratio, illustrating how more complex geometries can provide custom kinematic properties.

A picture of the experimental setup used to validate these joints is shown in Fig. 2C. Two “carrier” RCJs, each driven by an electric motor, secure a load cell against the end-effector of the joint under test. With the load cell held stationary by the motors, a known weight, connected via a circular pulley coaxial with the input noncircular pulley, applies a constant input torque. The load cell records the resulting output force, while the end-effector’s position is simultaneously tracked using visual markers.

As shown in Fig. 2 D and E, the measured position and force values align with the predicted results. Nevertheless, several unmodeled factors may introduce error and hysteresis. These factors include bending stiffness in the ligaments, extension in the tendon, friction at the idler bearing, deformation of the RCJ ligaments or body, and the nonzero thickness of both the tendon and ligament. Additional possible sources of error include fabrication imperfections, such as shrinkage in 3D-printed surfaces or misaligned wires, as well as discretization errors from approximating a continuous system with a finite number of decision points. The latter can be reduced by increasing the resolution of the optimization ($N_{\mathit{pts}}$).

Coupling.

The optimization framework discussed above can readily be adapted for passive rolling contact joints as well as actuated multijoint systems (SI Appendix, sections S1 and S4). Fig. 3A illustrates a passive RCJ whose usual idler bearing is replaced with an elastic tendon wrapped around two noncircular surfaces. Instead of providing a desired end-effector path, we specify a desired moment arm $r$ at each time step $k$, matching the elastic tendon’s radius of action on the RCJ. Note that the same optimization approach could also handle RCJs with elastic tendons anchored at fixed points or freely rotating RCJs, simply by removing or modifying certain decision variables and constraints. We focus on noncircular pulleys here because they are the most complex configuration and provide the greatest design freedom.

Fig. 3.

RCJ Coupling. (A) An illustration of a passive RCJ coupled with an elastic constraint. (B) Example passive joint behaviors that can be achieved by varying the moment arm of the elastic constraint. (C) An envelope of successfully optimized passive RCJs. (D–G) Measured (solid line) and predicted (scatter plot) force output of four passive RCJs. (H) An illustration of how multiple RCJs can be coupled to a single actuation input. (I and J) Envelopes of the design space for two coupled RCJs rotating in the same direction. Paths are shown in (I) and the normalized effective radius is shown in (J). (K and L) Envelopes of the design space for two coupled RCJs rotating in the same direction. Paths are shown in (K) and the normalized effective radius is shown in (L). Color corresponds to the joint behavior illustrated in (B).

Fig. 3B showcases the variety of joint behaviors achievable with a passive RCJ by adjusting the moment arm of the elastic tendon throughout its range of motion. If the elastic tendon passes exactly through the contact point of the two rolling surfaces (i.e., zero moment arm), it produces no net torque on the joint, merely pulling the surfaces together with constant force. This creates an alternative means of enforcing rolling contact without external ligaments and may allow for nonconvex RCJs (7). It can also be used to decouple the tendon’s tension from a joint’s motion, potentially useful when driving a chain of RCJs while keeping their individual motions independent. In contrast, if the elastic constraint has a nonzero moment arm, it generates a rotational moment on the RCJ, creating a nonlinear spring with customizable stiffness. Last, if the moment arm crosses zero, the joint exhibits stable or unstable equilibrium points depending on if the moment arm is increasing or decreasing as it passes through zero. This enables bistable and multistable joints with programmable locations of stability. An example envelope of passive RCJ designs that were successfully optimized is displayed in Fig. 3C.

Four representative designs were tested, with the results shown in Fig. 3D–G. The expected force output from each joint was calculated using the method detailed in SI Appendix, section S5. First, a monostable joint with nonlinear stiffness is tested. The joint returns to its neutral resting position when no external load is applied. Second, we test a link with zero bending stiffness, meaning the elastic ligament always passes through the contact point. Third, a bistable joint is tested with two stable equilibrium. Finally, we test a discontinuous trajectory that consists of an unstable equilibrium surrounded by regions of zero bending stiffness. The results show close agreement to the predicted force output in all cases. Compared to actuated joints, there is less potential for misalignment between the two pulleys to impact the force output, since both pulleys are connected rigidly to the surfaces and fabricated in a single 3D printed step. However, error caused by the assumption of a zero-thickness tendon may be increased, since the thickness of the elastic ligament (1 mm) is larger than that of the actuation wire (0.40 mm).

Rather than relying on passive coupling, one can also directly actuate multiple RCJs with a single degree of freedom by coupling them with an inextensible tendon (Fig. 3H). This greatly expands the RCJ design space, especially when path variations are considered. In these configurations, not only does surface geometry affect the path, but the relative rotation rates of the coupled RCJs also shape the overall end-effector trajectory. As illustrated in Fig. 3H, when the input pulley rotates counterclockwise, the middle link rotates counterclockwise as well, causing an extension of the coupling wire at the first joint. This, in turn, results in a contraction in the wire at the second joint, driving it clockwise. Reversing the wrapping direction around the distal idler bearing would instead cause both joints to rotate counterclockwise. Fig. 3I–L compares these two configurations and visualizes the corresponding design spaces.

Adding each additional coupled joint doubles the number of decision variables and constraints, increasing computational complexity and slowing solver convergence, as visualized in SI Appendix, Fig. S4. For the case where two joints were optimized to rotate in opposite directions, 364 out of 400 trials converged with an average solver time of 3.14 s. When both joints were optimized to rotate in the same direction, 384 out of 400 trials converged and the average solver time was 3.52 s. Both versions of the multijointed mechanism achieved better path-matching accuracy than the single-jointed case, with two joints rotating in opposite directions achieving the best results out of all three (SI Appendix, Fig. S4). At $\alpha =8$, the worst-case average objective values were 0.215 cm² and 0.035 cm² respectively, both significantly lower than the single-joint RCJ’s maximum of 0.623 cm². This improvement is expected given the increased degrees of optimization freedom. Optimized RCJs from these experiments are shown in Movie S2, along with their corresponding goal trajectory.

One explanation for the improved performance of multijointed RCJs that rotate in opposite directions instead of the same direction is the effect that modifying surface or pulley geometry has on the end-effector path. When both joints rotate in the same direction, geometric modifications to either surface or pulley tend to produce similar effects on the end-effector trajectory, limiting the optimizer’s ability to create complex paths, such as the looping motions shown in the Inset of Fig. 3K. However, when the joints rotate in opposite directions, changes to each joint’s geometry influence the end-effector in distinct and even orthogonal directions. This decoupling enables finer control over the trajectory, enhancing the ability to sculpt complex motion paths. However, this improved performance comes at the cost of added design complexity. Multilink RCJs fabricated in this way require careful consideration of how forces are applied, as to avoid putting a compressive load on a tendon which cannot be supported. Return springs or antagonistic pulleys may be required to support specific external loading conditions.

Artificial Knee Joint.

The trajectory-following capability of noncircular RCJs makes them particularly suitable for replicating the complex biomechanics of human and animal joints. This property is vital in designing assistive devices, such as braces, exo-suits, rehabilitative robotics, or prosthetics, that must move harmoniously with the body. Approximating biological joints as simple revolute joints often overconstrains natural motion, causing misalignment and unwanted forces at attachment points (44–46). RCJs, with their advantageous range of motion, ease of fabrication, low friction, and off-axis flexibility, have already found practical use in powered exoskeleton knee joints (20, 47), knee arthroplasty (48), the knees of humanoid robots (49), and prosthetic finger joints (11, 23).

Previous RCJ research has primarily focused on optimizing limited aspects of the rolling surface geometry, ranging from the radius of circular RCJs (20, 45), to selected geometric parameters (23), to complete noncircular surfaces (48). To showcase the flexibility of our optimization framework, we apply our method to design a passive RCJ intended to replicate the trajectory of a human knee (Fig. 4A). Given the knee’s inherent rolling and sliding joint surfaces (50) (Fig. 4A), it is an excellent candidate for RCJ-based assistance applications such as knee braces or prosthetics.

Fig. 4.

RCJ knee Joint. (A) An illustration of a knee brace using a rolling contact joint designed to follow the trajectory of a human knee. (B) An X-ray of a healthy human knee (X-ray of a normal knee by lateral projection, by Mikael Häggström licensed under CC0 1.0). (C) Schematic of optimized surfaces, along with graph of the knee-trajectory following ability of optimized noncircular RCJs, circular RCJs, and revolute joints. Experimental results are included for the noncircular RCJ. Image credit: Wikimedia Commons/Mikael Häggström.

Up to this point, the angle of the end-effector has been treated as a free variable that the optimizer can adjust, providing greater design flexibility, as only the X and Y positions of the end-effector are constrained. However, since a knee brace or assistive device requires multipoint attachment, the goal is to match a specific position to each corresponding leg flexion angle. Hence, it is essential to fix all three variables (X, Y, and the end-effector angle). We address this limitation by removing ${\theta }_D$ as a decision variable and instead constraining it to specific values at each timestep. Moreover, since this mechanism focuses solely on the end-effector trajectory and does not require noncircular actuation pulleys or actuation wires, we eliminate related constraints and decision variables. As a final modification, we adopt an objective function designed to reduce the maximum misalignment error, rather than minimizing the total error across all points. See SI Appendix, section S7 for details on these changes.

We use the equations presented by Walker et al. (51) (and listed below) to model the trajectory of a human knee during flexion ($\gamma$) from 0^° to 120^°. Walker et al. fit second-order polynomials to the experimentally measured X and Y positions of a fixed point on the femur, relative to a coordinate frame attached to the tibia (Fig. 4A).

$$\begin{array}{c}\hfill X=-0.05125\gamma +0.000308{\gamma }^2\end{array}$$ [14]

$$\begin{array}{c}\hfill Y=-0.0802\gamma +0.0000178{\gamma }^2\end{array}$$ [15]

To better align this trajectory with the initial guess used in our optimization routine, which is centered to the right of the origin, we reflect Eqs. 14 and 15 across the Y-axis and shift them 12 mm in the positive X-direction. To provide a benchmark for evaluating the artificial knee joint, we modified the optimization routine to also find the best-fitting circular surfaces and the best revolute joint for the given knee trajectory. This is achieved without major modifications to the optimization routine. For circular surfaces, we constrain the radius to remain constant and for the revolute joint, we restrict the rolling surfaces to a single contact point that is free to move. The results of these simulated joints, along with experimentally measured data from the fabricated noncircular joint, are shown in Fig. 4C and compared to the desired knee trajectory. We quantify the error as the Euclidean distance between the modeled knee path and the trajectory of the optimized (or measured) joint at each flexion angle from 0^° to 120^°. The optimized noncircular RCJ achieved an average error of 8.6 $\mathrm{\mu }$m, 99.6% lower than the revolute joint (2.34 mm) and 99.3% lower than the circular RCJ (1.31 mm).

The low misalignment error suggests that this optimization routine could be used to customize prosthetic devices or orthopedic braces to match the unique biomechanics of individual patients. Furthermore, by incorporating actuation pulleys into the optimization, it may be possible to design powered exosuits or legged robots with mechanical transmission ratios tailored to mimic human biomechanics, or designed to have constant transmission ratios while maintaining the desired trajectory. While knees are particularly well-suited to this approach due to their natural rolling motion, the method could also be extended to replicate the behavior of other human or animal joints using pure rolling, as demonstrated by Hong et al. with an RCJ finger implant (23).

Robotic Gripper.

RCJs offer several advantages for grasping tasks. First, their mechanical transmission ratio can be optimized to peak in regions of the workspace where object contact is most likely, enhancing grip performance. When paired with actuators that have limited stroke length, such as soft or linear actuators (52), this targeted efficiency reduces the actuator’s energy and power density requirements, allowing for the use of smaller, lighter, and more cost-effective actuators. Second, the off-axis flexibility of RCJs enables the gripper to passively conform to objects with unknown shapes or positional uncertainties, much like how human hands adapt to the contours of objects during grasping (53, 54).

We developed a two-link RCJ system coupled to a single pneumatic pouch motor actuator (55), and optimized to maximize mechanical transmission in the region of the workspace where grasping is expected Fig. 5A. Using a manual search to identify trajectories that exhibit high transmission ratios near the end of joint rotation and low transmission ratios near the start, the optimization achieved a maximum mechanical transmission ratio 11.7 times greater than the minimum. Two identical RCJ fingers were connected to a pneumatic pouch motor via a circular pulley coaxial with the noncircular input pulley. This setup was designed so that the actuator’s maximum stroke length matched the input required to rotate the input pulley through its full range of motion. During testing, the actuator was pressurized to 8 psi, and the gripper was manually aligned to perform an antipodal grasp on a container (Fig. 5 B and C). Silicone fingertips were added to increase friction. After the gripper statically held the container, weight was gradually added until the grasp failed. Across five trials, the gripper successfully held an average load of 231.9 g before failure, with a SD of 19.8 g. In some cases, the weight container slipped before a complete grasp failure occurred (Movie S3). In these instances, we report the final failure weight, as it represents the most clearly measurable point and best reflects the gripper’s performance.

Fig. 5.

Two finger RCJ gripper. (A) A schematic of the optimized noncircular surfaces, end-effector path, and mechanical transmission ratio of the gripper. (B) Picture of the optimized gripper holding a weight with a soft pouch motor actuated to 8 psi. (C) Enlarged image of the optimized RCJ linkage, highlighting the tendon path. (D) A schematic of the control gripper fabricated with circular RCJs. (E) Two overlapped photos of the control gripper attempting to hold the empty container weighing 65 g. (F) Enlarged image of the circular RCJ linkage, highlighting the tendon path.

We fabricated a control gripper using circular joints and pulleys to assess the impact of the optimized joints (Fig. 5D–F). The control gripper was designed to keep as many variables constant as possible, including: the actuator, the actuation stroke length and pressure, the starting and ending angles of each link, and the average radius of each surface. Despite these similarities, the optimized gripper achieved a maximum mechanical transmission ratio 2.73 times greater than the circular RCJ gripper at the same angular location. The difference in mechanical transmission ratio was apparent during testing; the control gripper was unable to hold the weight of the container (65 g), meaning the optimized gripper held at least 3.6 times as much weight. We expect the holding force of the gripper to scale linearly with the gripper’s output force since the task is dominated by the force of static friction. The optimized gripper outperformed this prediction, likely due to the decreasing force output of pneumatic pouch motors as a function of contraction length. Although both grippers had the same total actuation stroke length, the circular design’s larger initial mechanical transmission ratio means more stroke length is “consumed” early in the motion, potentially resulting in lower actuator force during the grasp, whereas the optimized design conserves actuator stroke until the grasping point, delivering higher force in conjunction with a more favorable transmission ratio.

An alternative explanation for the improved performance is that the optimized gripper made contact with the object at a different angular configuration compared to the circular version. Although this was not an intentional design choice, our framework enables the optimization of grasp configurations by guiding the RCJ chain’s path, meaning that it could be possible to achieve further improvements in our optimized mechanism. Future work could explore how variations in the RCJ path influence grasp success rates across different object sets, using the design framework presented in this work to construct and experimentally evaluate candidate mechanisms.

It should be noted that a trade-off may exist between optimizing grasp success for a known object set and maintaining performance on unknown objects. Designing for a high transmission ratio in specific regions of the workspace inherently reduces the ratio in other angular ranges. The more a robot designer understands about the intended use case of the gripper, the more precisely its performance can be tailored. However, even when the object set is entirely unknown, a variable transmission ratio may still enhance overall performance. A compelling indication of this comes from the fact that human hands exhibit changing transmission ratios throughout grasping (3), although further studies would be needed to confirm this effect in robotic systems.

Conclusion.

This work presents a flexible optimization framework to design noncircular RCJs tailored for a range of scenarios, including single and multilink actuated joints, passive rolling surfaces, and elastically coupled passive rotational springs. The flexibility of the optimization approach enables the inclusion of practical design constraints such as convexity, component size, and tendon moment arms into the design process.

Our results indicate that using this framework to tailor the mechanics of joints increases the performance of robotic linkages. Specifically, we demonstrated that a passive knee joint can closely match the biomechanics of a human knee, leading to a 99.6% reduction in misalignment error compared to a revolute joint, and an 99.3% reduction compared to circular rolling contact surfaces. Additionally by optimizing the mechanical transmission ratio of a pair of RCJs, we demonstrate a robotic gripper that can lift objects more than 3.5 times heavier than a comparable circular-jointed device.

The devices presented in this paper provide just two examples of how noncircular RCJs may improve the performance of robots. Such optimized joints could enable bioinspired robots to better emulate their biological counterparts, as well as more effective assistive devices, implants, and prostheses. Furthermore, the ability to fine-tune the mechanical transmission ratio makes noncircular RCJs promising for applications requiring variable force output, such as legged locomotion, manipulation, flapping flight, undulatory swimming, and other dynamic tasks.

While this work presents (to our knowledge) the first optimization framework capable of optimizing over both surfaces and pulleys, we identify some areas for future work. First, it is possible that alternative formulations to the optimization problem with a different choice of decision variables, constraint equations, or objective function could improve performance. Additionally, future work could address modeling assumptions made in this approach, such as the assumption that ligaments and tendons have zero width, are inextensible, and have zero bending stiffness. Accounting for these effects could increase simulation accuracy while allowing designers to choose off-axis stiffness properties. Moreover, extending the design space to consider spatially varying ligament properties, such as variable thickness, width, stiffness, and resting geometries, would introduce exciting possibilities for designing even more adaptable and robust mechanisms.

Durability and Design Considerations.

RCJs exhibit line contact during motion. Compared to alternative mechanisms that employ point contact, such as ball-bearings, the distribution of contact stress increases device lifetime under cyclic loading conditions (56). Previous studies have also found that RCJs can exhibit higher tensile and compressive stress limits than pin joints, especially at centimeter scales (19). Additionally, RCJ linkages made with flexible ligaments can dislocate and realign in the event of collision, making them highly robust to impacts (14, 18). These favorable wear properties enable RCJs to meet the high stress and loading requirements of devices such as knee braces (48).

Even with these advantages, it remains essential to select materials, joint geometries, and fabrication processes that align with intended loading conditions. In this work, FDM 3D printing with PLA and TPU was chosen for joint fabrication due to its accessibility and rapid prototyping capability, rather than for optimized mechanical performance. For many uses, this fabrication choice may satisfy design requirements. However, some applications will require more strength, durability, biocompatibility, or other specialized characteristics. Selecting the correct materials and processes will likely involve traditional approaches such as finite element analysis (FEA), analytical stress calculations with respect to material limits, and experimental validation of joint performance under various loading conditions.

One specific consideration for the design of noncircular RCJs, as opposed to traditional circular RCJs, is the emergence of surface stress concentrations arising from variations in local curvature and mechanical transmission ratio. We model these stress concentrations as a function of joint angle using Hertzian contact theory in SI Appendix, section S10, and find that areas of high curvature and wire tension increase surface stress. This curvature can be limited to acceptable levels during joint design by modifying the constraint bounds in Eq. 8.

To experimentally evaluate the limits of this design space, we fabricate a “worst-case” RCJ, a joint designed to have high surface curvature in one area of the workspace, combined with a low input-pulley radius in order to maximize wire tension. We calculate that this joint has higher surface stress concentrations than any of the joints tested in Fig. 2. This joint was then cyclically actuated for 1,000 cycles and subsequently subjected to impact loading for another 300 cycles before the testing apparatus failed. The force-transmission behavior and joint path were tracked throughout both tests, and no degradation was observed, demonstrating the durability of noncircular RCJs even before material or fabrication method optimization (SI Appendix, section S11).

In addition to surface stress, other design considerations include the stiffness, strength, and creep properties of the actuation tendon and ligaments, which should be chosen to satisfy device requirements.

Materials and Methods

Rolling contact joints in this work were fabricated using polylactic acid (PLA) filament (Sunlu, PLA+) on one of three commercially available 3D printers (Prusa Research, i3 Mk3s; Prusa Research, Prusa XL; Bambu Lab, X1Carbon) with 0.4 mm nozzles and 0.2 mm layer height. The ligaments and elastic elements were also 3D printed, using a flexible thermoplastic polyurethane (TPU) material with 85A Shore hardness (Ninjatek, Ninjaflex TPU 3DNF0417510). The ligaments used in this work were fabricated 20% shorter than their final path length, causing prestrain. Steel ball bearings with 5 mm outer diameter and 2 mm inner diameter were used to hold 3D printed wire guides for actuation. The actuation wires were terminated with a knot on 3D printed features, or with a screw used to adjust wire tension. Polytetrafluoroethylene (PTFE) coated fishing line (P-Line, PXBY8150-65) was used to actuate the joints. The pouch motor used for the two-finger gripper was heat-sealed using a desktop heat press (Digital Knight, DK20SP) and consisted of two layers of TPU coated Nylon Taffeta fabric (Trelleborg, 70D Nylon Taffeta with Single Coated Thermoplastic Polyurethane) with a paper-backed adhesive (Packzon, Sticker Printer Paper) to selectively prevent sealing, all cut with a laser cutter (Universal Laser Systems, Versa Laser 6.60). To increase friction, silicone (Smooth-On, Ecoflex 00-30) fingertips were molded onto the grippers in this work. The RCJ force output was measured with a uniaxial load cell (Transducer Techniques, MDB-2.5) paired with a signal conditioner (Futek, IAA100), then read by a data acquisition system (Digilent, Analog Discovery 2) on a bespoke testing platform. A camera (Logitech, C922x Pro Stream Webcam) was used to track the position and angle of the end-effector, as a motor with an embedded positional encoder (Dynamixel, XM430-W350-T) moved the load cell along the end-effector path. CSV files containing points corresponding to the tested surfaces and pulleys are supplied in Datasets S1–S4. Generative tools (OpenAI, ChatGPT-4o) were used to selectively refine language during manuscript preparation.

Supplementary Material

Appendix 01 (PDF)

Dataset S01 (CSV)

Dataset S02 (CSV)

Dataset S03 (CSV)

Dataset S04 (CSV)

Movie S1.

A passive RCJ with circular surfaces is shown, displaying low impedance to rolling and high off-axis flexibility. An example of an actuated RCJ with circular pulleys is also shown.

Movie S2.

Samples of simulated RCJs are animated, highlighting their goal path and speed along with the achieved results. Trials from a single joint, as well as two-joints with the same and opposite actuation directions are shown.

Movie S3.

A two-finger gripper with optimized coupled RCJs holds a box, actuated with a pneumatic pouch motor at 8 psi. Weight is added until the grasp fails. A comparable gripper with circular RCJs cannot hold the weight of the box alone.

Data, Materials, and Software Availability

Computer codes have been deposited in GitHub (DOI: 10.5281/zenodo.17015700) (57).