Abstract
Collaborative robots must simultaneously be safe enough to operate in close proximity to human operators and powerful enough to assist users in industrial tasks such as lifting heavy equipment. The requirement for safety necessitates that collaborative robots are designed with low-powered actuators. However, some industrial tasks may require the robot to have high payload capacity and/or long reach. For collaborative robot designs to be successful, they must find ways of addressing these conflicting design requirements. One promising strategy for navigating this tradeoff is through the use of static balancing mechanisms to offset the robot’s self-weight, thus enabling the selection of low-powered actuators. In this paper, we introduce a novel, two degrees-of-freedom static balancing mechanism based on spring-loaded, wire-wrapped cams. We also present an optimization-based cam design method that guarantees the cams stay convex, ensures the springs stay below their extensions limits, and minimizes sensitivity to unmodeled deviations from the nominal spring constant. Additionally, we present a model of the effect of friction between the wire and the cam. Lastly, we show experimentally that the torque generated by the cam mechanism matches the torque predicted in our modeling approach. Our results also suggest that the effects of wire-cam friction are significant for non-circular cams.
1 Introduction
1.1 Motivation.
The United States Bureau of Labor Statistics reports that in 2015, musculoskeletal disorders (MSDs), such as pinched nerves, herniated discs, and carpal tunnel syndrome, accounted for 31% of all days-away-from-work cases reported to the agency [1]. These work-related MSDs are caused by exerting strenuous and/or repetitive forces in awkward and non-ergonomic conditions [2]. While full automation could eliminate the risk completely, the complexity of certain industrial tasks, such as equipment maintenance and repair, may necessitate a worker to be physically present at the worksite to control critical aspects of the task.
In an attempt to address this issue, there has been a recent push to develop in-situ collaborative robots (ISCRs) that can help alleviate the physiological stresses of the aforementioned tasks by exerting the potentially harmful forces while being guided by the worker using physical human–robot interaction. Because of the worker’s proximity to the robot, it is absolutely imperative that the robot be endowed with both active and passive measures of safety. Active safety measures, such as collision detection and avoidance algorithms, are safeguards against injury coded into the robot’s control system. Passive safety measures, on the other hand, are features built into the robot’s mechanical design that minimize the likelihood of trauma in the case of a collision between the robot and the worker. Designing a robot with as low-powered actuators as possible can effectively increase the passive safety of the device, especially in the case of catastrophic control system failure. By removing the need for the actuators to lift the robot’s self-weight, static balancing mechanisms are a promising way of lowering actuator torque requirements and thereby achieving higher levels of passive safety [3].
1.2 Relevant Works and Summary of Limitations.
Static balancing mechanisms are typically realized using either counterbalancing masses (e.g., Refs. [4–6]), springs, or a combination of both [4,7]. Spring-based static balancing mechanisms offer the advantage of lower overall mass and inertia compared to counterbalancing, making them an attractive option for safety, but come at the cost of higher design and modeling complexity. Spring-based static balancing methods have mostly fallen into three categories: (1) spring-loaded linkages [8–14], (2) cams with spring-loaded followers [15–19], and (3) wire-wrapped cams [20–31]. Additionally, many simple balancing mechanisms are presented in Ref. [32].
It is possible to achieve exact static balancing of planar linkages by using zero-length springs and parallelogram linkages [11,33,34]. While these design solutions may yield acceptable embodiments for certain applications, it may be preferable to consider designs that do not increase the footprint of the manipulator’s linkages because of the restricted nature of some applications. A potential solution for this could be the more compact designs presented in Refs. [35,36]. However, these designs are very complex and add many components such as springs, bevel gears, linear bearings, and linear slides into the linkages of the robot. This increases the mass and inertia of the robot, thus reducing its safety. Therefore, there is interest in keeping as many balancing components as possible at the base of the robot.
This paper will focus on wire-wrapped cam methods. These mechanisms are typically realized with a wire that is fixed to the cam on one end, goes over an idler pulley, and is attached to a linear spring at the other end. When the cam rotates, it wraps the wire over the cam, inducing a torque on the cam from the resulting spring deflection. From a given spring constant, the cam profile can be specially designed to match a desired torque profile as a function of cam angle. Previous works in these areas have used either closed-form equations (e.g., Ref. [23]) or graphical synthesis (e.g., Ref. [28]) to determine the profile of the cam. These methods can provide a cam profile that will generate the desired torque profile exactly. However, they suffer from multiple drawbacks. For certain desired torque profiles, (1) the methods can return non-convex cams (which are needed for continued tangency between the wrapped wire and the cam), (2) there is no explicit guarantee the spring stay below the maximum allowable deflection and in many cases necessitate unrealistically large spring deflections, and (3) there is no consideration of how uncertainty in the spring stiffness may affect the balancing performance. Additionally, previous works have treated the torque on the cam as resulting from a point force applied at the wire-cam tangency point, ignoring the distributed friction, and pressure between the rest of the wire and the cam.
While it may be feasible to search for a set of input parameters that address the non-convexity problem of the closed-form solution, we believe this may not always be a feasible approach. The location and size of the idler are subject to space constraints. Similarly, the cam size is also governed by geometric constraints within a realistic design. The spring parameters that would satisfy cam convexity may also not be feasible when realistic spring design considerations are applied. Finally, this process of sampling the design space is essentially a time-consuming, ad-hoc design optimization process. In this paper, we offer an alternative approach that allows automatic consideration of geometric constraints, robustness to variation in spring parameters, and cam convexity. As such, our approach offers a design that considers realistic constraints while offering the best feasible approximation for the static balancing problem.
Design optimization for static balancing has been used in the past to determine which static robot configuration to perfectly balance to achieve the best balancing across its workspace [37] and to select torsional springs to balance a medical robot [38]. It has also been used to determine the Fourier coefficients of the cam dynamic equations [39] and to optimize mechanism parameters for multi-degrees-of-freedom (DOF) robots [40]. To the best of our knowledge, design optimization has not been used to address the concerns with wire-wrapped cams listed in the previous paragraph.
1.3 Contribution and Paper Organization.
The contribution of this paper relative to previous works is a cam design procedure that uses optimization to select modal coefficients for the cam shape and spring pre-extensions that minimize the difference between the actual cam torque and the desired cam torque. It also minimizes the sensitivity of the output torque to unanticipated changes in spring constant. The optimization problem is constrained to guarantee cam convexity and ensures the cam will not violate the maximum allowable spring deflections. As an additional contribution of this work, we model the effect of friction between the cam and the wire. This optimization routine is applied to a novel two DOF cam design. This two DOF system can be used for systems when the desired torque on one or both cams is a function of both cam angles. The major drawback of our method is that the torque on the cams is not guaranteed to exactly match the desired torque. However, for robotic applications, it is much more desirable to have a physically build-able cam system that approximately balances the static torques than a cam design that theoretically, exactly balances the static torques but violates the maximum allowable spring extensions and is not convex (i.e., does not work in practice).
This paper is organized as follows. In Sec. 2, we present the design of a novel one DOF cam system and solve for the wire tangency point. In Sec. 3, we solve for the spring extensions for the one DOF cam design and extend the equations for a two DOF cam design. Later, in Sec. 4, we calculate the torque on the one DOF cam system with and without ignoring frictional effects. In Sec. 5, we extend the equations for the two DOF cam design. Then in Sec. 6, we describe the modal basis chosen for the cam design and derive the conditions on the cam modal basis required for cam convexity. In Sec. 7, we derive a first-order approximation of the sensitivity of the cam torque to unmodeled changes in spring torque. The optimization problem used to design the cams is formulated in Sec. 8. In Sec. 10, we present the results of two simulation case studies, and in Sec. 11 our model for the torque on the cams is experimentally validated. Lastly, the results are discussed in Sec. 12 and conclusions are drawn in Sec. 13.
2 The Tangency Points in a Wire-Cam Mechanism
The calculation of spring extensions in wire-cam mechanisms requires knowledge of the tangency point between the idler and the cam. In this section, we present a numerical solution to finding the tangency point for a general wire-cam design. Specifically, we refer the reader to Fig. 1, where a follower idler is pressed horizontally against the cam using one spring and the wire rope is tensioned using another spring. Figure 1(a) shows the system with initial spring preload and zero cam rotation. Figure 1(b) shows the same system for a cam rotation θ > 0.
The spring extensions x1 and x2 shown in Fig. 1(b) are functions of the angles α and γ. These two angles characterize the location of the tangency point p on the cam and idler, respectively. To solve for these angles, we use the fact that the cam and idler are tangent to one another at p. To do this, we will calculate the unit vectors that are tangent to the cam and idler and set them equal.
3 Spring Extensions for Wire-Cam Mechanisms
Next, we will determine the spring deflections for design variants with one and two DOF.
3.1 One Degree-of-Freedom Wire-Wrapped Cams.
3.2 Two Degrees-of-Freedom Wire-Wrapped Cams.
A two DOF design variant is shown in Fig. 2. This system is designed so that the torque on the cams are functions of both cam angles (i.e., τ1(θ1, θ2) and τ1(θ1, θ2)). A planar RR (R refers to a revolute joint) manipulator arm is a potential application of this design variant. This system contains two cams with very similar structure to the one DOF cam system shown in Fig. 1. However, instead of being connected to ground, spring 2 connects the prismatic platforms for both cams. This means that changing θ1 affects the torque on cam 2 and vice versa. In this section, we again assume infinite friction between the cam and wire and assume that the springs are linear functions of displacement.
Because the angles α1 and γ1 are independent of θ2 and α2 and γ2 are independent of θ1, the equations in Sec. 2 can be directly applied to solve for the tangency angles of each cam.
4 One Degree-of-Freedom Wire-Wrapped Cam Torque
4.1 Wire-Wrapped Cam Torque With Finite Friction Coupling.
In this section, we explore the effect of finite capstan/wire-rope friction on the resulting torque on the cam. We model the forces at a quasi-static equilibrium where we assume θ is fixed.
4.1.1 Wire Model.
We will assume that the wire is inextensible and can only support tension forces (i.e., it cannot support transverse shear forces and bending moments). The tendon curve is assumed to match the planar cam shape parameterized by angle ϕ, as shown in Fig. 1(b).
4.1.2 Capstan Equation.
4.1.3 Distributed Wire Force and Cam Torque.
5 Two Degrees-of-Freedom Wire-Wrapped Cams
We will next present considerations for cam convexity, which will feed into the cam optimization process.
6 Modal Cam Shape and Condition for Convexity
In making the above choice of using a polynomial basis, we implicitly limit ourselves to cam representations with orders below n = 7 since polynomial bases are known to suffer from poor numerical conditioning for large powers (n > 7) [44].
7 Sensitivity to Changes in Spring Constant
8 Optimal Wire-Cam Mechanism Design
8.1 One Degree-of-Freedom Cam System.
8.2 Two Degrees-of-Freedom Cam System.
9 Wire-Cam Coefficient of Friction Characterization
This experiment was repeated ten times. The mean coefficient of friction was found to be 0.3273 with a standard deviation of 0.0274.
10 Simulation Case Studies
10.1 Comparison to Closed-Form Solution.
To demonstrate the benefit of our approach relative to the closed-form solution, we offer here an example of a single-DOF cam design for satisfying the desired torque function τd = 0.08θ2. Figure 6 shows a comparison of our method to the method of Ref. [23] with an assumed idler horizontal location of 0.1 m, McMaster–Carr spring (part number 5108N486), and idler radius of 0.01 m. For the method of Ref. [23], the spring was assumed to have no pre-extension.

(a) Friction characterization experimental setup: ① benchtop vise, ② cylindrical portion of cam material, ③ wire-rope, ④ basket for holding mass that creates f0, ⑤ basket for holding mass that creates f and (b) schematic of experimental setup showing terms used in Eq. (48)

(a) Friction characterization experimental setup: ① benchtop vise, ② cylindrical portion of cam material, ③ wire-rope, ④ basket for holding mass that creates f0, ⑤ basket for holding mass that creates f and (b) schematic of experimental setup showing terms used in Eq. (48)
![(a) Cam shapes and idlers for our method (c1 and I1) and Ref. [23] (c2 and I2) when θ = 0, (b) spring deflections: ① maximum allowable spring deflection, ② spring 2 deflection x2 for our method, ③ spring deflection in Ref. [23], ④ spring 1 deflection x1 for our method, and (c) desired versus actual torque cam torque for our method: ① our method, ② desired torque](https://asmedc.silverchair-cdn.com/asmedc/content_public/journal/mechanismsrobotics/16/2/10.1115_1.4056600/2/m_jmr_16_2_021001_f006.png?Expires=1739879302&Signature=kpfVfeIgZp8JWYiGpoCrGBhX6Qj~bzcKEebxyIe81fL~Gj3Z5ng8FqlccUa1-L3EuPqcHL2APZff8Mba7FfkUNW9QyU5EzorCh-KIBMmqi8rqIoECWcBzyjYSjF2HxjJZ3UPqjoIEynMtxPY8aLJQPXYefaiiPMPLUq7NP7nY28vA9yas6sk1U3SLMIMMJ5hrb2ujsMKLpxZ4RxLnOcaQ0QpI5Dm6FyK5Zd3dG-DXsZfYIV7qmYbtRQb~F2BT1R301m2uxWmiZJgl7k~TrDQd04uGAgYefGDpZgzLDlV3nnhkOBpjP3HQ25XvVi5cQ8~U-0vDNliQrd771y8UK5z8A__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
(a) Cam shapes and idlers for our method (c1 and I1) and Ref. [23] (c2 and I2) when θ = 0, (b) spring deflections: ① maximum allowable spring deflection, ② spring 2 deflection x2 for our method, ③ spring deflection in Ref. [23], ④ spring 1 deflection x1 for our method, and (c) desired versus actual torque cam torque for our method: ① our method, ② desired torque
![(a) Cam shapes and idlers for our method (c1 and I1) and Ref. [23] (c2 and I2) when θ = 0, (b) spring deflections: ① maximum allowable spring deflection, ② spring 2 deflection x2 for our method, ③ spring deflection in Ref. [23], ④ spring 1 deflection x1 for our method, and (c) desired versus actual torque cam torque for our method: ① our method, ② desired torque](https://asmedc.silverchair-cdn.com/asmedc/content_public/journal/mechanismsrobotics/16/2/10.1115_1.4056600/2/m_jmr_16_2_021001_f006.png?Expires=1739879302&Signature=kpfVfeIgZp8JWYiGpoCrGBhX6Qj~bzcKEebxyIe81fL~Gj3Z5ng8FqlccUa1-L3EuPqcHL2APZff8Mba7FfkUNW9QyU5EzorCh-KIBMmqi8rqIoECWcBzyjYSjF2HxjJZ3UPqjoIEynMtxPY8aLJQPXYefaiiPMPLUq7NP7nY28vA9yas6sk1U3SLMIMMJ5hrb2ujsMKLpxZ4RxLnOcaQ0QpI5Dm6FyK5Zd3dG-DXsZfYIV7qmYbtRQb~F2BT1R301m2uxWmiZJgl7k~TrDQd04uGAgYefGDpZgzLDlV3nnhkOBpjP3HQ25XvVi5cQ8~U-0vDNliQrd771y8UK5z8A__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
(a) Cam shapes and idlers for our method (c1 and I1) and Ref. [23] (c2 and I2) when θ = 0, (b) spring deflections: ① maximum allowable spring deflection, ② spring 2 deflection x2 for our method, ③ spring deflection in Ref. [23], ④ spring 1 deflection x1 for our method, and (c) desired versus actual torque cam torque for our method: ① our method, ② desired torque
The initial conditions of our optimization-based approach were x0 = [10, 10]T mm and β = [0.1, 0.1, 0.1, 0.1]T. The scalar weights were w1 = 10, w2 = w3 = 0. No size limit was placed on the cam.
Obviously, the closed-form cam profile
shown in Fig. 6(a) is not convex and would pose a problem for design implementation. Our method, on the other hand, remains convex (cam ). Additionally, as can be seen in Fig. 6(b), the method in Ref. [23] does not respect the extension limits of the spring. Our method, on the other hand, does not violate the extension limits. This more realistic design comes at the cost of 27.87 N mm of root-mean-square error (RMSE) between τ and τd.10.2 Desired Torque Functions.
Desired torque function constants
m1 (kg) | m2 (kg) | (m) | l1 (m) | (m) | θ |
---|---|---|---|---|---|
0.5 | 0.5 | 0.25 | 0.5 | 0.25 |
m1 (kg) | m2 (kg) | (m) | l1 (m) | (m) | θ |
---|---|---|---|---|---|
0.5 | 0.5 | 0.25 | 0.5 | 0.25 |
We will solve this optimization problem twice: once without trying to minimize the sensitivity of the cam torque to unmodeled changes in the spring constant and once allowing the optimization routine to minimize the aforementioned sensitivity. In the next section, we will discuss the constants and initial conditions used in both simulations.
10.3 Simulation Constants and Initial Conditions.
While in some cases, it may be desirable to let the optimization routine select the spring geometries, the springs in this case study were pre-selected and used as constants. The springs were chosen from McMaster–Carr stock springs. Spring 1 was selected to be part number 5667N212; spring 2 was selected to be part number 7749N634; and spring 3 was selected to be part number 1942N653. The rate and maximum extensions of these springs are shown in Table 2 along with the idler radii, coefficient of friction between the cam and the wire, minimum and maximum cam radii, and the idler vertical offsets. The optimization problems were solved using the active-set algorithm implemented in matlab™ 2021b’s fmincon function. The computer used for these simulations is running Windows 10 on an Intel i7-7700 3.6 GHz processor and has 16 GB of RAM. The initial conditions assumed for this simulation were 10 mm of pre-extension for all springs and β1 = β2 = [0.001, 0.001, 0.001, 0.001]T.
Simulation constants
k1, k2, k3 (N/mm) | x1,max, x2,max, x3,max (mm) | r1, r2 (mm) | μ | ρmin, ρmax (mm), | (mm) |
---|---|---|---|---|---|
1.10, 7.35, 0.58 | 57.66, 32.00, 105.00 | 20, 20 | 0.3273 | 25, 500 | 15, 15 |
k1, k2, k3 (N/mm) | x1,max, x2,max, x3,max (mm) | r1, r2 (mm) | μ | ρmin, ρmax (mm), | (mm) |
---|---|---|---|---|---|
1.10, 7.35, 0.58 | 57.66, 32.00, 105.00 | 20, 20 | 0.3273 | 25, 500 | 15, 15 |
10.4 Two Degrees-of-Freedom System Without Minimizing Sensitivity.
For this simulation, the scalar weights were chosen to be w1 = w2 = 10 and w3 = · · · = w8 = 0. Setting w3 through w8 to zero makes optimizer not attempt to minimize the sensitivity to changes in spring constant. Figure 8 shows the results of the simulation. Figure 8(a) shows a polar plot of the cam designs; Fig. 8(b) shows the difference between the desired cam 1 torque τd,1 and the actual cam torque returned by the optimizer τ1; Fig. 8(c) shows the difference between the desired cam 2 torque τd,2 and the actual cam torque returned by the optimizer τ2; Fig. 8(d) shows the deflection of springs 1 and 3 as a function of θ1 and θ2, respectively. Additionally, the plot shows the maximum allowable deflections in the springs; Fig. 8(e) shows the deflection in spring 2 as a function of θ1 and θ2. It also shows the maximum spring deflection. Table 3 shows the cam polynomial coefficients and spring pre-extensions returned by the optimization routine. Also, Table 3 shows the RMSE and maximum error between τ1 and τd,1 and between τ2 and τd,2. Our implementation generated these results in 8.60 min.

Two DOF simulation results without sensitivity minimization: (a) cam profiles, (b) τ1 versus τd,1, (c) τ2 versus τd,2, (d) x1, x1,max, x3, and x3,max, and (e) x2 and x2,max
Simulation results without minimizing sensitivity
Parameter | Value |
---|---|
β1 | [−0.0052, 0.0133, 0.0046, 0.0250]T |
β2 | [−0.0009, −0.0016, 0.0068, 0.0417]T |
x0 | [0, 9.33, 0] mm |
τ1 versus τd,1 RMSE | 243.12 Nmm |
τ1 versus τd,1 Max. error | 868.25 Nmm |
τ2 versus τd,2 RMSE | 124.04 Nmm |
τ1 versus τd,2 Max. error | 389.92 Nmm |
Parameter | Value |
---|---|
β1 | [−0.0052, 0.0133, 0.0046, 0.0250]T |
β2 | [−0.0009, −0.0016, 0.0068, 0.0417]T |
x0 | [0, 9.33, 0] mm |
τ1 versus τd,1 RMSE | 243.12 Nmm |
τ1 versus τd,1 Max. error | 868.25 Nmm |
τ2 versus τd,2 RMSE | 124.04 Nmm |
τ1 versus τd,2 Max. error | 389.92 Nmm |
10.5 Two Degrees-of-Freedom System With Minimizing Sensitivity.
For this simulation, the scalar weights were chosen to be w1 = w2 = 1 and w3 = · · · = w8 = 1000. This will cause the optimizer to minimize the sensitivity to changes in spring constant. These values were chosen so that the terms of Eq. (47) have similar magnitudes. Figure 9 shows the results of the simulation. Figure 9(a) shows a polar plot of the cam designs; Fig. 9(b) shows the difference between the desired cam 1 torque τd,1 and the actual cam torque returned by the optimizer τ1; Fig. 9(c) shows the difference between the desired cam 2 torque τd,2 and the actual cam torque returned by the optimizer τ2; Fig. 9(d) shows the deflection of springs 1 and 3 as a function of θ1 and θ2, respectively. Additionally, the plot shows the maximum allowable deflections in the springs; Fig. 9(e) shows the deflection in spring 2 as a function of θ1 and θ2. It also shows the maximum spring deflection. Table 4 shows the cam polynomial coefficients and spring pre-extensions returned by the optimization routine. Also, Table 4 shows the RMSE and maximum error between τ1 and τd,1 and between τ2 and τd,2. Our implementation generated these results in 5.67 min.

Two DOF simulation results with sensitivity minimization: (a) cam profiles, (b) τ1 versus τd,1, (c) τ2 versus τd,2, (d) x1, x1,max, x3, and x3,max, and (e) x2 and x2,max
Simulation results with minimizing sensitivity
Parameter | Value |
---|---|
β1 | [−0.0041, 0.0125, 0.0007, 0.0250]T |
β2 | [−0.0019, 0.0052, − 0.0014, 0.0251]T |
x0 | [0, 11.68, 0]T mm |
τ1 versus τd,1 RMSE | 415.00 Nmm |
τ1 versus τd,1 Max. error | 788.33 Nmm |
τ2 versus τd,2 RMSE | 384.84 Nmm |
τ2 versus τd,2 Max. error | 428.17 Nmm |
Parameter | Value |
---|---|
β1 | [−0.0041, 0.0125, 0.0007, 0.0250]T |
β2 | [−0.0019, 0.0052, − 0.0014, 0.0251]T |
x0 | [0, 11.68, 0]T mm |
τ1 versus τd,1 RMSE | 415.00 Nmm |
τ1 versus τd,1 Max. error | 788.33 Nmm |
τ2 versus τd,2 RMSE | 384.84 Nmm |
τ2 versus τd,2 Max. error | 428.17 Nmm |
10.6 Sensitivity Minimization Results.
To demonstrate that the method indeed minimizes the sensitivity of the cam torques, the spring constants shown in Table 2 were increased by 20%, 10%, and 5% and the torque on both cam designs was calculated with the increased spring constants. The RMSEs between the original cam torques and the cam torques with increased spring constants were calculated and the results are shown in Table 5. From these results, it is clear that the deviation in cam torque due to unexpected changes in spring constant is reduced by the optimization routine.
Deviation in cam torque after increase in spring constants
Without Sens. Min. (Fig. 8) | With Sens. Min. (Fig. 9) | Reduction in RMSE | ||
---|---|---|---|---|
20% increase | Cam 1 RMSE | 702.76 Nmm | 647.05 Nmm | 55.71 Nmm |
Cam 2 RMSE | 204.69 Nmm | 143.84 Nmm | 60.85 Nmm | |
10% increase | Cam 1 RMSE | 351.38 Nmm | 323.52 Nmm | 27.86 Nmm |
Cam 2 RMSE | 102.34 Nmm | 71.92 Nmm | 30.42 Nmm | |
5% increase | Cam 1 RMSE | 175.69 Nmm | 161.76 Nmm | 13.93 Nmm |
Cam 2 RMSE | 51.17 Nmm | 35.961 Nmm | 15.21 Nmm |
Without Sens. Min. (Fig. 8) | With Sens. Min. (Fig. 9) | Reduction in RMSE | ||
---|---|---|---|---|
20% increase | Cam 1 RMSE | 702.76 Nmm | 647.05 Nmm | 55.71 Nmm |
Cam 2 RMSE | 204.69 Nmm | 143.84 Nmm | 60.85 Nmm | |
10% increase | Cam 1 RMSE | 351.38 Nmm | 323.52 Nmm | 27.86 Nmm |
Cam 2 RMSE | 102.34 Nmm | 71.92 Nmm | 30.42 Nmm | |
5% increase | Cam 1 RMSE | 175.69 Nmm | 161.76 Nmm | 13.93 Nmm |
Cam 2 RMSE | 51.17 Nmm | 35.961 Nmm | 15.21 Nmm |
11 Experimental Verification
To experimentally validate our model of the cam torques, we manufactured a desktop prototype of the 2DOF cam system as shown in Fig. 10 using the springs listed in Sec. 10.3. The setup consists of 3D printed cams (Fig. 10① and ②) attached to Hebi™ X8-16 actuators (Fig. 10⑧). The cams roll against bearing mounted, 3D printed idlers (Fig. 10③ and ④). The idlers are mounted on linear bearings (Fig. 10⑩) between which spring 2 is mounted (Fig. 10⑥). The wire ropes are attached to the cams on one end, pass through grooves in the idlers, and are terminated on springs 1 (Fig. 10⑤) and 3 (Fig. 10⑦).

Experimental setup: ① cam 1, ② cam 2, ③ idler 1, ④ idler 2, ⑤ spring 1, ⑥ spring 2, ⑦ spring 3, ⑧ Hebi™ X8-16 actuator, ⑨ linear shaft, and ⑩ linear ball bearing
The torque on the cams is measured using the deflection in the series-elastic elements on the Hebi™ actuators. The spring constants of the series-elastic elements were calibrated by applying known moments to the actuators, recording the deflections, and finding the spring constants that best fit the data. The calibrated spring constant for the Hebi™ actuator attached to cam 1 was 337.83 Nm/rad and the calibrated spring constant of the Hebi attached to cam 2 was 148.68 Nm/rad.
According to the manufacturer, spring 2 has 23.22 N of pre-tension. However, the spring force given in Eq. (20) assumes there is no pre-tension in the spring. To account for this, the pre-extension in spring 2 given in Table 4 was lowered to 8.52 mm so that the force on the cams from spring 2 when θ1 = θ2 = 0 is equal to the force that was expected using Eq. (20).
As shown in this paper’s multimedia extension,3 both actuators start at . Cam 1 is kept fixed and cam 2 sweeps from to in 0.02 rad increments. At each increment, 20 torque measurements are taken on each cam. When , θ1 is incremented by 0.02 rad and then θ2 decreases to by 0.02 rad increments, taking 20 measurements at each increment. This process is repeated until . The results of this experiment are shown in Fig. 11. In this test, the RMSE between the model predicted torque and the actual torque for cam 1 was 353.0 Nmm. However, when friction was ignored, the RMSE increased to 518.6 Nmm. The RMSE between the model predicted torque and the actual torque for cam 2 was 166.3 Nmm. With infinite friction, the RMSE was 174.1 Nmm. The RMSE between the actual torque and τd,1 was 579.9 Nmm and 356.2 Nmm between the actual torque and τd,2.

Experimental cam torques compared to simulated cam torques with friction model and simulated cam torques with infinite friction: (a) cam 1 and (b) cam 2
Potential sources of error in this experiment include the accuracy and noise of the torque sensing on the Hebi actuators, friction in the linear bearings, the accuracy of the friction characterization experiment from Sec. 9, inexact pre-extension adjustment, and deflection/hysteresis in the wire rope. Additionally, the stiffness of springs 1–3 were assumed to be equal to the manufacturer-specified value. This assumption introduced additional error between the experimental and model predicted torques.
12 Discussion
While this method is guaranteed to return a physically realizable cam system, there is no guarantee that the resulting cam torques will exactly match the desired torque. We believe this is acceptable for robotic applications where the actuators are able to compensate for differences between the static torques and the torques provided by the cams. For discontinuous or highly non-monotonic desired torque profiles, the torque matching performance will degrade. Additionally, if the desired torque on one of the cams is a stronger function of the other cam angle than of its own angle, the optimization routine may struggle to match the desired torque profile. However, some of this can be alleviated by increasing |a0| (i.e., increasing the moment arm of spring 2’s force). Additionally, if the constraints on the cam design (e.g., ρmin, ~ρmax, etc.) are too narrow, the desired and actual torques may not match well. Additionally, this cam design only works for scenarios when the distance between the cam centers of rotation remains constant.
Because the partial derivative of the cam torques with respect to the spring constants (Eqs. (43) and (44)) are strong functions of the spring deflections and cam-idler contact point locations, minimizing the sensitivity to unexpected changes in spring constant is essentially equivalent to minimizing the size of the cams. This means that increasing values of w3 through w8 will sometimes decrease how well the torque matches the desired torque.
Lastly, the friction model presented in Sec. 4.1 has less impact on the torque for relatively circular cams. This is because the distributed forces that are locally perpendicular to the cam fc pass through the cam center of rotation and therefore cannot create a moment about the cam center of rotation. This can be seen in the experimental results presented in Sec. 11. In Fig. 11(a), the difference between the finite and infinite torque curves is much larger than in Fig. 11(b). This is because cam 2 is much closer to circular than cam 1, which can be seen in Figs. 9(a) and 10.
13 Conclusions
Collaborative robots must simultaneously pose minimal risk to workers and be powerful enough to assist workers in industrial tasks such as lifting heavy equipment. One method for navigating these conflicting design requirements is through the use of static balancing mechanisms to offset the robot’s self-weight, thus enabling the selection of low-powered (i.e., safer) actuators.
Because of their simple, lightweight, and compact design, wire-wrapped cam mechanisms are a promising option for static balancing. However, previous works on wire-wrapped cam mechanisms can return non-convex cams, require unrealistically long spring torques, and ignore the effect of friction between the wire and the cam. These methods are also sensitive to unmodeled deviations in spring constant.
To address these limitations, in this paper, we presented the design of a novel, two DOF wire-wrapped cam system where the torque on each cam is a function of both cam angles. We also presented a model of how the friction and distributed pressure between the cam and the wire affects the torque on the cams. This friction model takes into account the distributed friction and contact pressure between the wire and the cam. This relaxes key assumptions made in previous works that treated the torque on the cam as resulting from a single point force at the wire tangency point. Using this model, we presented an optimization-based cam design procedure that (1) ensured the cam is convex, (2) guaranteed the spring deflections stay below the maximum allowable values, and (3) minimized sensitivity to unexpected changes in spring constant.
Using this cam design method, we built a prototype of the proposed mechanism and experimentally determined that our model predicts the torque on the cams to within 353.0 Nmm of root mean square error. The results also indicate that the distributed force between the wire and the cam can have a significant effect on cam torque for more non-circular cams.
Footnotes
We use rx/y ≜ x − y where and are points.
Acknowledgment
This work was supported by NSF award #1734461 and by Vanderbilt internal university funds. A. Orekhov was partially supported by the NSF Graduate Research Fellowship under #DGE-1445197.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
No data, models, or code were generated or used for this paper.