Embedded Linear-Motion Developable Mechanisms on Cylindrical Surfaces

A demand exists to push mechanisms to be more capable and compact. Increasing functionality by embedding mechanisms onto the peripheral surface of a product shows promise for meeting these demands. For example, a medical device with a cylindrical shaft designed for minimally invasive surgery could have increased utility while being less invasive if embedded mechanisms allowed the surgical tool to enter the body as a cylinder and then deploy and perform additional functions. Specific design criteria or constraints may call for the generation of linear motion, such as a controlled linear-guided biopsy tool. This article pursues a synthesis of traditional kinematics and developable mechanisms to facilitate mechanisms that are (a) more compact, (b) stowable in cylindrical surfaces, and (c) capable of advanced functionality such as generation of linear motion.

Straight-line and linear-motion linkages have been a fundamental building block in machine design since the eighteenth century to generate straight (or approximately straight) motion to perform a useful function. Recent discoveries and growing literature around developable mechanisms examine how traditional mechanisms can be geometrically constrained to a curved or developable surface [1–3]. This research investigates various classical linear-motion linkages and their potential to be embedded onto developable surfaces, such as cylinders, to open the door for more sophisticated machines.

This is done by introducing an approach for mapping and embedding linear-motion mechanisms onto right cylinders. The subclass of kinematic design introduced here is called linear-motion cylindrical developable mechanisms. This article serves as a design tool to aid the designer in understanding the potential of developable mechanisms and help them in the selection process of straight-line mechanisms for use on cylindrical developable surfaces.

A developable surface is a type of surface that can be flattened into a plane without creasing, tearing, or stretching [4]. The four subclasses of developable surfaces are as follows: (a) planar, (b) generalized cylinders, (c) generalized cones, (d) and tangent developable [1]. A right circular cylinder is a type of generalized cylinder whose side is perpendicular to its closed circular base, and all points on the base are at a fixed distance from the axis of the cylinder. Right circular cylinders and tube-based designs are the manufactured embodiment of products such as surgical instruments [5], consumer products, mining equipment, wheels, pipes, rocket fuselages, pressure tanks, and utility poles [6]. In this article, the word “cylinder” refers to a right circular cylinder.

Developable mechanisms are linkages that conform to or emerge from curved, or developable surfaces [1]. Nelson et al. proposed three conditions to describe a developable mechanism: a mechanism that is (i) contained within or conforms to developable surfaces when both are modeled with zero thickness, (ii) has mobility, and (iii) does not require the developable surface to deform to enable the mechanism’s movement [1]. Cylindrical developable mechanisms are a subclass of developable mechanisms that are specifically embedded into a cylinder [6]. Any planar linkage which, when in a specific configuration, is completely concealed in a cylinder is referred to as a cylindrical developable mechanism (as shown in Fig. 1). The hyper-compact nature of developable mechanisms provides benefits for many applications, such as increased functionality of a minimally invasive surgical tool [5] or a wheel that also morphs to have walking abilities [1].

Compliant mechanisms are flexible mechanisms that achieve force and motion transmission through elastic body deformation rather than traditional motion elements (e.g., hinges or bearings) [7]. These mechanisms are advantageous because they integrate high functionality into fewer parts resulting in lower manufacturing costs and improved performance. Compliant mechanisms have been used to further enhance the performance of developable mechanisms [1,5]. Through rigid-body replacement, developable mechanisms can become compliant, providing the benefits of reduced degrees-of-freedom (DOFs) in unconstrained linear-motion linkages and allowing for practical actuation.

During the first industrial revolution (1760–1840), motion control problems demanded more sophisticated machine design to perform more complex tasks [8]. Without the precision and planar machining capabilities of today’s era, reliable sliding components were infeasible [9]. The development of a mechanism that could produce a straight motion became a century-long focus for many engineers, mathematicians, and machinists, even coining the phrase “The Straight Line Problem” [10]. James Watt (1736–1819) became the first to discover a four-bar linkage that could sufficiently approximate a straight line. The Watt’s linkage (1784) was used to guide a steam engine piston stroke in a straight line. It was not until 1864, 80 years after Watt’s discovery, that M. Peaucellier discovered an exact straight-line mechanism with eight bars [11].

Approximate straight-line mechanisms are generally straightforward linkages that generate an approximated straight-line path over a short distance, while exact straight-line linkages are more complex. Exact straight-line mechanisms exist in two subcategories: those with sliders and those without.

A linear-motion mechanism, or linear stage, is a broader category of mechanisms that require added support structures, called a shuttle, to guide a point of the mechanism along a linear path [12]. Rectilinear-motion mechanisms exist when all points of a body move together in a straight line, such as an entire link.

Modern machining advances such as linear bushings, planar guide surfaces, and precision linear stages have resulted in a disregard for classical pinned straight-line mechanisms. However, they are still used in applications such as automobile suspension systems, robotic precision devices [13], and exercise equipment [9]. Linear sliding devices, such as the crank-slider, do not work with the geometric constraints of developable surfaces. Therefore, classical pinned straight-line linkages are re-evaluated as a key resource for creating cylindrically conforming linear-motion mechanisms.

Linear-motion cylindrical developable mechanisms include both the benefits of developable and linear-motion mechanisms. This enables cylinders to have a mechanism concealed within the surface while adding the capability of deployment and generation of linear motion outside, inside, or tangent to the cylindrical body. Useful applications include orthogonal injectors and needles, flow control valves in pipes, alignment and tube centering devices, disk ejectors, and cylindrically conformed wiper mechanisms.

Constructing developable mechanisms is formally characterized by two conditions [1,6]. When in the conformed position (when the links and joints are aligned with the curvature of the reference surface), developable mechanisms are constrained to have their joint axes coincident with the ruling lines on a developable surface [14]. Nelson et al. refer to this constraint as the hinge-axis ruling condition [1]. Traditional rigid-link kinematics assumes that links in a mechanism are rigid, and only the relative distance between joint axes is important while the geometry of the link between two joints does not matter [15]. With developable mechanisms, however, the shape of the link is essential to ensure the mechanism properly conforms to the reference geometry at a certain point in its motion. Greenwood et al. refer to this constraint as the link-shape condition [6]. Additional conditions and guidelines are also relevant for the special branch of linear-motion developable mechanisms. The first is the requirement that the mechanism must be capable of generating linear motion at a point during a region of travel. This will be called the straight-line condition. It is possible that the point of interest of a developable mechanism synthesis problem does not overlap with a joint axis. In the case of straight-line mechanisms, the point of interest, the tracer point (the point of the mechanism that traces a straight line), is often not located at a joint axis. This is always the case for four-bar approximate straight-line mechanisms that have their tracer points located at the midpoint of the coupler link such as the Watt’s and Chebyschev linkages. When the tracer point is colinear with two joint axes it is impossible to have all three points be coincident with a cylindrical ruling line. For exact conformance, the tracer point as well as the joint axes must align with the ruling line of the developable surface. This will be called the tracer point ruling condition.

The conditions are summarized for linear-motion developable mechanisms and are visualized for a right cylinder in Fig. 2.

• Condition 1: Hinge-axis ruling condition [1]: The joint axes must be coincident with ruling lines on the developable surface (see Figs. 2(a) and 2(b)).

• Condition 2: Link-shape condition [6]: The links must conform to the developable surface (see Fig. 2(c)).

• Condition 3: Straight-line condition. The mechanism must have at least one point that travels in a linear path during a discrete region of travel in its rigid and straight kinematic form.

• Condition 4: Tracer point ruling condition. For exact conformance, the tracer point must also be coincident with the ruling lines on the developable surface.

It should be noted that in practice, where surface thicknesses are nonzero, not all conditions must be perfectly met to satisfy design requirements, but can be approximated. For example, it is feasible that a straight-line mechanism that does not satisfy the tracer point ruling condition conforms close enough to avoid any object interference. Trade-offs can also exist between these conditions. For example, a straight-line mechanism can deviate from the maximum accuracy of straight-line output to better satisfy the tracer point ruling condition.

In satisfying the link-shape condition, the individual links of a cylindrical developable mechanism are circular arcs with equal radii to the reference cylinder. There are nine different types of kinematically equivalent link-shape configurations shown in Fig. 3 that can be used in a cylindrical developable mechanism. The arcs can be curved clockwise or counterclockwise around the circle, fully enclose the circle, or even extend beyond the joint axis location in either direction. Although the motion synthesis of the link joints would not change, the overall behavior and workspace of the mechanism and motion of various points of interest change significantly depending on link geometry and directionality. Structural integrity, manufacturability, assembly, and interference prevention are also factors in these design choices.

This section evaluates the ability of various classical straight-line mechanisms to be embedded in a cylinder and analyzes useful characteristics that aid the designer to make guided decisions during the linkage selection process. Selected results are outlined in a design reference in Table 1.

The following characteristics of linear-motion cylindrical developable mechanisms are evaluated and used as comparison references:

1. Linkage type

2. Ability to conform

3. Accuracy of straight-line path output

4. Relative location of straight-line relative to reference cylinder

5. Relative location of straight-line initiation location from conformed position

All mechanisms explored in this article are planar linkages (all links move in parallel planes [16]) and developable mechanisms on circular right cylinders. Further modeling and analysis of planer cylindrical developable mechanisms can be accomplished using established methods for other planar linkages [17] and developable mechanisms [1–3,6]. The material included in this article is not a complete investigation of all known kinematic configurations of linear-motion mechanisms that may be embedded in a cylinder. Modifying link lengths affect mechanism characteristics such as linearity and relative location of straight line on the reference circle. Existing research exploring the mathematical models for the synthesis of straight-line linkages can be referenced to determine variables such as linearity and length of the straight-line segment produced [9,18,19].

With simple mechanisms, constraining a linkage in a CAD software and making the vertices coincident with a circle while rotating the linkage helps to visualize if the mechanism can be sufficiently contained and which configuration shows the most promise for the highest degree of conformance. This process is outlined in Fig. 4(a). However, a tool for more direct comparison is needed to quickly and accurately see how well a linkage can conform in each step during its motion.

To categorize and quantify a classical mechanism’s ability to be mapped onto a cylinder, a nondimensional parameter is needed to measure how uniformly all vertices (joints and tracer point) can fit to the ruling lines of a cylinder and in a configuration that minimizes variance. This will be called the conformance factor. To calculate this nondimensional number, a best-fit circle is fit to the data to provide a radius and coordinates for the center point. This is calculated by minimizing the sum of the squared values of the normal distance from each vertex to the curve of best-fit circle. If all vertices can fit perfectly onto a circle, then the function will eventually arrive at zero.

Although the conditions for a developable mechanism state that all joint axes must be coincident with the ruling lines of the surface, this is not necessary when dealing with nonzero-thickness surfaces. As long as the joints can still be concealed within the wall, they can still be considered “developable.” Before this numerical approach for mapping a linkage onto a best-fit circle, previous examples have had to select which joints to exclude from the ruling line, usually, the tracer point [5]. Though it may be possible to align four of the five points of interest on the ruling lines, having all joints equally spaced from the ruling lines can result in a mechanism with thinner walls and better conformance.

The Watt’s linkage shown in Fig. 6 is an important first linkage to consider because it not only was the first straight-line linkage recorded [11] but also serves as a basis of inspiration for many other straight-line linkages.

The Watt’s linkage is transmobile: as the mechanism deploys from the conformed position, one part of the mechanism moves toward the interior of the circle, while another part of the mechanism moves away from the center of the circle [2].

Another possible limitation of this linkage is that the generated straight line does not include the concealed position. Some initial actuation must occur before the tracer point begins tracing the straight-line path.

Various modifications can be made to the link relationships that still result in an approximate straight line. Mathematical models for determining the linearity accuracy and length of the straight line produced for the Watts linkage can be referenced in existing literature [18]. One interesting configuration is when the ground link is equal to the diameter of the reference circle, which results in the tracer point passing through the center of the circle.

The Chebyshev linkage is a Grashof double-rocker, and similar to the Watt’s and Roberts mechanisms, there are two equal-length rocker segments with a traversing bar that has the tracer point positioned at the midpoint.

When embedding the Peaucellier Linkage onto a reference circle, it can entirely meet the two conditions of a developable mechanism with a caveat. Unlike the previously mentioned mechanisms, even the tracer point is simultaneously aligned when concealed. One drawback of this mechanism is that it may be challenging to actuate due to the two degrees-of-freedom that exist when in the concealed state (Fig. 10(a)). To be concealed, it must pass through a change point (Fig. 10(c)) before it can reach the proper configuration (Fig. 10(d)). Once the mechanism is deployed, a straight line is traced outside the cylinder, and the linkage exhibits one degree-of-freedom again. Butler et al. further analyze and classify the challenges that exist in the design of developable mechanisms when dealing with change points [2].

Hoeken’s linkage is an example of a popular straight-line linkage that cannot be inherently embedded onto a reference right cylinder circular profile. Although the base four-bar linkage meets all conditions for a developable mechanism, it fails to satisfy the tracer point ruling condition. It is possible to add another subembedded developable mechanism onto the base four-bar linkage that can initially deploy to obtain the mechanism to meet the geometric requirements of the Hoeken’s Linkage. This extra deployment is demonstrated in Fig. 11. Adding extra deployment to transform a linkage into a new geometry offers flexibility in adapting wide ranges of mechanisms onto cylinders. This example also demonstrates other design considerations that can be made such as using a prismatic joint that runs perpendicular to the cylinder ruling lines in addition to revolute joints.

Another method of achieving a linear path is with a kite linkage. If two fixed symmetric links move with equal velocity in opposite directions, the free end will move in a straight line. Fixing two contra-parallelograms to a kite mechanism makes this possible [11]. This intricate linkage cannot follow the hinge-axis ruling condition for a single cylinder, but all vertices can, however, align on two distinct concentric cylinders as shown in Fig. 12. In this example, an extra link is added to the mechanism to connect the outer cylinder to the mechanism. When the outer cylinder rotates counterclockwise relative to the inner cylinder, the mechanism is deployed.

A basic kite configuration has the benefit of generating a straight line that passes through the center of the cylinder. The Watt’s linkage is also capable of generating linear motion through the center of the reference circle, but the straight line does not initiate from the ruling lines of the reference circle. This mechanism is the only straight-line mechanism explored in this article that can internally span the whole diameter (or longer) of the reference circle.

The use of compliant mechanisms to induce symmetrical link rotation is an alternative approach to reversing angular velocity by adding an additional linkage as shown with the contra-parallelogram in Fig. 12. Figure 13 shows that replacing revolute joints with small flexural pivots using rigid-body replacement models [7] produces stabilizing forces that keep the tracer point moving in a straight path. Manufacturing the flexures in a symmetrical initially curved configuration results in a mechanism bias, which causes the linkage to move with equal velocities in opposite directions when the mechanism is “released.” This only applies to one-direction deployment, and a reset mechanism must be added to pull/push back the mechanism to its stressed position. One possible method to release this strain energy is a tube-in-tube lock. When an outer or inner tube is removed, the link quickly moves into the biased position following a linear path. Alternatively, a cable that runs in line with the linear motion to release the tension on the linkage produces the desired results. The cable has the benefits of being able to both deploy and return the mechanism into its biased and conformed configuration.

The basic kite methodology of using two symmetric links moving with equal velocity in opposite directions to generate linear motion can be expounded upon to generate rectilinear motion. Adding a fifth rigid link with revolute joints also results in a third undesirable degree-of-freedom. Torsional springs aligned with the joint axes, or replacing revolute joints with compliant flexural pivots, stabilizes the rectilinear-motion mechanism. This spring force induces equal and opposite forces in the symmetric linkage halves, which keep it contained on its linear path as long as there is no perpendicular loading.

Linear-motion mechanisms may also aim to exhibit intramobile behavior while generating a linear path through the center of the cylinder. Greenwood et al. define intramobility as a mechanism’s ability to completely enter (or remain on the edge of) the interior of the reference surface when moved from the conformed position. Extending the coupler link past the center point facilitates this behavior. By making the coupler link at least 90 deg to the straight line, all links will simultaneously move inward during deployment. This behavior is highlighted in Fig. 14.

To demonstrate the practicality of using the concepts introduced in this article, an unmet problem is presented that is uniquely suited to benefit from the kinematic synthesis of linear-motion developable mechanisms and inspire the proposed solution.

Minimally invasive surgery has revolutionized the medical industry. Laparoscopic surgery, a minimally invasive procedure, enables surgeons to operate through a few small keyhole incisions in the abdominal cavity. Benefits of laparoscopic surgery include a shorter hospital stay, faster recovery, reduced pain and bleeding postoperation, and reduced scarring. A rigid endoscope camera, called a laparoscope, is the physician’s eyes into the surgical site.

Minimally invasive surgeons worldwide perform 13 million laparoscopic surgeries every year [22,23]. Recurrent visual impairments from smudged camera lenses plague the surgeons in most of these procedures [22–25]. Condensation (fog), particulate debris (fat tissue and cautery), and blood cause these visual disruptions by accumulating on scope lenses during surgery. Dirty scope optics hamper the surgeon’s view of the surgical area, diminishing critical operative awareness, efficiency, and safety [26].

Surgeons often continue operating despite occluded visualization, spending an average of more than 20 min. of an hour-long surgical case with compromised vision [24]. Visual quality eventually deteriorates so much that it forces the surgeon to halt the operation and lose critical focus of the surgical area. The surgeon restores the surgical field of vision by removing the scope from the patient, manually wiping the lens, reinserting the scope, relocating the surgical site, and then resuming the surgical case. Surgeons repeat this process an average of six times per hour, prolonging procedures, increasing the likelihood of patient complications, frustrating surgical staff, and increasing operating costs [24]. In extreme cases, surgeons may spend more time cleaning the scope than performing the procedure [27] or resort to open surgery if lens clarity cannot be maintained.

Smudged laparoscope lenses squander 336,000 hours of operating room (OR) procedure time every year in the United States, costing the US healthcare system $1.25B annually [24,28]. Advanced ex vivo lens cleaning systems, the current industry standard for maintaining visualization, do not eliminate the worst consequence of obfuscated optics: the pain of removing the scope from the patient for manually cleaning the lens [29].

Linear-motion cylindrical developable mechanisms provide the framework for a minimalistic and innovative mechanical wiper that satisfies the unique design requirements for dirtied laparoscope lenses. The proposed design is a disposable attachment that slides over and attaches onto a 5 or 10 mm laparoscope with a wiper mechanism on the distal end that is actuated by an ex vivo, cable-driven trigger. It would allow surgeons to quickly clean fog, bodily fluid, and fat debris from the lens in vivo (inside the abdomen), without removing the scope to clean and hindering workflow.

The developable mechanism design bypasses the two geometric constraints of a laparoscope attachment: it (i) circumvents obstruction of the scope’s field of view when idle and (ii) avoids interference with the cylindrical trocar channel during scope insertion and removal.

The base linkage is a four-bar slider mechanism (see Fig. 15(a)) that is designed to be intramobile, meaning none of the links extend beyond the outer diameter during deployment [6]. Intramobility is possible when θ₃ ≤ 360 deg, where the x-axis is perpendicular to the direction of travel as shown in Fig. 15(a). When the angular velocities of R₂₋₁ = R₂₋₂ and R₃₋₁ = R₃₋₂ are equal and opposite, the connecting link R₅ moves in a rectilinear motion that uniformly transverses the face of a lens as shown in Fig. 15(b). The straight links are then curved to be concentric to the reference cylinder in the concealed position as shown in Fig. 15(c), following the link-shape condition for developable mechanisms.

The intramobility of the design enables one direction of rotation of θ₂ and θ₄ during deployment, which is essential for actuation when the revolute joints are replaced with torsion joints as shown in Fig. 15(b). This behavior also prevents a failure mode of the wiper mechanism from becoming stuck in the deployed or semi-deployed position, restricting withdrawal through the trocar channel or interfering with nearby instruments and organs. It even allows the surgeon to clean the scope inside the trocar or close to other instruments without interference.

This rigid mechanism with revolute joints shown in Figs. 16(a)–16(c) has six links and six lower kinematic pair revolute joints, resulting in a three-degree-of-freedom mechanism from using the Gruebler’s equation. Adding torsion springs to the pin joints reduces the mobility to a single biased stable position by mitigating the undesired degrees-of-freedom. Designing the mechanism to be manufactured and assembled in the deployed position elastically constrains the connecting link. A cable perpendicular to the connecting link at the midpoint actuates the mechanism by adding or releasing tension, causing it to move in a rectilinear motion. The crossed spring-loaded configuration stabilizes the deployment of the connecting link, ensuring rectilinear motion and only one preferred resting position (Fig. 16(c)).

To simplify the design and improve performance, traditional rigid links and joints were transformed with compliant mechanisms. This provides benefits such as reduced part count, the ability to be miniaturized, and more reliable force outputs. The torsion springs and revolute joints were replaced with small-length flexural pivots (shown in Figs. 16(d)–16(f)), and then two crossed specially formed initially curved large flexural pivots (shown in Figs. 16(g)–16(i)).

The final concept depicted in Fig. 17 comprises two symmetrically crossed flexures with a rigid connecting link functioning as the wiper blade. The flexures are manufactured and assembled in the deployed position (see Fig. 17(b)). A centered tensioned cable pulls back on the connecting link until it reaches the closed position (see Fig. 17(a)) and is held in place with a stronger counter spring until a trigger releases tension. Figure 18 demonstrates a physical prototype removing obstructing fluid from the lens.

This application and prototype physically demonstrate the utility of linear-motion developable mechanisms. Initial modeling and testing indicate that his design shows promise to satisfy the problem requirements and retain linear-motion capabilities. A relationship between the tangential deployment force and linear actuation displacement is determined by using the principle of virtual work in the context of the pseudo-rigid-body model. The virtual work analysis evaluated and optimized the deployment force of each symmetric linkage, ensuring the force response is equivalent, causing each arm of the mechanism to move with an equal and opposite velocity and pushing the wiper to move in a rectilinear motion. An analysis of compliant joint stiffnesses and maximum joint stresses was also conducted using the pseudo-rigid-body model, treating each joint as small-length flexural pivots to predict mechanism reliability. Fatigue testing of large-scale prototypes has been conducted to verify the performance and has achieved a life cycle of over 15,000 wipes. Future work will entail at-scale (for a 5–10 mm laparoscope) fabrication and testing.

Analyzing various classical straight-line mechanisms resulted in developing a mechanism synthesis design reference for linear-motion cylindrical developable mechanisms. Table 1 is a design guide for selecting from various straight-line linkages to identify which are best suited for a specialized design problem.

The table shows information for each linkage, including a conformance factor: a measure of how closely all vertices co-align with a cylindrical ruling line, accuracy of the straight-line path, relative location of the straight line generated, and initiation location of straight line relative to the conformed position.

The exploration of embedded linear-motion developable mechanisms expands the possibilities of machine design and mechanism synthesis to achieve greater sophistication in many applications. The concepts introduced in this article were physically demonstrated with an application for a linear motion in vivo wiper mechanism to clean obstructed laparoscope lenses during surgery while inside the body. The design inspiration and guidelines in this research enable a designer to select from various linear-motion cylindrical developable mechanisms by comparing design parameters and making guided decisions regarding which linkage adequately meets a design problem. Developable mechanisms present a new way to think about engineering design and the nonobvious possibilities of a product’s shape to house functional mechanisms. Developable mechanisms, and especially linear-motion developable mechanisms, show promise to provide increased solutions to specialized problems.

• The National Science Foundation (NSF Grant Nos 1663345 and 1926024).

• The BYU Technology Transfer Bridging Fund.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.