## Abstract

Many robotic systems require linear actuation with high forces, large displacements, and compact profiles. This article presents a series of mechanisms, termed double-helix linear actuators (DHLAs), designed for this purpose. By rotating the fixed end of a double-helix linear actuator, its helix angle changes, displacing at the free end. This article proposes two concepts for DHLA designs, differing in their supporting structure, and derives kinematic and geometric models for both. Prototypes of each concept are presented, and for the more promising “continuous-rails” design, hardware tests are conducted that validate the actuator’s kinematic model and characterize its force transmission properties. The final prototypes can exert both tension and compression forces, can displace up to 75% of their total length, and show consistent trends for torque versus force load. These designs have the potential to overcome the force and displacement limitations of other linear actuators while simultaneously reducing size and weight.

## 1 Introduction

Modern robotic systems often require compact linear actuators that have large displacements. Mobile robots that are actuated by cables must house their mechanisms within their structure and are subject to many design constraints related to volume and desired actuator performance. For example, tensegrity (“tension-integrity”) robots locomote by changing the lengths of the cables that hold their structures together [1–4]. These robots can be multiple meters tall and require cable displacements on the order of tens of centimeters [5,6]. Other large cable-driven exoskeletons [7] or human-scale biped robots [8] face similar size constraints and so are often limited to tethered operation. New actuators that allow for untethered, onboard cable actuation would make these robots practical for more applications.

A variety of possible designs exist for linear actuation onboard a robot, each of which come with limitations. Linear motors, belts, and lead screw actuators are heavy and bulky, with displacements limited to a fixed length [9]. Twisted-string actuators (TSAs) and twisted-cable actuators (TCAs) are compact and lightweight but have small maximum displacements of roughly 40% for the best designs in the literature ([20]). Spooled cable around rotary motors suffers from poor modeling, friction, and cable routing issues [6]. Thermally activated materials have been used in soft robots and robot hands [10,11], but face challenges in measuring internal actuator states (temperature) unless external hardware exists.

Reference | L_{0}, cm | L_{f}, cm | $%D$ | Source? |
---|---|---|---|---|

This work | 40.5 | 10.5 | 75% | – |

Sonoda and Godler [20] | 2.5 | 1.5 | 40% | Fig. 3 |

Palli et al. [21] | 5.0 | 3.3 | 34% | Fig. 1 |

Guzek et al. [22] | 41. | 28. | 32% | Fig. 5(b) |

Suzuki and Ichikawa [23] | 10. | 7.5 | 25% | Fig. 4 |

Park and SunSpiral [24] | 16.9 | 13. | 23% | Fig. 4 |

Gaponov et al. [25] | – | – | 18% | Fig. 11 |

Singh et al. [26] | 33. | 27.0 | 18% | Fig. 6 |

Mennitto and Buehler [15] | 38.4 | 33.7 | 12% | Figs. 7, 9 |

Shoham [12] | 1.0 | 0.96 | 4% | Fig. 7 |

Reference | L_{0}, cm | L_{f}, cm | $%D$ | Source? |
---|---|---|---|---|

This work | 40.5 | 10.5 | 75% | – |

Sonoda and Godler [20] | 2.5 | 1.5 | 40% | Fig. 3 |

Palli et al. [21] | 5.0 | 3.3 | 34% | Fig. 1 |

Guzek et al. [22] | 41. | 28. | 32% | Fig. 5(b) |

Suzuki and Ichikawa [23] | 10. | 7.5 | 25% | Fig. 4 |

Park and SunSpiral [24] | 16.9 | 13. | 23% | Fig. 4 |

Gaponov et al. [25] | – | – | 18% | Fig. 11 |

Singh et al. [26] | 33. | 27.0 | 18% | Fig. 6 |

Mennitto and Buehler [15] | 38.4 | 33.7 | 12% | Figs. 7, 9 |

Shoham [12] | 1.0 | 0.96 | 4% | Fig. 7 |

Note: This work reports a purely kinematic unloaded displacement, but does demonstrate a significant increase over the literature.

This article proposes a series of *double-helix linear actuators* (DHLAs) for mobile cable-driven robots, simultaneously addressing space and displacement constraints. These actuators transform rotational motion into linear motion via a changing helix angle (Fig. 1) so that the helical “rails” wrap into themselves, making efficient use of their space. By separating the rails at a distance, displacement is significantly increased over the similar concept of TSAs/TCAs, up to 75% of total length for the prototypes in this article. The DHLAs presented here also exert both compression and extension forces, unlike TSAs/TCAs, eliminating the need for multiple juxtaposed actuators [7,12–14]. Though helical geometries have been previously exploited in various other mechanisms [15–19], this article is the first to characterize exactly two double-helix rails for purposes of continuous force-torque transmission. This article also shows how the kinematics of DHLA designs unite seemingly dissimilar helical-actuator models from prior work.

The following sections derive mathematical models for two DHLA concepts (Sec. 3), examine their theoretical performance (Sec. 4), demonstrate proof-of-concept prototypes (Sec. 5), and perform a design study to characterize the prototypes (Sec. 6). The final set of actuators demonstrate robust, adaptable designs with large displacements. The displacements and force/torque capabilities of these actuators offer significant improvements over the state-of-the-art in compact linear actuation (c.f. Table 1). Double-helix linear actuators have the potential to make mobile cable-driven robots much more practical, enabling numerous applications across the spectrum of modern robotics.

## 2 Concept and Background

The conceptual basis for DHLAs lies among a number of other twisting actuators but is distinct due to the manner in which the helical structure is used. Here, we introduce the working principle of the design, its relationship to prior work, as well as a key set of modeling assumptions. This article expands upon the concept proposed by the authors in an earlier conference paper [27] and patent [28].

### 2.1 Working Principle.

Double-helix linear actuators are comparable to (and motivated by) a number of concepts previously proposed in the literature, such as TSAs/TCAs. TSAs and TCAs employ two or more strings attached between a rotating element (such as a motor) and a load on which to pull. As the rotation twists the strings around themselves, their length is shortened (Fig. 2(a)). The strings in these actuators are, commonly, deformed into helices. The displacement of a TSA is determined by the radius of its helix. Conceptually, a DHLA simply increases this radius by separating the two wound elements and maintaining the helix radius throughout the mechanism. We denote these wound elements as “rails” and their separating supports as “rungs.” With an increased radius, the displacement also dramatically increases (Fig. 2(b)).

### 2.2 Relationship to Prior Work.

Twisted string and twisted cable actuators have long been used successfully for small displacements. With stiff cables, the kinematics of a helix accurately describe small length changes [12]. More commonly, models of the force carried in the strings are required [21,22,29,30] and accounting for transverse stress-strain within fibers improves the models further [25]. No matter the model, when strings or cables are wrapped closely around themselves, actuator displacement is limited by the small radius of the resulting helices (Fig. 2, Table 1). Finally, Since TSAs/TCAs can only exert tension forces—informally, one “cannot push a rope”—multiple antagonistic actuators are often required [31], although some recent work uses semi-stiff mechanisms for passive returning forces [32].

Alternatively, more displacement is generated when flexible cables are separated at a distance at some location within the mechanism. Cables can be separated at both the rotation and translation end, wrapping around each other in the middle section [26] or can meet at the translational end also [20]. The greatest displacement occurs when the cables do not meet at any point in the mechanism, keeping the helix radius large, such as with the hoops of the LADD actuator [15,33]. Cables separated at a distance can also be used for the opposite goal of this work: applying a displacement which results in a torque [18].

Prior work has also used stiffer but still flexible material as the mechanisms’ rails, as in collapsing frames [34,35] and deployable structures such as antennas [17]. Although these concepts use curved strips of material as rails, none demonstrates force/torque transfer as a linear actuator. Certain designs have exploited helical strips of metal to create multi-stable mechanisms [19] but have not been evaluated for continuous linear actuation.

Finally, recent research has explored thermally-actuated materials in similar helical geometries. Shape-memory alloy wires can be manufactured into spring-like helices with large displacements, but are limited to small size scales and small forces [36]. Twisted-and-coiled actuators employ a cable twisted around its longitudinal axis, not around another cable, with the changing helix angle resulting from internal stresses [10,16,37]. Some designs include one cable with a coating, termed a “double helix” [38], though there is only one composite thread of material. In any of the above, actuation occurs via heating, whereas our designs consider the conversion of torque/rotation into force/translation, allowing for geometry to be scaled and customized for larger loads.

### 2.3 Rail Geometry: Continuous Or Discretized.

The mechanism designs discussed above can be grouped into two categories, depending on a core assumption about the twisting elements (strings, rails, etc.). In all designs, some component enforces a radius at certain points along the device. In the case that the strings or rails are flexible and in contact with each other, or if the rails are stiff and held apart, the mechanism’s kinematics may (approximately) follow the equations of a helix throughout the structure [21]. In contrast, if the supports for rails are intermittent, and cables are flexible, they may form (approximately) straight lines between rungs in three-dimensional space [15]. This article considers each of these two assumptions and demonstrates that both give valid DHLA models. We also present realizations of each assumption as a DHLA prototype, so as to give intuition as to which model may be appropriate for a given mechanical design. We term these two concepts *continuous* rails (C-rails) and *discretized* rails (D-rails).

In the following sections, analytical models for both concepts provide kinematics and design constraints and a proof demonstrates that the D-rails concept becomes the C-rails concept in the limit of increasing number of rungs. Prototypes and testing demonstrate that both concepts require rungs to rotate with respect to rails, and that D-rails concepts become unstable without this rotation. Hardware tests validate the kinematics of the C-rails prototype, for which mechanical designs more readily allow rung rotation, and characterize its force/torque transmission properties. Our concept for the discretized-rails DLHA will be shown to be kinematically equivalent to a LADD with exactly two cables [15], potentially simplifying design and fabrication, though instabilities may result.

## 3 Modeling

The following section derives the kinematics of both the discretized-rail and continuous-rail models and discusses some design constraints for each. It will be shown later that a kinematic approximation (stiff rails) is valid under some circumstances; consequently, we leave questions of stress for future work.

### 3.1 Displacement.

Both models employ a helix radius of *r* = *W*/2, where *W* is the width of a rung and therefore also the transverse distance between the helices. The total angle of twist of the structure (the input into the actuator) is *θ* (rad). There are *N* rungs that separate the rails. The initial length of the actuator (length of a rail) is *L*_{0}. The length of the actuator, given an input rotation, is *L*(*θ*), and so displacement is *D* = *L*_{0} − *L*(*θ*).

#### 3.1.1 Discretized-Rails Actuator Displacement.

The discretized-rails model (Fig. 3) considers the actuator as a stacked series of *i* = 1, …, *N* − 1 cells, where a cell is one pair of rungs. The model is parameterized by the angle of twist within a cell, *θ*_{i}. The length of the rail segment in one cell is *ℓ*_{i}. It is assumed that the rails are spaced evenly throughout the actuator (each cell has the same *ℓ*_{i}) and that the total angle of twist is distributed equally between all cells:

Denote the displacement between two rungs as *z*_{i} (Fig. 3). At *θ* = 0, then *ℓ*_{i} = *z*_{i}. To calculate the relationship between *ℓ*_{i} and *z*_{i}, first note that the outer point of any rung is always at distance *r* = *W*/2 from the center (0, 0). The arc drawn out by a moving rung consequently has radius *r*. Onto this arc, a chord can be drawn between the (*x*, *y*) positions of two sequential rungs. This chord, between rungs *i* and *i* + 1, has length *c*_{i} (Fig. 3).

*ℓ*

_{i},

*z*

_{i}, and

*c*

_{i}form a triangle, and the chord length

*c*

_{i}on the base circle can be found by the law of sines and various trigonometric identities,

*N*− 1 cells between rungs, the total length of the actuator as a function of the input rotation,

*L*(

*θ*), is found by substituting for the initial length of the actuator via Eq. (1):

#### 3.1.2 Continuous-Rails Actuator Displacement.

The equations of a helix are used as the kinematic model of the continuous-rails actuator (Fig. 4, rungs omitted for clarity), since it is assumed that the rails are inextensible. Though this assumption of an infinitely stiff helix has been shown to be insufficient by various authors for twisted string actuators [25], our designs are instead chosen such that the helices’ rails do not deform axially, and therefore the simple equations of a helix are a reasonable approximation. This prevention of deformation is crucial to the concept of double-helix linear actuators: the rungs that separate the rails prevent the helices’ base circles from varying along the actuator’s length, i.e., from compressing radially as in more common twisted-string actuators [25].

*L*(

*θ*,

*t*) with angle of rotation

*θ*and arc length

*t*. The total length of one rail is always

*L*

_{0}, the unactuated length of the helix. The arc drawn out on the circular base of the helix’s cylinder is

*s*=

*rθ*, so considering the triangle on the cylindrical face gives

*t*=

*L*

_{0}is then

### 3.2 Transmission Ratio.

For various design tasks, the differential kinematics of the structure—i.e. transmission ratio—helps determine a desired actuator geometry. This quantity functions similarly to the gear ratio on a rotary actuator. The following analysis therefore provides intuition about comparable applications of the D-rails and C-rails.

#### 3.2.1 Discretized-Rails Actuator Transmission Ratio.

*θ*

^{max}). Intuitively, the actuator shortens as input torque is applied. Finally, although

*d*

^{2}

*L*/

*dθ*

^{2}and further derivatives are uninformative. and therefore not stated here, numerical plots imply that Eq. (8) may be either convex or concave based on actuator geometry.

#### 3.2.2 Continuous-Rails Actuator Transmission Ratio.

*θ*≥ 0).

*θ*≤

*θ*

^{max}(see Sec. 3.3.2), the denominator is positive. Therefore, the C-rails actuator length is a concave function, and transmission ratio will become more negative as the actuator displaces. Consequently, though these actuators theoretically have very large displacements, their useful range may be limited by the amount of torque required as the actuator becomes significantly retracted. For example, electromechanical devices that apply a torque will likely stall as

*θ*→

*θ*

^{max}independent of loading conditions.

### 3.3 Design Constraints and Fundamental Limits.

Each of these two designs has fundamental limits on the amount of retraction and required geometry. The following subsection introduces these considerations, as well as two constants (the cell aspect ratio and overall aspect ratio) that allow for scale-independent comparisons of geometry for the actuators.

#### 3.3.1 Discretized-Rails Actuator Constraints and Limits.

Constraints on the actuator’s total input rotation are dependent on *β*. Consider two cases, where *β* > 1 versus *β* < 1, that is, when the spacing between rungs is greater than or less than the rung width. When *β* > 1, the rails would cross over each other at *θ*_{i} > *π*, and this model breaks down. Such a situation is the intended result of the work in Ref. [26], but since those designs required contact modeling, they are avoided here.

*β*< 1, then there is a maximum

*θ*

_{i}when the rail itself becomes the chord of the base circle, at which point the actuator collapses down to its minimum length. Equating the chord length from Eq. (2) with

*ℓ*

_{i}gives an expression for this maximum angle between rungs as well as the total maximum input rotation,

For example, with an aspect ratio of *β* = 0.5, the input rotation at zero actuator length is (*N* − 1)60 deg.

Finally, observe that the angle between a given rung’s two rail connection points inherently changes as the actuator moves. The two rail segments twist in opposite directions (Fig. 3). Intuitively, then, if the rail segments are to remain straight lines, the rung must somehow rotate to allow the rails to move freely. This observation is implicitly made in Ref. [15], where the cables in the LADD actuator become sigmoids upon actuation due to the constraint between the cables and their circular “rungs.” Since the LADD contains many cables (more than just two) per cell, the requirement of *rotating rungs* may be less clear.

Consequently, constructing a D-rails double-helix linear actuator prompts the question: if the LADD actuator did not have rotation between rail attachment points and performed well with sigmoidal cable deformations, would the same approach work when only two rails (cables) are present instead?

#### 3.3.2 Continuous-Rails Actuator Constraints and Limits.

*N*does not appear in either Eqs. (7) or (9). The rungs here only serve to keep the two helices separated by a distance. Since the cell aspect ratio

*β*does not apply, the total actuator aspect ratio can be used instead to compare designs:

*L*(

*θ*

^{max}) = 0 from Eq. (7). This angle (in radians) is exactly twice the aspect ratio:

*α*≥ 1, the rails simply become arcs of the base circle at

*L*(

*θ*

^{max}) = 0. For

*α*< 1, the rails become interleaved during motion, but do not collide in space, so the model is again still valid.

### 3.4 Relationship Between Discretized-Rails and Continuous-Rails Models.

Although this article motivates the term “discretized” by visual inspection, there is a deeper relationship between the two models.

*The displacement, transmission ratio, and maximum input angle of the discretized-rails model approach that of the continuous-rails model as**N* → ∞.

*u*=

*N*− 1, and various applications of limit properties and L’Hopital’s rule,

*N*as

*f*(

*θ*,

*N*) and Eq. (7) as

*g*(

*θ*). The above proof gives lim

_{N→∞}

*f*(

*θ*,

*N*) =

*g*(

*θ*). By the definition of the partial derivative, expressed as limits, and since limits are linear functionals,

*θ*

^{max}of Eq. (13) approaches Eq. (15) by the same limit arguments as above.

## 4 Theoretical Performance Comparison

The above equations for lengths and transmission ratios (4, 8, 7, 9) were used to simulate the properties of example actuators of both types. The following section uses these simulations to compare the expected performance of each concept. However, as mentioned above, these models are purely kinematic and assume perfectly inextensible rails; therefore, force transmission is not discussed here. All software and data have been made publicly available.^{2}

### 4.1 Reference Designs.

Three reference geometries are used to compare actuators in terms of the aspect ratios *β* and *α* (Table 2). A range of three values of *β* are first chosen (designs I-III), since the D-rails actuator has more stringent constraints on aspect ratio than the C-rails actuator. The C-rails reference designs (IV–VI) are chosen such that *α* arises from the *L*_{0} and *r* values from the corresponding D-rails actuator.

Design | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Type | D | D | D | C | C | C |

L_{0} (cm) | 30.48 | 22.86 | 26.67 | 30.48 | 22.86 | 26.67 |

r (cm) | 2.54 | 2.54 | 1.905 | 2.54 | 2.54 | 1.905 |

N | 7 | 7 | 8 | – | – | – |

α | – | – | – | 6 | 4.5 | 7 |

β | 1 | 0.75 | 1 | – | – | – |

θ^{max} (rad) | 18.85 | 10.18 | 22 | 12 | 9 | 14 |

Design | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Type | D | D | D | C | C | C |

L_{0} (cm) | 30.48 | 22.86 | 26.67 | 30.48 | 22.86 | 26.67 |

r (cm) | 2.54 | 2.54 | 1.905 | 2.54 | 2.54 | 1.905 |

N | 7 | 7 | 8 | – | – | – |

α | – | – | – | 6 | 4.5 | 7 |

β | 1 | 0.75 | 1 | – | – | – |

θ^{max} (rad) | 18.85 | 10.18 | 22 | 12 | 9 | 14 |

*N*must be an integer. Given a

*β*, the relationship between

*N*and

*L*

_{0}is given via Eq. (11) as

*L*

_{0}must be a dividend of 2

*rβ*. All geometries in Table 2 satisfy this condition. Figure 5 shows a visualization of designs I and IV at varying displacements for intuition on the movements of comparable D-rails and C-rails designs.

### 4.2 Length and Displacement Comparison.

Figure 6 plots the lengths *L* (*a*) and displacements *D* = 1 − *L*/*L*_{0} (*b*) from 0 ≤ *θ* ≤ *θ*^{max} for each of the reference designs. As suggested by the analytical models, the C-rails designs (IV–VI) displace “faster” than their D-rails counterparts: for the same *θ*, the C-rails are always shorter. In addition, the C-rails actuators all have the same normalized displacement, implying that all have similar qualitative properties.

However, the D-rails actuator displacements differ according to *β*, even when normalized. Tuning *β* by reducing the number of rungs *N* can make the actuator response more linear throughout its range of motion (Fig. 6(a), c.f. designs I and III versus II). In contrast, as *N* increases, then *β* decreases via Eq. (11), and the D-rails models approach the C-rails models.

### 4.3 Transmission Ratio Comparison.

Figure 7 plots the transmission ratios for each design, including when normalized by *θ*^{max} (*b*). As anticipated by Eq. (10), the C-rails designs suffer from a sharp decrease in their transmission ratio regardless of geometry. However, the D-rails actuators do not have the same limitation: for *β* = 1, the actuator response does not drop off. It therefore appears possible to choose designs in which either $lim\theta \u2192\theta maxdL/d\theta =\u2212\u221e$, or instead levels out to a constant.

### 4.4 Design Implications.

The relationship between actuator geometry and behavior leads to a variety of design guidelines. First, the induced *θ*^{max} from the actuator geometry has a significant impact on the practical range of motion. Given an *L*_{0}, adjusting *α* or *β* determines how much input rotation is required to achieve a certain length.

In addition, different designs may be more appropriate for different goals; there is no single optimal design. For example, assuming that a motor drives one end of the actuator, a larger range-of-motion would generally be achieved by a D-rails actuator with higher *β* and therefore less rapid change in *dL*/*dθ*. C-rails designs are likely to stall a driving motor at high displacements. In comparison, if high-speed but small-displacement motions are required, the “faster” displacement of the C-rails actuator may be preferable.

For the discretized-rails actuators, the number of rungs *N* acts as a tuning constant (via the cell aspect ratio *β*). The D-rails actuator properties are sensitive to *β*, in particular with the bifurcation that appears in the transmission ratio somewhere between 0.75 ≤ *β* ≤ 1. A driving motor with a constant gear ratio would favor designs with relatively constant transmission ratios, motivating a choice of *N* such that *β* = 1 or similar. Such designs give the most linear response while preventing rail collisions.

## 5 Proof-of-Concept Prototypes

A variety of prototypes of double-helix linear actuators were constructed in order to confirm feasibility of design concepts, as well as to validate the modeling from Secs. 3–4. The analysis in this section concludes that although the D-rails DHLA designs are presumably more favorable for common hardware use cases, the C-rails designs can be constructed more practically and agree with their models more reliably.

### 5.1 Discretized-Rails Proof-of-Concept Prototype.

The transmission ratio observations from Sec. 4 motivate the use of a D-rails model. However, it was noted in Sec. 3.3.1 that the rungs of any model must be able to internally rotate: the angles of each rail segment turn in opposite directions on each side of a rung. Other related actuators (e.g., the LADD [15]) do not include this degree-of-freedom, and instead, the rails to deform into sigmoid shapes. For a design with only two rails (our double helix), would this also be an acceptable approximation for easier-to-construct designs? Qualitative experimentation with a prototype gave a definitive *no* to this question, motivating the focus on C-rails prototypes for the remainder of the article.

#### 5.1.1 D-Rails Prototype Mechanical Design.

The D-rails prototype is inspired by the LADD actuator, which threads flexible cables through rigid flat rings. As opposed to those ring-like plates, we use a single 6061 aluminum dowel (0.64 cm diameter) as a rung, sized to create a *r* = 2.54 cm helix (Fig. 8). Two 18-8 stainless steel wire ropes (0.119 cm diameter) were passed through holes drilled into the cylindrical face of each rung. The flat faces of the rung were tapped for 10-32 soft point set screws to constrain the rope.

This design is simple to prototype, though assembly requires careful alignment of rungs before tightening screws. Consistent positioning of the rungs was achieved using an assembly jig, with a spacing of *ℓ*_{i} =3.81 cm between rungs. Therefore, for this design, *β* ≈ 0.75.

#### 5.1.2 D-Rails Prototype Qualitative Testing.

Initial tests by hand were sufficient to conclude that D-rails designs without internal rotation are not practical for use (Fig. 9). Under tension, the actuator became unstable at relatively small input angles (here, *θ* = 450 deg). This can be attributed to the deformations induced in the wire rope. Various other experiments by hand demonstrated similar behavior. Whereas similar past work on actuators with three or more rails have out-of-plane forces that stabilize the three-dimensional moments in the mechanism, rungs *must* be able to rotate in a double-helix design.

Though there are potential remedies for this failure mode, such as including rotational bearings inside each rung, the continuous-rails paradigm offers simpler solutions. We therefore chose not to pursue additional D-rails designs, which are left for future work.

### 5.2 Continuous-Rails Proof-of-Concept Prototype.

Mechanical designs of a continuous-rails double helix linear actuator must be able to resist some stress along their edges in order to retain their helical shape. A rail must have an area moment of inertia that allows bending in the radial direction of the base circle, to form the helix, while resisting deformations along the axis of the actuator. Thin ribbons provide one such geometry, and in addition, can easily be attached to freely rotating rungs. This section introduces a prototype with thin ribbon rails, discusses a number of qualitative observations about this class of designs, and presents a simple experiment validating the prototype against the kinematic model from Sec. 3.

#### 5.2.1 C-Rails Prototype Mechanical Design.

The flat face of a sheet ribbon provides a number of possible ways to attach a rung. One of the simplest is to manufacture a series of holes in the ribbon, at regular intervals, and attach dowels through those holes. Ensuring that the dowel rungs can rotate freely becomes a matter of ensuring a loose, but secure, fit. For the first continuous-rails prototype, a series of snap rings and shims held the ribbon rail in place, with wave springs used to minimize unnecessary movement (Fig. 10).

A number of materials were initially considered for the ribbon rails, including nylon and various metals. After rapid prototyping and experimentation by hand, spring steel was chosen for further analysis. Softer materials were apt to have plastic deformations in the rails under moderate loading, even with favorable cross-sectional geometries.

The first prototype (Fig. 11) used rails that were 1.27 cm high by 0.018 cm thick (1/2 in. by 0.007 in.) 1095 blue-tempered spring steel, with a hole pattern cut via water jet at 2.54 cm (1 in.) intervals. Rungs were fabricated from 5.08 cm long by 0.64 cm diameter (2 in. by 1/4 in.) 6061 aluminum rods, with a step in the shaft (shown in Fig. 10) of 0.43 cm length, reducing the rung to a diameter of 0.48 cm (3/16 in.). Note that, as per the model, the spacing of rungs has no effect on C-rails prototypes as long as there are a sufficient number of rungs to ensure that the rails remain helical.

The actuator was attached to a test fixture that uses a fixed slide to constrain its free end (Fig. 12). A handle is attached to the fixed end to manually apply an input rotation *θ*. Note that the fixed slide does not itself retract as the actuator changes length, reducing the compact-profile benefits of these designs in comparison to, e.g., a linear belt. There are many ways to overcome this limitation; it is not inherent to the concept of DHLAs but was instead a rapid-prototyping trade-off.

#### 5.2.2 C-Rails Prototype Qualitative Testing.

Figure 12 demonstrates three different poses of the actuator, all of which visually confirm the helical path of each rail.

Unlike the D-rails prototype, internal stresses in the ribbon rails resisted twisting and would untwist back to a flat position when released. We define this behavior as the presence of *restorative torque* in the actuator, i.e., internal stresses that restore the actuator to its equilibrium length, similar to Ref. [32]. It is therefore possible for a C-rails DHLA to apply a compression force in addition to a tension force. Investigating the extent of this restorative torque within the actuator itself, under no loading, would allow for design guidelines about using the actuator in both push and pull. Modeling of rail stress is left for future work, but may be adapted from prior theory in thin ribbons [19,39].

#### 5.2.3 C-Rails Prototype Quantitative Testing: Kinematics Verification.

Since qualitative testing of the C-rails prototype visually confirmed the model (c.f. Fig. 5), a simple test of the structure’s kinematics was performed. This prototype had *N* = 17 rungs, for a resting length of *L*_{0} = 40.64 cm (16 in.). The handle on one end of the test setup was manually turned in 90 deg (*π*/2 rad) increments, and its length was measured from the center of the first rung to the center of the last rung, using a metric ruler with 1 mm demarcations.

Figure 13 shows this data plotted alongside the kinematics model for a helical actuator with the same geometry. This test was only performed once, so statistics are not provided here. However, the data are clear: the kinematics model provides an excellent prediction of the unloaded actuator behavior. The root-mean-squared error between the model and experiment, *RMSE* = 0.74 cm, indicated a particularly accurate model given the noisy data collection procedure of visually identifying rotation increments.

The minimum and maximum lengths observed in this test were 40.5 cm and 10.0 cm, leading to a total displacement of 75.3% length. However, data collection near the actuator’s *θ*^{max} = 16 radians was affected by the friction in the slide. Intuition from the transmission ratio plots (Fig. 7) concurs with this observation: the large transmission ratio of the C-rails actuator creates challenges for use at high displacements.

## 6 Actuator Characterization

Proof-of-concept testing demonstrated the feasibility of constructing continuous-rails double helix linear actuators and verified the unloaded kinematics of an example design. However, in order to use these actuators in practical settings, models of the actuators’ force, torque, and displacement relationships are needed. Constructing an analytical model that relates inputs (angle *θ* and torque *τ* at the fixed end) to outputs (length *L* and force *F* at the free end) would require modeling the stress in the actuators’ rails, and accounting for the numerous frictional effects present throughout the mechanism. Instead, we provide a series of hardware experiments, varying different design parameters and loads, which can be used to inform future design decisions or create numerical models. We leave investigation of the D-rails actuators for future work, given the instabilities of D-rails prototypes.

This section introduces the test setup and test procedures, presents data on three prototypes, then discusses implications for using each design. These initial data demonstrate that, although the kinematics of the unloaded actuator may match the model, the force-torque transmission properties only partially agree with the shape of the *dL*/*dθ* curves from Sec. 3. Generally, both friction and internal stresses that create restorative torques have significant impacts on the actuator’s behavior.

### 6.1 Actuator Characterization Test Setups.

The tests presented in this section study the relationship between input angle (*θ*) and torque (*τ*) as compared to applied load at the free end (*F*). Various hand testing demonstrated that the length (*L*) only deviates from the nominal kinematic model by small amounts upon loading. Since deviations were small and measurements were noisy, it was assumed that Eq. (7) holds in all tests.

Our tests use the actuator as per its intended application in cable-driven robots: a cable was attached to the free end of the actuator, and a load applied via that cable. Two different test stands were used. The design with the sliding friction rail (Fig. 12) was used for an unloaded test of the initial prototype. Subsequent testing of different prototypes used a revised mechanism with two linear bearings for reduced friction (Fig. 14(a)). The use of two supporting bearings also reduced three-dimensional torques that caused friction (and therefore noisy measurements) with the single-slide design. A load was applied by attaching various amounts of hanging weight to the free end’s cable (Fig. 14(b)).

For each test, a given prototype geometry was placed in the test stand, a weight was hung from the cable, and a digital torque meter was attached to the handle at the fixed end via a three-jaw chuck. As with the kinematics test, the handle was then rotated in intervals of *π*/2 radians, determined visually, and the torque sensor’s reading was recorded. Five tests were performed for each loading condition.

Three different prototypes were tested in this manner (Table 3), varying the thickness of the sheet metal rails as well as the attachment method of the rungs to rails. All used the same number of rungs (*N* = 17) and rail length (*L* = 40.6 cm) as the initial prototype.

Prototype | Rail thickness | Rung style | Test stand(s) |
---|---|---|---|

(i) | 0.018 cm | Pretensioned | Bearings, slide |

(ii) | 0.018 cm | Loose Fit | Bearings |

(iii) | 0.038 cm | Loose Fit | Bearings |

Prototype | Rail thickness | Rung style | Test stand(s) |
---|---|---|---|

(i) | 0.018 cm | Pretensioned | Bearings, slide |

(ii) | 0.018 cm | Loose Fit | Bearings |

(iii) | 0.038 cm | Loose Fit | Bearings |

Prototypes (ii) and (iii) used a different rung design than the first prototype. The first prototype included wave springs to press the rails against the rungs; we hypothesized that this would be necessary to reduce backlash and noise in the actuator’s response (due to, for example, unmodeled friction). However, the custom rung geometry made it impractical to rapid-prototype rails of different thicknesses. Motivated by the original D-rails concept, the prototypes (ii) and (iii) used a simpler rung design, where an off-the-shelf threaded standoff was placed between the rails, with bolts loosely screwed in at each end to hold the rails in place. Tests with this “loose fit” rung helped determine the relationship between design complexity and backlash/noise for C-rails DHLAs.

### 6.2 Actuator Characterization Tests.

Figure 15 gives results for tests of each of the three designs. The plot for prototype (i) includes data from the unloaded test of the actuator on the original slide setup (Fig. 12) as well as the lower-friction bearing setup. The lower-friction setup was used for all loaded tests. Loads correspond to hanging weights converted to force via *F* = *mg*.

Required torque increases with increasing load and with increasing angle, as expected from the *dL*/*dθ* plots from Sec. 3. Thicker rails required more torque, but could also be tested at higher loads. However, this relationship was not monotonic: a peak in torque appears in the unloaded actuator around *θ* = 2*π* for both prototypes (i) and (ii), and around *θ* = (5/2)*π* for prototype (iii). Kinematically, these input rotations correspond to 9% and 13% displacements, respectively.

These plots indicate how the internal stresses in the rails dominate at low loads and small displacements, whereas high loads (in comparison to rail geometry) dominate behavior at large displacements. The extreme drop-off in theoretical transmission ratio (Sec. 3) is apparent with large loads in prototypes (i) and (ii). The thicker-rails prototype (iii) was primarily governed by internal stresses, indicating the possibility of actuating much higher applied loads than were tested here.

Overall errors are reasonably small, implying that numerical models created from this data could be used with some confidence for future hardware designs. The variance in measurements is smaller as a percent of total torque for higher loads. Although not evident from the plot, error bars for prototype (iii) are of the same magnitude as the other prototypes. Noise from our test setup was therefore likely a larger source of error than inherent behavior of the designs, since noise did not scale with rail geometry.

Prototype (ii) demonstrated larger amounts of noise at high displacements, indicating that a loose fit for the rungs may introduce some minor backlash and therefore modeling errors in comparison to a pretensioned rung design. In contrast, prototype (iii) carried sufficient internal rail stress to keep the rails in contact with the loose-fit rungs’ outer bolts under all loading conditions; therefore pretension may only be needed with thinner rail geometry.

All designs were relatively robust, and could handle a variety of loads. However, under repeated testing, all rails experienced some amount of helical plastic deformation, particularly noticeable with the thicker (0.38 mm) rails. Future work with internal stress modeling in the rails (such as from Ref. [39]) may be able to provide a set of constraints on actuator load to prevent permanent deformation.

### 6.3 Actuator Experimental Efficiency.

Practical use of actuator designs on mobile robots benefits from knowledge of mechanical efficiency, where constraints such as onboard battery life are relevant. This section presents estimates of the efficiency of the C-rails DHLA prototypes during the force/torque characterization tests.

*θ*

_{k}is

*L*(

*θ*) were used as a reasonable estimate of length change for the numerator. The input work was approximated by a trapezoidal integration of the data from Fig. 15 as

*k*. Here, $\mu \tau i\u22121$ is the mean of the five recorded observations of

*τ*

_{t−1}. Calculations were performed iteratively from

*k*= 1 to

*k*=

*K*where

*θ*

_{K}=

*θ*

^{max}, giving five values for each

*η*

_{k}. Figure 16 plots these calculations with the mean and standard deviation.

Efficiencies from these hardware experiments were generally lower than common transmission designs, but appear to be mostly dictated by the small applied loads in these tests. The required torque for motion can be attributed in part to the applied load but also in part to deformation in the rails. Therefore, with small loads, efficiencies are expected to be low, since most input energy goes toward rail deformations. The minima in Fig. 16 appear at similar input angles as the local maxima in Fig. 15, where the rails exert the maximum restorative torque, suggesting the significant role of rail stress in efficiency.

## 7 Discussion and Conclusion

This article proposes the concept of double-helix linear actuators (DHLAs) examines two different design paradigms for possible mechanisms, introduces analytical models and physical prototypes, and evaluates actuator performance. The final concept of the continuous-rails double-helix linear actuator was a compact design that showed large unloaded displacements of up to 75% of its length, with kinematics that align well with analytical models, and relatively consistent force/torque transmission between different prototypes and tests. Prototypes showed the ability to produce both tension and compression forces due to energy storage in the sheet metal rails as the actuator moves.

Multiple conclusions arise from the models and prototypes. First, the discretized-rails actuator may have more beneficial properties in theory, but will be unstable in hardware if rungs are unable to rotate with respect to rails. Future work may produce designs without the unstable deformations in the rails, using for example ball joints at the rail-to-rung connections. In addition, future work will consider an energetic model of stress that may be able to predict these instabilities, particularly if different D-rail cells are allowed to rotate at different angles.

Second, the continuous-rails actuator requires careful choice of rail thickness and material so as to avoid energy loss due to deformation. Efficiencies were somewhat low, possibly due to testing parameters (such as low applied loads) but also possibly due to permanent deformation in the rails. Several of the designs (particularly with thicker rails) were visibly deformed at the completion of tests. However, it is possible that with different choices of materials and geometry, C-rails DHLAs could be used for consistent application of large loads and displacements. For purposes of position or force control, geometry would need to be chosen such that the prototypes’ torque varied monotonically with input angle. Future work will examine procedures to optimize efficiency as a function of geometry.

Third, an analytical model of the stress in the rails would allow us to relax our kinematic assumptions, and would provide estimates of force/torque transmission. With the relatively small applied loads examined in Sec. 6, the actuator’s linear deformations under load were also small. All displacements were essentially kinematic at the level of our measurement capabilities. Future characterization will require kineto-static stress models, which will be adapted from the literature for our device [19,39] and will consider designs for which our inextensible-rails assumption does not hold. The data in this article may still be useful in parameter selection for future designs by calibrating numerical models.

Future work will incorporate these double-helix linear actuators into mobile cable-driven robot designs. Varying the actuator geometry would allow for placement within, for example, tensegrity robots of many different sizes. Such work will require more investigation of pairing motors with DHLAs and required torque for different cable tensions, as well as design options for rail-to-rung attachment. Finally, practical designs will replace the linear bearings or linear slide of these prototypes with another method to constrain rotation of the free end, such as telescoping support structures.

## Footnotes

## Acknowledgment

This work would not have been possible without the many members of the Berkeley Emergent Space Tensegrities Lab at UC Berkeley and the Dynamic Tensegrity Robotics Lab at NASA Ames Research Center’s Intelligent Robotics Group. In particular, thanks to Vytas SunSpiral and Terry Fong at NASA Ames. This research was supported by NASA Space Technology Research Fellowship No. NNX15AQ55H, NASA ESI Grant No. NNX15AD74G, NSF Graduate Research Fellowship no. DGE 1106400, and the NASA Advanced Studies Laboratory at NASA Ames Research Center and UC Santa Cruz.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data and information that support the findings of this article are freely available online.^{3} The authors attest that all data for this study are included in the paper.

## References

*Climbing and Walking Robots*

*United States Patent US2564669A*.