Large-Range Rotation-to-Translation Compliant Transmission Mechanism

Transmission mechanisms, which transform the input motions of certain constituent bodies into desired output motions of other constituent bodies, are some of the oldest [1] and most ubiquitous mechanisms ever used. Examples of such mechanisms include levers [2], gears [1,3], pulleys [3], and screws [4]. However, most transmission mechanisms are used to either amplify or attenuate a displacement or a load, some transmission mechanisms transform the very nature of an input motion (e.g., they transform an input translational motion into an output rotational motion) or they relocate and/or change the direction of the input motion at their output [5].

More recently, compliant transmission mechanisms [6–9], which consist of flexible joints that deform to achieve desired transmission capabilities, have been pursued to include the benefits of compliant systems. These benefits include the ability to (i) guide complex motions with high precision (i.e., repeatability) free of friction and hysteresis, (ii) avoid backlash and wear, and (iii) reduce part count, weight, and assembly requirements [10]. Compliant transmission mechanisms have also been used to transform input motions into complex output motion paths [11–13], adopt the resolution, stroke, and load capacity of precision actuators (e.g., piezo actuators) [14–17], and transform the single-degree-of-freedom (DOF) motions of multiple decoupled input bodies into the control of a single multi-DOF output body [18]. Compliant transmission mechanisms are also commonly used in micro-electromechanical systems devices [19–21] since compliant elements that deform are one of the few bearing options that can be miniaturized while still permitting the intended motions.

Two of the largest challenges that beset compliant transmission mechanisms, however, are that their range is often limited and their transmission ratio (i.e., the ratio between the transmission’s output displacement to its input displacement) often changes in undesirable ways as the mechanism deforms over its short range. These challenges result from the fact that compliant elements tend to yield after relatively small deformations, and such deformations tend to be nonlinear [10].

In this work, we introduce a large-range compliant transmission mechanism (Fig. 1) that can be tuned to transform an input rotation into an output translation (Figs. 1(a) and 1(b)), or an input translation into an output rotation (Figs. 1(c) and 1(d)), with a desired transmission ratio (i.e., the ratio between the translation’s linear displacement and the rotation’s angular displacement or vice versa depending on which is the input) that remains constant over the mechanism’s full range regardless of the direction in which the input body is actuated. The mechanism can be used on large and small scales to amplify or attenuate the displacement, load, or resolution of a high-precision rotary or linear actuator while transforming its motion into a collinear translation or rotation respectively. This paper explains how the mechanism works, how it was designed, and how its geometry and material properties can be tuned to achieve desired transmission ratios. A specific design was selected to fabricate and test, and the results are discussed.

The second compliant module that constitutes the transmission mechanism of Fig. 1 is shown in Fig. 2(d). It consists of a cross-pivot flexure [31–35], which uses two blade flexures that directly join a grounded body, labeled “G,” to a rigid stage, labeled “S.” The blade flexures constrain the stage to only rotate about the rotational axis (i.e., the red line shown in Figs. 2(d) and 2(e)) along the intersection of both blade-flexure planes.

The third and final compliant module that constitutes the transmission mechanism of Fig. 1 is shown in Fig. 2(f). It consists of three identical and axisymmetric compliant elements consisting of a series of triangle-shaped deformable systems that join a grounded body, labeled “G,” to a rigid stage, labeled “S.” These flexures, which are inspired by previous large-deflection designs [36,37], collectively constrain the module’s stage to only translate in a single direction (i.e., the black arrow shown in Figs. 2(f) and 2(g)) along the stage’s axis.

When the three compliant modules are correctly combined, as shown in Fig. 2(h), the resulting mechanism achieves the desired transmission capabilities. Specifically, the stage of the second module (Fig. 2(d)) must be joined to the grounded body of the first module (Fig. 2(a)) so that the combined body is constrained to only rotate about the coincident axes of the first module’s screw DOF and the second module’s rotational DOF. Moreover, the stages of the third module (Fig. 2(f)) and the first module (Fig. 2(a)) must be merged so that the resulting body is constrained to only translate parallel to the axis of the first module’s screw DOF. In other words, as long as the two bodies labeled “G” in Fig. 2(h) are held fixed, the new stage, labeled “S,” will translate when the new intermediate body, labeled “I,” is rotated about the red line shown. If the grounded bodies are joined together, handles are added to the intermediate body to aid with its rotation, and a rigid shaft with a block at its end is added to the mechanism’s stage as shown in Fig. 2(i), this transmission behavior can be seen using the displacement color coding of Fig. 2(j). Note that as long as all the flexures are modeled as ideal constraints [29,30] (i.e., the constraint lines that model their geometry are assumed to resist extension and compression along their axes with infinite stiffness but can freely deform in all other directions with infinite compliance), the transmission ratio of the resulting mechanism, Δx/Δα, (Fig. 2(j)) will be equal to the negative pitch, −p, of the first module’s screw DOF (Figs. 2(a),–2(c)). If the mechanism is back-driven such that its rotational input is now the output and its translational output is now the input, the transmission ratio will be the inverse of this negative pitch, −1/p. Thus, the transmission ratio of the mechanism of Figs. 2(i) and 2(j) can be tuned as desired, regardless of what end is the actuated input, by adjusting the geometry of the three bent-blade flexures of the first module (Fig. 2(c)) according to Eq. (1). Note, however, since the flexures of the mechanism only approach the theoretical behavior of ideal constraints, the negative of the pitch of the first module’s screw DOF, or its inverse, will only approximate the actual mechanism’s transmission ratios. Stiffness matrices are required to calculate the exact transmission ratios of the mechanism as explained in Sec. 2.4.

This section briefly reviews the relevant principles of the freedom and constraint topologies (FACT) design approach [38–40] that are necessary to understand how the flexure topology of the transmission mechanism of Fig. 1 was generated. The FACT approach uses a library of geometric shapes (Fig. 3) that visually represent the mathematics of screw theory [41–43]. One set of shapes, called freedom spaces, represent all the ways a compliant system could freely deform (i.e., the linear combination of its DOFs). They consist of red lines that represent rotations, green lines that represent screws, and black arrows that represent translations. Freedom spaces are organized into the columns of the FACT library (Fig. 3) according to the number of their DOFs and they are arranged on the left side of each column. Another set of shapes, called constraint spaces, represent the space within which flexures could be located and oriented such that they achieve the motions within their complementary freedom space. Constraint spaces consist of blue lines that represent pure-force constraint lines and they are positioned to the right of their complementary freedom spaces in the FACT library of Fig. 3. The detailed geometry of each of the FACT shapes is rigorously described in previous publications [30] but they are visually depicted in Fig. 3 as a reference for the purposes of this paper. When designing compliant mechanisms using the geometric shapes of the FACT library, it is important to recognize that the freedom spaces of flexures arranged in series linearly combine or sum together, while the constraint spaces of flexures linearly combine when arranged in parallel [40].

This section details how each of the three compliant modules that constitute the mechanism of Fig. 1 was designed using the FACT approach reviewed in Sec. 2.2. The bent-blade flexures of the first module of Fig. 2(a) consist of two serially stacked blade flexures. The freedom and constraint space type that pertains to each of these blade flexures is 3 DOF Type 1 from the FACT library of Fig. 3, which is shown enlarged in Fig. 4(a). The freedom space consists of every red rotation line that lies on the plane of the blade flexure and a black translation arrow that is perpendicular to the flexure’s plane. The complementary constraint space is a plane of blue constraint lines that are coplanar with the red lines of the freedom space. Since the two blade flexures that constitute a bent-blade flexure are arranged in series as shown in Fig. 4(b), their freedom spaces must be summed together to identify the freedom space of the stage, labeled “S.” The resulting freedom space would be the 5 DOF Type 1 freedom space from the FACT library of Fig. 3, which is shown enlarged in Fig. 4(c). This freedom space consists of an infinite number of red planes of rotation lines that all intersect along the bend line of the bent-blade flexure. It also contains a disk of black translation arrows that are perpendicular to the axis of the bend line and screws that lie on the surfaces of circular hyperboloids and disks as described in previous works [30]. The complementary constraint space of this freedom space is a single blue constraint line that is collinear with the bend line of the bent-blade flexure (Fig. 4(d)). Thus, this single blue constraint line is the effective constraint space of a bent-blade flexure. Since the first module consists of three identical bent-blade flexures arranged in parallel, as shown in Fig. 4(e), each of their blue constraint lines can be summed together to identify the constraint space of the entire module. The resulting constraint space is a circular hyperboloid of blue constraint lines as shown in Fig. 4(f). The complementary freedom space is the 3 DOF Type 7 freedom space from the FACT library of Fig. 3 and is shown enlarged in Fig. 4(g). It consists of red rotation lines that lie on the same circular hyperboloid of the complementary constraint space but are angled in the opposite direction. The freedom space also contains many screws described in previous works [30], including the critical screw of the module shown in Fig. 2(a), which lies along the axis of the freedom space’s circular hyperboloid as shown labeled with its pitch, p, in Fig. 4(g). Thus, the first module not only achieves the screw DOF shown in Fig. 2(a), but it also achieves all the other rotations and screws shown in the freedom space of Fig. 4(g).

The FACT approach could be used to design an alternative module that only achieves the single screw DOF desired (e.g., the over-constrained module consisting of three pairs of identical axisymmetric bent-blade flexures shown in Fig. 4(h)), but such designs would increase the size, weight, and complexity of the overall mechanism and would unnecessarily over-constrain it, which would reduce its precision. Thus, the compact and exactly-constrained [29,30] module of Fig. 2(a) was designed to produce the desired large-range high-precision screw DOF required for the mechanism’s transmission capabilities. Note that the extra unwanted DOFs of the module of Fig. 2(a) are eliminated when the module is combined with the other two modules within the overall transmission mechanism of Fig. 2(i), so they don’t introduce any negative issues such as under-constraint [30] in the final mechanism.

The FACT approach was also used to design the second module shown in Fig. 2(d). The module’s two blade flexures each possess the same freedom and constraint spaces shown in Fig. 4(a). And since they are arranged in parallel, the constraint spaces of the blade flexures can be summed together (Fig. 5(a)) to produce the module’s effective constraint space. The resulting freedom and constraint space type of the module is thus 1 DOF Type 1 from the FACT library of Fig. 3, shown enlarged in Fig. 5(b). The constraint space consists of all the blue planes of constraint lines that intersect along the axis where the planes of the two blade flexures intersect. The complementary freedom space is a single red rotation line (Fig. 5(b)) that is collinear with the aforementioned intersection line (Figs. 2(d) and 2(e)). Note that the resulting module is technically over-constrained by one independent constraint line within the two blade flexures collectively, but this degree of over-constraint is necessary to achieve symmetry. A notch could be cut into one of the blade flexures to alleviate the over-constraint if precision takes priority over symmetry and fabrication simplicity.

The FACT approach was also used to design the third module shown in Fig. 2(f). As mentioned previously, the module consists of three axisymmetric flexures, which in turn, are made of eight triangular systems. A single triangular system within the module (Fig. 6(a)) is comprised of a blade flexure and a bent-blade flexure arranged in parallel. Thus, the constraint spaces of both the blade flexure (Fig. 4(a)) and the bent-blade flexure (Figs. 4(c) and 4(d)) can be summed to produce the effective constraint space of the triangular system of Fig. 6(a), which is 2 DOF Type 2 in the FACT library of Fig. 3, and is shown enlarged in Fig. 6(b). The constraint space consists of a blue plane of constraint lines and an infinite box of parallel constraint lines. The constraint space’s complementary freedom space consists of a plane of red rotation lines that are parallel to the blue constraint lines in the constraint space’s box and lie on the same plane as the constraint space’s blue plane. The freedom space also possesses a black translation arrow that is perpendicular to the plane of parallel red rotation lines. The plane of the effective freedom space of the single triangular system of Fig. 6(c) lies on the plane of the blade flexure and the freedom space’s parallel red rotation lines are parallel to the blue constraint line of the bent-blade flexure shown in Fig. 6(a). When two triangular systems are arranged in series such that the blade flexures are coplanar as shown in Fig. 6(d), their individual freedom spaces can be summed together to produce the new element’s effective freedom space, which is the same freedom space shown in Fig. 6(b). Thus, as more triangular systems are joined together in this way, the resulting element will possess the same freedom space as shown in Fig. 6(e). If, however, the element of Fig. 6(e) is serially arranged with itself such that the red rotation lines of both spaces are still parallel but no longer coplanar, as shown in Fig. 6(f), the freedom spaces of each element (Fig. 6(e)) can be summed (Fig. 6(f)) to produce the resulting element’s freedom space. This space belongs within 3 DOF Type 2 in the FACT library of Fig. 3, but it is shown enlarged in Fig. 6(g). The freedom space is an infinitely large box of red rotation lines that are parallel to the red rotation lines shown in Fig. 6(f). The space also consists of a disk of black translation arrows that point in all directions perpendicular to these rotation lines. The freedom space’s complementary constraint space is also an infinite box, but it contains blue constraint lines that are all parallel to the red rotation lines of the freedom space. Since the third module (Fig. 6(h)) consists of three axisymmetric elements of the kind shown in Fig. 6(f) that are arranged in parallel, their effective constraint spaces can each be summed together (Fig. 6(h)) to produce the constraint space of the module itself. The resulting constraint space belongs within 1 DOF Type 3 in the FACT library of Fig. 3, shown enlarged in Fig. 6(i). It consists of an infinite number of stacked parallel blue planes that consist of constraint lines. The constraint space’s complementary freedom space is a single black translation arrow that is orthogonal to the blue planes of the constraint space. This freedom space is shown in Figs. 2(f) and 2(g). Note that although the resulting module of Fig. 2(f) is over-constrained by the three axisymmetric elements of Fig. 6(f) with four independent constraint lines, the over-constraint provides important symmetry that guides the desired translational DOF over its full range of motion without parasitic error (i.e., drift of the stage in unwanted directions).

Once all three modules are assembled together within the transmission mechanism of Figs. 2(h) and 2(i), the final system is not under-constrained [30]. In other words, if the body, labeled “I” in Fig. 2(h) is held fixed, all the other rigid bodies in the system are fully constrained to not move with any stray DOFs. Likewise, if the body, labeled “S” in Fig. 2(h), is held fixed, all the other rigid bodies are also unable to move. Thus, an actuator attached to either the body labeled “I” or “S” can fully control the motion of all the other moving bodies in the system. Moreover, system’s that are not under-constrained are not as susceptible to unwanted vibrations or other dynamic issues as systems that possess under-constraint. Regardless of which of the rigid bodies labeled “I” or “S” in Fig. 2(h) is considered the system’s output stage, the final transmission mechanism of Fig. 2(i) is over-constrained by seven independent constraints. This over-constraint provides important symmetry that guides all the moving bodies with their intended motions over their full range.

This section provides the theory necessary to calculate the transmission ratio of the mechanism of Fig. 1 using stiffness matrices. Although Eq. (1) can be used to rapidly determine a good starting point for setting the geometric parameters of the mechanism during the initial phase of the design process, the transmission ratio that results from using this equation will only be a rough approximation of its actual transmission ratio. The reason is that no flexure truly behaves like an ideal constraint as described in Sec. 2.1. Thus, once the rough topology is set using the ideal constraint assumptions of Eq. (1), a local optimization sweep of the geometric parameters of the topology can be performed to refine its geometry (i.e., the lengths, widths, and thicknesses of its flexures) so that the mechanism achieves the exact transmission ratio desired once a material has been assigned using stiffness matrices.

According to the updated labeling in Fig. 7(d), where T_i is a 6 × 1 input displacement twist vector that describes the translational displacement of the intermediate body, labeled “I” in Fig. 7(d), and T_o is a 6 × 1 output displacement twist vector that describes the rotational displacement of the output stage, labeled “S” in Fig. 7(d). The new transmission ratio of the mechanism of Fig 7(d) in units of rad/m could be calculated by selecting the third component of the T_o vector that results from inputting T_i = [0, 0, 0, 0, 0, 1 m]^T into Eq. (5), according to the coordinate system shown in Fig. 7(d). Note that the inverse of this new transmission ratio is different from the previous transmission ratio when the mechanism was driven with the rotational body instead of the translational body as the input. This observation becomes obvious with the recognition that the stiffness matrix [k₂] is not even applicable in the scenario of Eq. (3) and similarly, the stiffness matrix [k₃] is not applicable in the scenario of Eq. (5).

This section describes how the transmission mechanism of Fig. 1 was fabricated and tested. The mechanism was assembled from 15 additively fabricated parts (Fig. 8), 9 of which are unique, using the fused filament fabrication method. The parts were made of Vanilla White Prusament polylactic acid using an Original Prusa i3 Mk3S + printer with a 0.4 mm nozzle. The parts come complete with press-fit features to help facilitate their Lego-like assembly, but they were also glued together using a cyanoacrylate adhesive to further prevent slip during use. The parts were designed so that the theoretical pitch, p, of the screw mechanism of Fig. 2(c) would be −300/π mm/rad (i.e., the pitch that would be achieved if the mechanism’s flexures were ideal constraints) with geometric parameters d and θ equal to 300/π mm and 3π/4 rad respectively according to Eq. (1).

Data were collected from the assembled mechanism of Fig. 1 by mounting it to an 18ʺ × 18ʺ aluminum optical breadboard (Newport SA2) using bolts. The mechanism was first driven at the end that rotates in a counterclockwise (CCW) direction by using two weight buckets tied to cables that attach to its handles as shown in Fig. 9(a). One of the cables was wrapped over a pulley with high-performance roller bearings so that when equal weights would be placed in the buckets, a pure moment input load would be imparted on the mechanism’s rotational end. Two variance indicators (Mitutoyo 543-792) were mounted 70 mm from the center of rotation as shown in Fig. 9(a) to measure the input’s resulting rotational angle, Δα. The translational displacement, Δx, that resulted at the output translational end of the mechanism was measured using another pre-compressed variance indicator as shown in Fig. 9(b). Measurement increments were taken after 100 g weights were put into each bucket and after a heavy mass in a guided thumper (Fig. 9(b)) was lifted and dropped three times so that the polymer chains within the freshly deformed 3D-printed flexures could settle to a stable measurement. The mechanism was then driven at the end that rotates in a clockwise (CW) direction by using the same two weight buckets tied to cables that attach to its handles as shown with the altered configuration shown in Fig. 9(c). The same procedure was conducted to measure the input rotational angle and the corresponding output translational displacement (Fig. 9(d)) for this CW loading scenario.

The mechanism was then driven at its other end, which translates, with a pushing direction as the input by using a linear actuator optical XY mount (Thorlabs XYF1) as shown in Fig. 9(e). A variance indicator was used to measure the input’s resulting translational displacement, Δd. Two other variance indicators were mounted 70 mm from the center of the output rotational body as shown in Fig. 9(e) to measure the output’s resulting rotational angle, Δα, in the CCW direction. Measurement increments were taken after ∼0.5 mm input displacement steps with the thumper mass lifted and dropped three times between each step. The optical mount actuator was then pre-extended and attached to a tab on the same translational end of the mechanism to drive the mechanism with a pulling displacement as shown in Fig. 9(f). The same procedure was conducted to measure the resulting input translation displacement and the corresponding output CW rotational angle (Fig. 9(d)) for this pulling scenario.

This section discusses the results of the experimental tests detailed in Sec. 2.5. The results of the transmission mechanism’s translation displacement, Δx, plotted against its rotation angle, Δα, for the four different loading scenarios of Fig. 1, but measured using the test setups shown in Fig. 9, are shown in Fig. 10. Specifically, the measured increment results of the input CCW rotational setup of Figs. 9(a) and 9(b) are shown as blue dots in Fig. 10. A solid blue line with a slope of 1.367 mm/deg was fit to the data with an R² value of 0.9999. The measured increment results of the input CW rotational setup of Figs. 9(c) and 9(d) are shown as purple dots in Fig. 10. A solid purple line with a slope of 1.344 mm/deg was fit to the data with an R² value of 0.9998. The measured increment results of the input-pushing translational setup of Fig. 9(e) are shown as red dots in Fig. 10. A solid red line with a slope of 1.998 mm/deg was fit to the data with an R² value of 0.9997. The measured increment results of the input-pulling translational setup of Fig. 9(f) are shown as orange dots in Fig. 10. A solid orange line with a slope of 1.791 mm/deg was fit to the data with an R² value of 0.9999.

Note from the results of Fig. 10 that regardless of which end of the mechanism is driven as the input and regardless of which direction that input end is driven, the mechanism achieves remarkable linearity over its full range of deformation such that its transmission ratios (i.e., the slopes of the blue and purple lines and the inverse slopes of the red and orange lines in Fig. 10) remain largely constant. Additionally, note that as long as the same end of the mechanism is driven as the input, the transmission ratios are similar for both directions of loading (i.e., CCW or CW for the rotational input and pushing or pulling for the translational input). The slight differences between the slopes of the two input rotations and the slopes of the two input translations are most likely the result of how the mechanism was actuated using different setups for each of the two directions in each scenario. The transmission ratio achieved by driving the mechanism from one end is, however, substantially different than the inverse of the transmission ratio achieved by driving the same mechanism at the other end. The amounts that these values differ (i.e., the slopes of the blue and red lines of Fig. 10 and the slopes of the purple and orange lines) are a measure of how far the flexures in the mechanism stray from being ideal constraints as defined in Sec. 2.1. If the flexures were ideal constraints, they would achieve the translation-displacement-to-rotation-angle response shown as the dashed black line in Fig. 10. The transmission ratio of such an ideally constrained mechanism driven at the rotation end would be 1.667 mm/deg and the transmission ratio of the same mechanism driven at the translation end would be equal to the inverse of the same value. Note that this value (i.e., 1.667 mm/deg or 300/π mm/rad) is equal to the negative pitch, −p, of the module shown in Figs. 2(a),–2(c) with its final fabricated geometric parameters provided in Sec. 2.5.

A compliant transmission mechanism was introduced that can convert large-range rotational motion into large-range translational motion and vice versa with a desired and unchanging transmission ratio over its full range of deformation. The FACT synthesis approach was used to generate the design’s topology and a simple equation was provided to help designers select key geometric parameters within the topology to ensure that the resulting initial version of the design will approximate the transmission ratios desired. An analytical stiffness-matrix-based approach was also provided to then enable the rapid optimization of all the topology’s geometric parameters to refine them around the initial solution such that a final solution could be generated that achieves the exact transmission ratios desired. The approach was also used to prove why the transmission ratio of the mechanism driven at one end can never equal the inverse of the transmission ratio of the same mechanism driven at its other end. An example design was fabricated and tested to measure the different transmission ratios achieved by driving the mechanism at its two different ends in either direction. Impressive linearity was demonstrated in all cases, suggesting that the mechanism’s transmission ratios would not appreciably change over the mechanism’s full range of motion. Thus, this work paves the way for enabling the clean amplification or attenuation of the displacement, load, or resolution of high-precision rotary or linear actuators without sacrificing their precision.

This work was supported in part by solidworks and their parent company, Dassault Systèmes, under fund number 442561-HJ-79007. This work was also supported in part by the Air Force Office of Scientific Research (AFOSR) under fund number FA9550-22-1-0008. Program officers Shrikant Savant and Byung “Les” Lee are gratefully acknowledged respectively.

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.