## Abstract

Compliant mechanisms gain motion through the elastic deformation of the monolithic flexible elements. The geometric design freedom of metallic additive manufacturing enables the fabrication of complex and three-dimensional (3D) compliant elements within mechanisms previously too complicated to produce. However, the design of metallic additive manufactured mechanisms faces various challenges of manufacturing restrictions, such as avoiding critical overhanging geometries and minimizing the amount of support structure, which has been reported in a few cases. This paper presents a synthesis approach for translational compliant elements, involving building blocks based on leaf-type springs and covering building orientations between 0 deg and 90 deg. In particular, this range is approached by the synthesis of self-supported 3D building blocks with orientations of 0 deg, 45 deg and 90 deg. The compliant elements are built based on linear and circular plane curves and compared numerically according to their mechanical performance to create preferable building blocks. The applicability of the presented procedure and the manufacturability of the compliant mechanisms are proven by printing individual 3D building blocks and their serial aggregation with laser-based powder bed fusion. Consequently, several prototypes are demonstrated, including a bistable switch mechanism and a large displaceable rotational spring joint. In addition, a small-scale highly maneuverable segment of a surgical instrument with a grasping mechanism at the distal end is proposed.

## 1 Introduction

Compliant mechanisms (CMs) offer several advantages such as reduced part number and assembly effort, which make them widely sought in various applications such as a minimally invasive surgery [1], robotic actuators [2], or space deployable structures [3], and the ability to eliminate the presence of friction, wear, and backslash [4], which give them an edge over rigid-body mechanisms. These advantages are driven by the monolithic integration of flexible beam structures called compliant elements (CEs), with thin cross-sections that undergo bending and elasticity-dominated deformation. Although the motion is obtained through the deformation of CEs, it is limited by the elastic strength of the deformed material [4,5].

The spatial freedom in additive manufacturing (AM) produces parts in a single step by repeated layer-wise addition of a metallic or polymer material [6]. The “free complexity” of AM enables three-dimensional (3D) and complex CEs (usually the combination of simpler or more complex CEs [79]) that were previously too complicated to fabricate using traditional manufacturing processes [10], including computer-controlled numerical (CNC) milling, wire electrical discharge machining (EDM), or laser cutting, in which the material is continuously machined out of bulk material. Several spatial additive manufactured mechanisms were demonstrated using polymer-based AM techniques such as material extrusion [1114], selective laser sintering [1517], or vat photopolymerization [1820]. However, metallic AM, such as laser-based powder bed fusion (PBF) or electron beam melting (EBM), has been realized only in a few cases [2124]. For example, for the design of a spacecraft antenna, Merriam et al. [25] developed a two-degrees-of-freedom (2-DOF) pointing mechanism using EBM and titanium alloy (Ti6Al4V), showcasing added values with possibilities that are often seen as a potential for metallic AM [2628].

The lack of metallic printed CMs could be attributed to manufacturing restrictions with respect to the critical overhanging design geometries and the reduction of the amount of support structure, which compromise the design freedom and build direction of metallic AM [29,30]. The support structure is made of the same material as the part and enables heat transfer from the laser melt pool to the build platform. It becomes problematic to remove these structures manually during post-processing, which is a time-consuming and expensive process, aside from it possibly causing damages to thin and fragile flexible elements [31]. It thus becomes necessary to take into account the process-specific design guidelines for the manufacture of a CE without a support structure [29,30]. The literature provided in this paper demonstrates very well that the build angle between a downward-faced surface and the x–y plane of the build platform is self-supported (see Fig. 1(a)) [29]. Specifically, the downward-faced surface depends on the CE orientation (δOri) on the build platform [3234], which is defined in Fig. 1(a) as the clockwise and parallel rotational angle of the build direction (z-axis) to the x–y plane of the build platform.

Merriam et al. approached the design of the 2-DOF pointing mechanism through the assembly of several 3D CEs [25]. The elements were manufactured by choosing the well-known cross-axis flexural pivot with $δOri=0deg$ to the build direction. However, 3D CEs with both a CE orientation and a translational displacement (axis of motion) in 0 deg, 45 deg, and 90 deg have not yet been synthesized for metallic AM. Figure 1(a) illustrates very well with the helical spring that the amount of required support structure is related to its spatial alignment to the build direction and that such a relation applies to both the profile and base of the spring (Fig. 1(b)). The profile undergoes elastic deformation and can be considered a simple building block of the serial aggregated CE, whereas the base serves either as a load introduction or as a rigid body.

This paper utilizes the above definitions to present an approach for a building block design based on leaf-type flexure beams to synthesize 3D CEs by addressing the geometrical overhanging constraint of metallic AM. It especially considers translational CEs in the orientation of $δOri=0deg$, $δOri=45deg$, and $δOri=90deg$ and regards the coverage of the orientation between $0deg$ and $90deg$. The design of the building block synthesis is founded on creating self-supported DEs acting as a basic two-dimensional (2D) building unit for the formation of the CEs. Figure 2 gives an overview of the organization of this paper and the steps involved in approaching the building block synthesis. In the first step, DEs are derived from planar linear and circular curves (Sec. 2.1). Subsequently, a library of base and profile curves is created by selecting and combining the DEs (Sec. 2.2). The solid geometry of the 3D building blocks is constituted by the base and profile curves (Sec. 3.1). The building blocks are then evaluated and compared numerically with respect to their elastic displacement yields, applied compressive force, and stored energy within the overall material volume (Sec. 3.2) and the manufacturability of the PBF printed elements is ensured both as a building block and as a unit for serial aggregation. In addition, several physical prototypes and application ideas demonstrating the applicability of the building blocks are presented (Sec. 4). Conclusive statements and an overview of future work are provided at the end (Sec. 5).

## 2 Self-Supported Two-Dimensional Design Elements

### 2.1 Analytical Derivation of Design Elements.

A self-supported design element in metallic AM is defined by the geometry of its curves. The build angle $γ≥45deg$ restricts the curve’s inclination between element boundaries and defines the curve’s tangential angle at starting point A (θA) and ending point B (θB) (see Fig. 3). The plane curves Cn defined with an angle of $θA≥θB≥γ≥45deg$ ensure the self-supported character of the design element (DE). The curve selection in this work is focused on the first (n = 1) and second (n = 2) degrees of algebraic, both linear and circular, curves. A pair of DEs is based on linear curves C145 and C190, and another pair is based on circular curves C245 and C290, with a constant height H along the z-axis between the bounded points. Here, the minimum and maximum self-supported curves can be expressed by enclosing the tangential angles θA and θB between 45 deg and 90 deg. DEs C145 (Fig. 3(a)) and C190 (Fig. 3(b)) are assigned the tangential angles of $θA=θB=45deg$ and $θA=θB=90deg$, respectively, as described in a Cartesian coordinate system with Eq. (1). More importantly, DEs C145 and C190 can be described by choosing xA and zA as starting points, with the curve inclination $m=cot(θA)$.
$x=cot(θA)⋅(z−zA)+xA$
(1)
Figures 3(c) and 3(d) show the curve inclination angles of two circular DEs at points A and B. Element C245 starts with a tangential angle $θA=45deg$ and ends with $θB=90deg$, whereas C290 begins with $θA=90deg$ and ends with $θB=45deg$. Both curves are defined in a Cartesian coordinate system using the equation of a circle in Eq. (2) with a center point (xM, zM) and a radius R. Here, coordinates x and z are taken in a set containing all points on the curve C245 (Eq. (3)) with the starting point A(xA, zA), whereas Eq. (4) reveals the points on the curve C290 with xA and zA.
$(x−xM)2+(z−zM)2=R2$
(2)

$(x−(xA−Hcos(θA)⋅sin(θB)))2+(z−(zA+H))2=(Hcos(θA))2$
(3)

$(x−(xA+Hcos(θB)))2+(z−zA)2=(Hcos(θB))2$
(4)

### 2.2 Library of Profile and Base Curves.

The base and profile curves of the CEs are constructed by selecting and combining the DEs and their reflective counterparts (C145′, C245′, and C290′) resulting in a library of self-supported base and profile curves (see Figs. 4 and 5). The profile library is built by serially connecting two DEs as exemplarily shown in Eq. (5) with the help of the profile curve C145 − C145′.
$f(z)={(z−zA)+xAfor0≤z≤H2−(z−zB)+xBforH2≤z≤H$
(5)
Here, the starting point A(xA, zA) of the second DE C145′ is added to the ending point B(xB, zB) of the first element C145, and the profile height H between the starting point A(xA, zA) and the ending point C(xC, zC) is kept equal to the height of the DEs. A combination of more than two elements is conceivable, but it is omitted in this work for the sake of clarity. Accordingly, the profile library of the DE curves extends to 36 profile curves, but several curve combinations are excluded because of their similarity with other combinations (more faintly illustrated in Fig. 4), bringing the total down to 18 profile curves. The excluded combinations represent a redundant reflection either along the x-axis (e.g., C245 − C145′) or the z-axis (e.g., C190 − C290′).

By compiling the base library, the resulting profiles are mirrored at their starting point (A-base), connecting point (B-base), and ending point (C-base) along the z-axis. Similar to the profile library construction, redundant base curves are excluded (e.g., C: C290 − C245′). The equal distance from the axis of motion (translational displacement) to the vertices of the base is assumed by regarding rotational symmetric base curves having concyclic vertices inscribed in a circle. The circumcenter is concurrent with both the axis of motion and the centroid point of the base curve, as shown in Fig. 6(b). Curves not fulfilling this assumption promote deviation from the desired axis of motion. The resulting neglected bases (e.g., A: C245 − C290 or A: C290 − C245′) are illustrated more faintly in Fig. 5. For reasons of traceability, the synthesis method is demonstrated in the next sections using the base B: C145 − C145′. The applicability of base A: C145 − C145′ is also proven in Sec. 4 for the prototype of a surgical instrument.

## 3 Synthesis of Three-Dimensional Building Blocks

### 3.1 Forming Three-Dimensional Geometrical Compliant Elements.

The geometry of the 3D building blocks was assembled using the thickened base and profile curves. The solid geometry of the profile curves was built with a constant, rectangular beam cross-section, where each beam element has out-of-plane and in-plane wall thicknesses of w = 2 mm and t = 0.5 mm, respectively (Fig. 6(a)). As shown in Eq. (6), the displacement Δs of a rectangular beam cross-section is inversely proportional to the thickness t. Thus, choosing a minimum thickness of t = 0.5 mm could provide an additional safety margin of 0.1 mm to the minimum PBF-manufacturable element size of 0.4 ± 0.02 mm [30]. The ratio of the material yield strength σyield and Young’s modulus E was kept equal. Consequently, the 3D building blocks differed in the length l of their profile beams.
$Δs∼2⋅l2⋅σyield3⋅t⋅E$
(6)
Similarly, the base was thickened by a 2-mm out-of-plane thickness for it to be aligned with the projected outer contour w. The geometry of the base B: C145 − C145′ is shown in Fig. 6(b), and CE90 deg is formed through the assembly of base B: C145 − C145′ and profile C245 − C290, as depicted in Fig. 6(c). The starting and ending points of the profiles are enclosed by edges E1E4, with the starting point A attached to the rigid base and the ending point C mounted to the load base, creating a rise in the profile along the base curve to the center point M. The base plane is perpendicular to the CE’s orientation, and the parallel and rotational symmetric arrangement of the four beam elements ensures the alignment of the axis of motion through the segment [M1M2] (Fig. 6(d)).
Despite the self-supported characters of the profile and the base beam, the self-supported geometry of the building blocks was defined by the beam inclination angle α and the base leg angle β, which can be visualized for all three CEδOri in Figs. 7(a)7(c). The self-supported building space was represented by a conical surface, where the angle between this surface and the build platform is $γ≥45deg$. Given both angles δOri and γ, the ranges of the minimum and maximum values of α and β were obtained using Eq. (7), as summarized in Table 3. The trigonometrical derivation of Eq. (7) is described and visualized in Fig. 14.
$sin(δOri)⋅sin(α)=cos(δOri)⋅cos(β)⋅cos(α)+cos(γ)2−(sin(β)⋅cos(α))2$
(7)
The inclination angle α is the sharpest angle that is included in the rigid base plane and the profile line [AB] that inclines along the conical surface. All of the values of α chosen in this work for orientating CE0 deg, CE45 deg, and CE90 deg, which are related to the leg angle β of the bases, can be found in Table 1. Specifically, β was defined as the mid-angle between the two rays [M1E1] and [M1E2] was valued originally by $β=45deg$ because of the symmetric character of the rays.

The original base angle of $β=45deg$ resulted in $α=0deg$ for CE90 deg with both bases in contact and with no motion to be realized. Consequently, β was reduced by 5 deg to $β=40deg$, leading to an increased $α=22deg$. Note that a smaller base angle led to an increased height along the z-axis, which, in turn, increased the building time. Thus, a wide combination of α and β could be selected to form this CE.

Accordingly, CE45 deg resulted in $α=90deg$ and $β=0deg$, as shown in Fig. 7(b). Here, the two rays of the base merged to form two profile beams.

As shown in Eq. (7), the integration range of $δOri=0deg$ to $δOri=90deg$ can be covered by CE0 deg, CE45 deg, and CE90 deg by adjustment of the design angles of α and β or by extending the process limitations of the build angle γ. Varying α or β reduces the compliance (or its inverse stiffness) of the CE. Thus, it is neither recommended nor followed in this work. However, the integration range was encompassed by lowering the build angle from $γ≥45deg$ to $γ≥25deg$, allowing CE0 deg to cover the range of $δOri=0deg$ to $δOri=25deg$. Meanwhile, CE45 deg enclosed the orientation from $δOri=25deg$ to $δOri=75deg$. With CE90 deg spanning the field from $δOri=60deg$ to $δOri=90deg$, an overlap of $15deg$ was attained by both CE45 deg and CE90 deg. A summary of the covered ranges is presented in Table 1.

### 3.2 Numerical Comparison of the Building Blocks.

The compliance of the 3D building blocks was numerically evaluated and compared using finite element analysis (FEA). The compressive force applied at the load base, defined as F, was continuously increased until the 0.2% yield stress before the plastic deformation of the CEs was reached. From the analysis, an elastic stress-strain material relationship could be assumed. Because of the large deformation of the bending dominated beam profile, a nonlinear FEA was implemented using the structural mechanics solver of ansys workbench. The surface of the 3D model was meshed with a 0.167-mm finely sized tetrahedral element (10-noded) that divides the thickness of the profile beams into three mesh elements. The translation of the load base, defined as Δs, was taken as the displacement of the loaded base along the axis of motion between the center point M1 of the rigid base and center point M2 of the loading base. Additionally, the building blocks were compared using Δs and F, with the plotted results for CE0 deg (Fig. 8) and CE90 deg (Fig. 9) considering all of the eligible profiles from Fig. 4. Furthermore, the performance of these building blocks was compared with the average strain energy density, which represents the overall stored energy per building block volume and which indicates the material utilization of the CE [35,36].

For the numerical evaluation of the CE, the martensitic precipitation-hardening stainless steel 17-4 PH was chosen as a suitable material candidate because of its high modulus of resilience, described by a low Young’s modulus E and a high yield strength σyield. As Eq. (6) suggests, materials with higher resilience can be defined as optimum CE materials considering their enlarged elastic deflections [4]. The material parameters given by the feedstock supplier are presented in Table 2 [37].

Among the 18 building blocks of the CE0 deg, the profile C145 − C145′ achieved the highest displacement of Δs = 4.84 mm with an applied compressive force of F = 36.9 N and an average strain energy density of 3.72 × 10−1 mJ/mm3. The nearly linear force-displacement relation (or stiffness behavior) is shown complementarily in Fig. 15. Moreover, the CE0 degC145 − C145 (marked with a red star in Fig. 8) could not be deformed to the yield strength because of its instability caused by buckling.

The same instability was observed among some of the 14 CE90 deg building blocks (Fig. 9). A low average strain energy density was especially depicted in the building block of the profiles C145 − C145 and C245 − C290, which is a similar case of the block’s buckling instability caused by the rigid boundary condition of the profiles. For these unstable building blocks, the maximum displacement was calculated by continuously increasing the applied load until the first buckling mode with the lowest critical load was reached. As a result, they were not loaded to the yield strength. Nevertheless, this instability could be made steady by changing the boundary condition. A serial connection of two identical CEs (2x C145 − C145 and 2x C245 − C290) enabled a stabilized displacement to the yield strength with a raised amount of the overall stored energy density. The degressive stiffness behavior of C245 − C290 was linearized by the serial connection of 2x C245 − C290 (see Fig. 15). In particular, CE 2x C245 − C290 yielded a 9.6-fold increase in its displacement, 1.2-fold decrease in force, and three-fold increase in its averaged strain energy density. The Δs of 2x C145 − C145 rose from 0.05 to 15.27 mm and F fell from 99 to 17 N with a stored energy density of 2.71 × 10−1 mJ/mm3. In a separate scenario, CE C190 − C290 (marked with a blue star in Fig. 9), despite its high Δs, exhibited a low average strain energy density, which is attribute to the short distance [M1M2]. Consequently, the rigid and loading base made contact before the yield strength was reached. However, several building blocks (e.g., C190 − C245, C245 − C145, and C190 − C145) showed promising and preferable results with high elastic displacement and large storable energy per material volume. In contrast to CE90 deg, the synthesis of the CE45 deg resulted in only one building block, having a translation of Δs = 4.47 mm, an applied force of F = 10.2 N, and a stored energy of 1.63 × 10−1 mJ/mm3 (based on the profile C145 − C145′), whereas all of its remaining building blocks could not ensure translational symmetric displacement along the axis of motion and were consequently ineligible for such an orientation. Moreover, the drift was perpendicular to the axis of motion, in a direction parallel to the upper edge of the base. The slightly linear stiffness behavior of the preferable CE45 deg building block is described together with a selected set of CE0 deg and CE90 deg building blocks in Fig. 15.

## 4 Manufacturability and Potential Use-Cases of the Building Blocks

The printability and applicability of the CEs were validated using various examples of PBF CEs fabricated both as individual or serial aggregations of the 3D building blocks. The building blocks of CE0 deg, CE45 deg, and CE90 deg were produced with the PBF machine Mlab Cusing R of Concept Laser using the stainless-steel material 17-4 PH and with a machine build angle of $γ≥45deg$. Some of these PBF building blocks are displayed in Fig. 10(a).

The additive manufactured CE90 degC245 − C290 demonstrated an unstable buckling behavior (Fig. 10(b)), which is consistent with that observed in the numerical simulations. As mentioned in the previous section, this behavior could be stabilized by changing the boundary condition of the building blocks and by letting the profile beam be displaced in the orientation of the CE. The motion of the unstable and stabilized CE90 deg is shown in Fig. 10(c). With regard to such an unstable character of certain CEs, it can be potentially utilized in the form of a bistable mechanism that works as a switch, latch, or release device in space systems. A certain prototype of a bistable switch mechanism is proposed in Fig. 11. Here, the CE90 deg is integrated monolithically in a cage-like base housing, halved for the purpose of presentation. The deflection from the as-fabricated position (first stable position) into the second stable position beyond the threshold distance [M1M2] was realized by replacing the rigid base B: C145 − C145′ with A: C145 − C145′. The first and second stable positions of the beam are visualized in the top view in Figs. 11(a) and 11(b), respectively.

Accordingly, the serial aggregation of CEs was realized by chaining up the building blocks, thereby increasing the overall displacement of the CEs. Figure 12 describes the resulting motion for a serially aggregated CE90 deg based on the profile C245 − C290 and the base B: C145 − C145′. An enlarged displacement is shown for translational (Fig. 12(a)) and rotational (Fig. 12(b)) movements. The rotation was actuated with a torque wrench such that the circularly grouped building blocks were deformed, a mechanism that works ideally for the design of robotic spring actuators where a monolithic and compact mechanical device with reduced assembly complexity is demanded. The desired compliance can be customized for the user.

Furthermore, the serial arrangement of CEs could extend the number of DOFs of the translational motion from one to three, with the latter (3-DOF) consisting of up/down, forward/backward, and left/right movements. This mechanism has a potential use in a large maneuverable surgical instrument in minimally invasive surgery requiring high steerability for the formation of a continuous curvature. As illustrated in Fig. 13(a), a small-sized prototype of a flexible segment for a surgical instrument could be designed by considering the profile C245 − C290 and the base A: C145 − C145′. Here, a maximal curved angle of nearly 90 deg (Fig. 13(a)) is actuated by pulling four 0.1-mm outer-diameter Nitinol tendon wires. Additionally, a gripping mechanism (Fig. 13(b)) is ideally attached at the distal end.

## 5 Conclusion and Future Work

This paper has presented the synthesis method of a building block for metallic additive manufactured CEs with both the orientation and axis of motion in $δOri=0deg$, $δOri=45deg$, and $δOri=90deg$ by addressing the manufacturing restrictions of the build angle and by minimizing the amount of support structure. The self-supporting characteristic of the spatial CEs was based on linear and circular plane curves, from which various 3D building blocks were synthesized and compared numerically according to their translational motion, applied compressive force, and averaged strain energy density before the yield strength was exceeded. For both CE0 deg and CE45 deg, the building blocks based on the profile C145 − C145′ resulted in maximum compliance (or its nearly linear inverse stiffness) and in a high overall stored energy of the utilized material volume, making them preferable. By contrast, several building blocks for CE90 deg resulted in high translation and distribution of the stored energy, offering a larger selection of promising building blocks with degressive stiffness behavior. The three orientated CEs covered the entire range of $δOri=0deg$ to $δOri=90deg$ via the reduction of the build angle from $γ≥45deg$ to $γ≥25deg$. However, the manufacturability of the surfaces up to $γ≥25deg$ has not yet been reported in the literature and has restricted the findings of this paper. Notwithstanding its limitations, a building angle of $γ≥30deg$, as confirmed in the literature [29,38], resulted in an excluded range of 0 – 90 deg within $δOri=20deg−30deg$.

The experimental characterization of the 3D building blocks and its robustness with the presented numerical simulation is a part of future work. In particular, the geometric accuracy of the printed CEs and its effects on the mechanical performance with respect to manufacturing errors (e.g., surface roughness and thermal stresses), reported in a few cases [24,26], to vary spatially, will be investigated and tested accordingly. Furthermore, the influence of process parameters (e.g., laser power and scanning speed) and post-processing (e.g., heat and surface treatment) on the mechanical characteristics of the CE will be analyzed [24,39]. Despite the foreseen comprehensive testing, the purpose of the presented numerical simulation was limited to comparing and focusing on favorable 3D building blocks.

The manufacturability and applicability of the CEs, both as a building block and as a unit for serial aggregation, were proven using PBF and the stainless-steel material 17-4 PH. The serial arrangement of DEs allowed large displacements and extended the original translational motion by two additional DOFs. The 3-DOF mechanism prototype can be described as a tendon-wire-steered segment of a surgical instrument for minimally invasive surgery. Moreover, the rotational movement demonstrated by a circular arrangement of the CE90 deg building blocks could be potentially applied for a customizable robotic spring actuator. Further, the bistable character of CE90 degC245 − C290 as discussed could be proposed as a bistable switch mechanism for space systems. An extension of the presented design principle to cover the synthesis of rotational CEs for orientations $δOri=0deg,δOri=45deg$, and $δOri=90deg$ is another consideration for a future work.

## Acknowledgment

The author would like to thank Mr. Manuel Biedermann and the anonymous reviewers for their valuable feedback and comments, as well as Mr. Luca Niederhauser for his valuable comments on the design of the steerable surgical instrument.

## Funding Data

• Swiss National Science Foundation (SNSF, Div. II) under Grant No. 178689; Funder ID: 10.13039/501100001711.

## Conflict of Interest

There are no conflicts of interest.

## Nomenclature

• l =

length, mm

•
• w =

out-of-plane thickness, mm

•
• t =

in-plane wall thickness, mm

•
• A =

starting point

•
• B =

ending and connecting point

•
• C =

ending point

•
• E =

Young’s modulus, GPa

•
• F =

compressive force, N

•
• H =

height, mm

•
• α =

beam inclination angle, deg

•
• β =

base leg angle, deg

•
• γ =

build angle, deg

•
• θA =

tangential angle of the curve at starting point A, deg

•
• δOri =

orientation of the compliant element, deg

•
• σyield =

yield strength, MPa

•
• Δs =

translational displacement, mm

•
• ()n =

order of plane curve

### Appendix

Mathematical Derivation of Eq. (7). Figure 14 shows all of the parameters for the derivation of Eq. (7) by means of the CE90 deg building block. This 3D building block is an exemplary synthesis with the profile C145 − C145 and the B-base C145 − C145′. Using trigonometric expressions for the angles α, β, γ and δOri, the following equations can be obtained:
$la1=cos(α)⋅l$
(A1)

$la2=sin(α)⋅l$
(A2)

$lb1=cos(β)⋅la1$
(A3)

$lb2=sin(β)⋅la1$
(A4)

$lc1=cos(δOri)⋅lb1$
(A5)

$f=cos(γ)⋅l$
(A6)

$c=f2−lb22$
(A7)

$sin(δOri)⋅la2=lc1+c$
(A8)
Using substitution, Eqs. (A1)(A8) can be organized to deduce Eqs. (A9)(A12). For instance, Eq. (A9) is derived by including Eqs. (A2) and (A6), and Eq. (A10) can be formulated by substituting Eq. (A3) into Eq. (A9). Equation (A11) is an extended form of Eq. (A10) following the inclusion of Eqs. (A1) and (A7). Further, Eq. (A12) is established by adding Eqs. (A2), (A5), and (A7) in Eq. (A11). Note that dividing Eq. (A12) by l results to Eq. (7).
$sin(δOri)⋅sin(α)⋅l=cos(δOri)⋅lb1+c$
(A9)

$sin(δOri)⋅sin(α)⋅l=cos(δOri)⋅cos(β)⋅la1+c$
(A10)

$sin(δOri)⋅sin(α)⋅l=cos(δOri)⋅cos(β)⋅cos(α)⋅l+f2−lb22$
(A11)

$sin(δOri)⋅sin(α)⋅l=cos(δOri)⋅cos(β)⋅cos(α)⋅l+cos(γ)2−(sin(β)⋅cos(α))2⋅l$
(A12)

Self-Supported Range for α and β. Table 3 provides a list of the respective ranges of α and β assuming $γ≥45deg$ and δOri. For CE0 deg, the profile beam remains in the self-supported building space for $α=45deg$ and $α=90deg$. It should be regarded that increasing α would result in reduced compliance caused by the shortened profile length l. In addition, for $β≤22.5deg$, the neighboring profile beams merge with each other to lower the translational compliance, ranged as a minimum. Similarly, β is defined between $22.5deg$ and $45deg$ for CE90 deg, and the beam profile is bounded within the conical space for $0deg≤α≤40deg$. Moreover, CE45 deg is self-supported for $α=0deg$ or $α=90deg$. Otherwise, the inclination of the profile beam lies below 45 deg. Furthermore, the leg angle is set to 0 deg to be built within the conical surface.

Force-Displacement Relation of the Building Blocks. Figure 15 presents the stiffness curves of a selected group of CE0 deg, CE45 deg, and CE90 deg building blocks. The force-displacement characteristics of the CE0 deg and CE45 deg building blocks (based on the profile C145 − C145′) give a linear regression equation with a correlation of 99.75%. The CE90 deg building blocks C190 − C145, C190 − C245, C245 − C145, and C245 − C290 results in a degressive curve that fits a quadratic regressed equation with the determination coefficient between 98.65% and 99.83%. The serial connection of two C245 − C290 (2x C245 − C290) purely linearizes the degressive curve of a single C245 − C290 and correlates with a linear regression equation to 100%. The degressive curve of CE90 deg is caused by the loading type and boundary condition of the CE90 deg building blocks.

## References

References
1.
Arata
,
J.
,
Kogiso
,
S.
,
Sakaguchi
,
M.
,
,
R.
,
Oguri
,
S.
,
Uemura
,
M.
,
Byunghyun
,
C.
,
Akahoshi
,
T.
,
Ikeda
,
T.
, and
Hashizume
,
M.
,
2015
, “
Articulated Minimally Invasive Surgical Instrument Based on Compliant Mechanism
,”
Int. J. Comput. Assist. Radiol. Surg
,
10
(
11
), pp.
1837
1843
. 10.1007/s11548-015-1159-4
2.
Palli
,
G.
,
Berselli
,
G.
,
Melchiorri
,
C.
, and
Vassura
,
G.
,
2011
, “
Design of a Variable Stiffness Actuator Based on Flexures
,”
J. Mech. Robot
,
3
(
3
), p.
034501
. 10.1115/1.4004228
3.
Zirbel
,
S. A.
,
Tolman
,
K. A.
,
Trease
,
B. P.
, and
Howell
,
L. L.
,
2016
, “
Bistable Mechanisms for Space Applications
,”
PLoS One
,
11
(
12
), p.
e0168218
. 10.1371/journal.pone.0168218
4.
Howell
,
L. L.
,
Magleby
,
S. P.
, and
Olsen
,
B. M.
,
2013
,
Handbook of Compliant Mechanisms
,
John Wiley & Sons
,
New York
.
5.
Wang
,
H. V.
,
2005
,
A Unit Cell Approach for Lightweight Structure and Compliant Mechanism
,
Georgia Institute of Technology
,
Atlanta, GA
.
6.
Ngo
,
T. D.
,
Kashani
,
A.
,
Imbalzano
,
G.
,
Nguyen
,
K. T. Q.
, and
Hui
,
D.
,
2018
, “
Additive Manufacturing (3D Printing): A Review of Materials, Methods, Applications and Challenges
,”
Composites, Part B
,
143
, pp.
172
196
. 10.1016/j.compositesb.2018.02.012
7.
,
D.
,
Tolou
,
N.
, and
Herder
,
J. L.
,
2012
, “
The Scope for a Compliant Homokinetic Coupling Based on Review of Compliant Joints and Rigid-Body Constant Velocity Universal Joints
,”
Proceedings of the ASME 2012 IDETC/CIE Conference
,
Chicago, IL
,
August
, pp.
379
392
.
8.
Yu
,
J.
,
Pei
,
X.
,
Sun
,
M.
,
Zhao
,
S.
,
Bi
,
S.
, and
Zong
,
G.
,
2009
, “
A New Large-Stroke Compliant Joint & Micro/Nano Positioner Design Based on Compliant Building Blocks
,”
2009 ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots
,
London
, pp.
409
416
.
9.
Safai
,
L.
,
2018
,
Fatigue Testing of 3D-Printed Compliant Joints: An Experimental Study
,
Delft University of Technology
. http://resolver.tudelft.nl/uuid:7df15c93-e792-4a35-9b16-a74b0fdb7ea7
10.
Hao
,
G.
,
2017
, “
Determinate Synthesis of Symmetrical, Monolithic Tip-Tilt-Piston Flexure Stages
,”
ASME J. Mech. Des.
,
139
(
4
), p.
042203
. 10.1115/1.4035965
11.
Mutlu
,
R.
,
Alici
,
G.
,
In het Panhuis
,
M.
, and
Spinks
,
G. M.
,
2016
, “
3D Printed Flexure Hinges for Soft Monolithic Prosthetic Fingers
,”
Soft Robot
,
3
(
3
), pp.
120
133
. 10.1089/soro.2016.0026
12.
Zhang
,
Z.
,
Liu
,
B.
,
Wang
,
P.
, and
Yan
,
P.
,
2016
, “
Design of an Additive Manufactured XY Compliant Manipulator with Spatial Redundant Constraints
,”
2016 35th Chinese Control Conference (CCC)
,
Chengdu, China
,
July
, pp.
9149
9154
.
13.
Guang
,
C.
, and
Yang
,
Y.
,
2018
, “
An Approach to Designing Deployable Mechanisms Based on Rigid Modified Origami Flashers
,”
ASME J. Mech. Des
,
140
(
8
), pp.
7
13
. 10.1115/1.4040178
14.
Bilancia
,
P.
,
Berselli
,
G.
,
Magleby
,
S.
, and
Howell
,
L.
,
2020
, “
On the Modeling of a Contact-Aided Cross-Axis Flexural Pivot
,”
Mech. Mach. Theory
,
143
, pp.
103618
. 10.1016/j.mechmachtheory.2019.103618
15.
Megaro
,
V.
,
Zehnder
,
J.
,
Bächer
,
M.
,
Coros
,
S.
,
Gross
,
M.
, and
Thomaszewski
,
B.
,
2017
, “
A Computational Design Tool for Compliant Mechanisms
,”
ACM Trans. Graph.
,
36
(
4
), pp.
1
12
. 10.1145/3072959.3073636
16.
Krieger
,
Y. S.
,
Kuball
,
C. M.
,
Rumschoettel
,
D.
,
Dietz
,
C.
,
Pfeiffer
,
J. H.
,
Roppenecker
,
D. B.
, and
Lueth
,
T. C.
,
2017
, “
Fatigue Strength of Laser Sintered Flexure Hinge Structures for Soft Robotic Applications
,”
IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
,
,
September
, pp.
1230
1235
.
17.
Abdelaziz
,
S.
,
Esteveny
,
L.
,
Barbé
,
L.
,
Renaud
,
P.
,
Bayle
,
B.
, and
De Mathelin
,
M.
,
2014
, “
Design of a Magnetic Resonance Imaging-Compatible Cable-Driven Manipulator with New Instrumentation and Synthesis Methods
,”
ASME J. Mech. Des.
,
136
(
9
), p.
091006
. 10.1115/1.4027783
18.
Clark
,
L.
,
,
B.
,
Zhong
,
Y.
,
Tian
,
Y.
, and
Zhang
,
D.
,
2016
, “
Design and Analysis of a Compact Flexure-Based Precision Pure Rotation Stage Without Actuator Redundancy
,”
Mech. Mach. Theory
,
105
, pp.
129
144
. 10.1016/j.mechmachtheory.2016.06.017
19.
Qiu
,
C.
,
Qi
,
P.
,
Liu
,
H.
,
Althoefer
,
K.
, and
Dai
,
J. S.
,
2016
, “
Six-Dimensional Compliance Analysis and Validation of Orthoplanar Springs
,”
ASME J. Mech. Des.
,
138
(
4
), pp.
1
9
. 10.1115/1.4032580
20.
Santer
,
M.
, and
Pellegrino
,
S.
,
2011
, “
Concept and Design of a Multistable Plate Structure
,”
ASME J. Mech. Des.
,
133
(
8
), pp.
1
7
.
21.
Hu
,
Y.
,
Zhang
,
L.
,
Li
,
W.
, and
Yang
,
G.-Z.
,
2019
, “
Design and Fabrication of a 3-D Printed Metallic Flexible Joint for Snake-Like Surgical Robot
,”
IEEE Robot. Autom. Lett.
,
4
(
2
), pp.
1557
1563
. 10.1109/LRA.2019.2896475
22.
Pham
,
M. T.
,
Teo
,
T. J.
,
Yeo
,
S. H.
,
Wang
,
P.
, and
Nai
,
M. L. S.
,
2017
, “
A 3-D Printed Ti-6Al-4V 3-DOF Compliant Parallel Mechanism for High Precision Manipulation
,”
IEEE/ASME Trans. Mechatron.
,
22
(
5
), pp.
2359
2368
. 10.1109/TMECH.2017.2726692
23.
Fowler
,
R. M.
,
Maselli
,
A.
,
Pluimers
,
P.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2014
, “
Flex-16: A Large-Displacement Monolithic Compliant Rotational Hinge
,”
Mech. Mach. Theory
,
82
, pp.
203
217
. 10.1016/j.mechmachtheory.2014.08.008
24.
Khurana
,
J.
,
Hanks
,
B.
, and
Frecker
,
M.
,
2018
, “
Design for Additive Manufacturing of Cellular Compliant Mechanism Using Thermal History Feedback
,”
ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
,
August
, pp.
1
15
.
25.
Merriam
,
E. G.
,
Jones
,
J. E.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2013
, “
Monolithic 2 DOF Fully Compliant Space Pointing Mechanism
,”
Mech. Sci
,
4
(
2
), pp.
381
390
. 10.5194/ms-4-381-2013
26.
Wei
,
H.
,
Wang
,
L.
,
Niu
,
X.
,
Zhang
,
J.
, and
Simeone
,
A.
,
2018
, “
Fabrication, Experiments, and Analysis of an LBM Additive-Manufactured Flexure Parallel Mechanism
,”
Micromachines
,
9
(
11
), p.
572
. 10.3390/mi9110572
27.
Merriam
,
E. G.
,
Jones
,
J. E.
, and
Howell
,
L. L.
,
2014
, “
Design of 3D-Printed Titanium Compliant Mechanisms
,”
Proceedings of the 42nd Aerospace Mechanisms Symposium
,
Baltimore, MD
,
May
, pp.
169
174
.
28.
Gao
,
W.
,
Zhang
,
Y.
,
Ramanujan
,
D.
,
Ramani
,
K.
,
Chen
,
Y.
,
Williams
,
C. B.
,
Wang
,
C. C. L.
,
Shin
,
Y. C.
,
Zhang
,
S.
, and
Zavattieri
,
P. D.
,
2015
, “
The Status, Challenges, and Future of Additive Manufacturing in Engineering
,”
Comput.-Aided Des.
,
69
, pp.
65
89
29.
Calignano
,
F.
,
2014
, “
Design Optimization of Supports for Overhanging Structures in Aluminum and Titanium Alloys by Selective Laser Melting
,”
Mater. Des
,
64
, pp.
203
213
. 10.1016/j.matdes.2014.07.043
30.
Thomas
,
D.
,
2009
,
The Development of Design Rules for Selective Laser Melting
,
University of Wales Institute
,
Cardiff
.
31.
Coemert
,
S.
,
Traeger
,
M. F.
,
Graf
,
E. C.
, and
Lueth
,
T. C.
,
2017
, “
Suitability Evaluation of Various Manufacturing Technologies for the Development of Surgical Snake-Like Manipulators From Metals Based on Flexure Hinges
,”
Procedia CIRP
,
65
, pp.
1
6
. 10.1016/j.procir.2017.03.108
32.
,
G. A. O.
, and
Zimmer
,
D.
,
2015
, “
On Design for Additive Manufacturing: Evaluating Geometrical Limitations
,”
Rapid Prototyp. J.
,
21
(
6
), pp.
662
670
. 10.1108/RPJ-06-2013-0060
33.
Allen
,
S.
,
1994
, “
On the Computation of Part Orientation Using Support Structures in Layered Manufacturing
,”
International Solid Freeform Fabrication Symposium
,
University of Texas, Austin, TX
,
September
, pp.
259
269
.
34.
Leutenecker-Twelsiek
,
B.
,
Klahn
,
C.
, and
Meboldt
,
M.
,
2016
, “
Considering Part Orientation in Design for Additive Manufacturing
,”
Procedia CIRP
,
50
, pp.
408
413
. 10.1016/j.procir.2016.05.016
35.
Krishnan
,
G.
,
Kim
,
C.
, and
Kota
,
S.
,
2013
, “
A Metric to Evaluate and Synthesize Distributed Compliant Mechanisms
,”
ASME J. Mech. Des.
,
135
(
1
), p.
011004
. 10.1115/1.4007926
36.
Patiballa
,
S. K.
, and
Krishnan
,
G.
,
2017
, “
Estimating Optimized Stress Bounds in Early Stage Design of Compliant Mechanisms
,”
ASME J. Mech. Des.
,
139
(
6
), p.
062302
. 10.1115/1.4036305
37.
Concept Laser GmbH
,
2019
,
.
38.
Han
,
Q.
,
Gu
,
H.
,
Soe
,
S.
,
Setchi
,
R.
,
Lacan
,
F.
, and
Hill
,
J.
,
2018
, “
Manufacturability of AlSi10Mg Overhang Structures Fabricated by Laser Powder Bed Fusion
,”
Mater. Des.
,
160
, pp.
1080
1095
. 10.1016/j.matdes.2018.10.043
39.
Qi
,
X.
,
Feng
,
H.
, and
Liu
,
L.
,
2019
, “
Microstructure and Mechanical Properties of 316L Stainless Steel Produced by Selective Laser Melting
,”
AIP Conf. Proc.
,
2154
(
1
), p.
020019
. 10.1063/1.5125347