## 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 [7–9]) 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 [11–14], selective laser sintering [15–17], or vat photopolymerization [18–20]. However, metallic AM, such as laser-based powder bed fusion (PBF) or electron beam melting (EBM), has been realized only in a few cases [21–24]. 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 [26–28].

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 (*δ*_{Ori}) on the build platform [32–34], 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 $\delta Ori=0deg$ 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 $\delta Ori=0deg$, $\delta Ori=45deg$, and $\delta Ori=90deg$ 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.

*θ*

_{A}) and ending point B (

*θ*

_{B}) (see Fig. 3). The plane curves

*C*

_{n}defined with an angle of $\theta A\u2265\theta B\u2265\gamma \u226545deg$ 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

*C*

_{1}45 and

*C*

_{1}90, and another pair is based on circular curves

*C*

_{2}45 and

*C*

_{2}90, 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

*θ*

_{A}and

*θ*

_{B}between 45 deg and 90 deg. DEs

*C*

_{1}45 (Fig. 3(a)) and

*C*

_{1}90 (Fig. 3(b)) are assigned the tangential angles of $\theta A=\theta B=45deg$ and $\theta A=\theta B=90deg$, respectively, as described in a Cartesian coordinate system with Eq. (1). More importantly, DEs

*C*

_{1}45 and

*C*

_{1}90 can be described by choosing

*x*

_{A}and

*z*

_{A}as starting points, with the curve inclination $m=cot(\theta A)$.

*A*and

*B*. Element

*C*

_{2}45 starts with a tangential angle $\theta A=45deg$ and ends with $\theta B=90deg$, whereas

*C*

_{2}90 begins with $\theta A=90deg$ and ends with $\theta B=45deg$. Both curves are defined in a Cartesian coordinate system using the equation of a circle in Eq. (2) with a center point (

*x*

_{M},

*z*

_{M}) and a radius

*R*. Here, coordinates

*x*and

*z*are taken in a set containing all points on the curve

*C*

_{2}45 (Eq. (3)) with the starting point

*A*(

*x*

_{A},

*z*

_{A}), whereas Eq. (4) reveals the points on the curve

*C*

_{2}90 with

*x*

_{A}and

*z*

_{A}.

### 2.2 Library of Profile and Base Curves.

*C*

_{1}45′,

*C*

_{2}45′, and

*C*

_{2}90′) 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

*C*

_{1}45 −

*C*

_{1}45′.

*A*(

*x*

_{A},

*z*

_{A}) of the second DE

*C*

_{1}45′ is added to the ending point

*B*(

*x*

_{B},

*z*

_{B}) of the first element

*C*

_{1}45, and the profile height

*H*between the starting point

*A*(

*x*

_{A},

*z*

_{A}) and the ending point

*C*(

*x*

_{C},

*z*

_{C}) 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.,

*C*

_{2}45 −

*C*

_{1}45′) or the

*z*-axis (e.g.,

*C*

_{1}90 −

*C*

_{2}90′).

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*: *C*_{2}90 − *C*_{2}45′). 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*: *C*_{2}45 − *C*_{2}90 or *A*: *C*_{2}90 − *C*_{2}45′) are illustrated more faintly in Fig. 5. For reasons of traceability, the synthesis method is demonstrated in the next sections using the base *B*: *C*_{1}45 − *C*_{1}45′. The applicability of base *A*: *C*_{1}45 − *C*_{1}45′ 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.

*w*= 2 mm and

*t*= 0.5 mm, respectively (Fig. 6(a)). As shown in Eq. (6), the displacement Δ

*s*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

*σ*

_{yield}and Young’s modulus

*E*was kept equal. Consequently, the 3D building blocks differed in the length

*l*of their profile beams.

*w*. The geometry of the base

*B*:

*C*

_{1}45 −

*C*

_{1}45′ is shown in Fig. 6(b), and CE

_{90 deg}is formed through the assembly of base

*B*:

*C*

_{1}45 −

*C*

_{1}45′ and profile

*C*

_{2}45 −

*C*

_{2}90, as depicted in Fig. 6(c). The starting and ending points of the profiles are enclosed by edges

*E*

_{1}−

*E*

_{4}, 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 [

*M*

_{1}

*M*

_{2}] (Fig. 6(d)).

*α*and the base leg angle

*β*, which can be visualized for all three

*CEδ*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 $\gamma \u226545deg$. Given both angles

_{Ori}*δ*

_{Ori}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.

*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 CE

_{0 deg}, CE

_{45 deg}, and CE

_{90 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 [M

_{1}E

_{1}] and [M

_{1}E

_{2}] was valued originally by $\beta =45deg$ because of the symmetric character of the rays.

The original base angle of $\beta =45deg$ resulted in $\alpha =0deg$ for CE_{90 deg} with both bases in contact and with no motion to be realized. Consequently, *β* was reduced by 5 deg to $\beta =40deg$, leading to an increased $\alpha =22deg$. 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, CE_{45 deg} resulted in $\alpha =90deg$ and $\beta =0deg$, 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 $\delta Ori=0deg$ to $\delta Ori=90deg$ can be covered by CE_{0 deg}, CE_{45 deg}, and CE_{90 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 $\gamma \u226545deg$ to $\gamma \u226525deg$, allowing CE_{0 deg} to cover the range of $\delta Ori=0deg$ to $\delta Ori=25deg$. Meanwhile, CE_{45 deg} enclosed the orientation from $\delta Ori=25deg$ to $\delta Ori=75deg$. With CE_{90 deg} spanning the field from $\delta Ori=60deg$ to $\delta Ori=90deg$, an overlap of $15deg$ was attained by both CE_{45 deg} and CE_{90 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 Δ*s*, was taken as the displacement of the loaded base along the axis of motion between the center point *M*_{1} of the rigid base and center point *M*_{2} of the loading base. Additionally, the building blocks were compared using Δ*s* and *F*, with the plotted results for CE_{0 deg} (Fig. 8) and CE_{90 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 *σ*_{yield}. 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 CE_{0 deg}, the profile *C*_{1}45 − *C*_{1}45′ achieved the highest displacement of Δ*s* = 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/mm^{3}. The nearly linear force-displacement relation (or stiffness behavior) is shown complementarily in Fig. 15. Moreover, the CE_{0 deg}*C*_{1}45 − *C*_{1}45 (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 CE_{90 deg} building blocks (Fig. 9). A low average strain energy density was especially depicted in the building block of the profiles *C*_{1}45 − *C*_{1}45 and *C*_{2}45 − *C*_{2}90, 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 (2*x C*_{1}45 − *C*_{1}45 and 2*x C*_{2}45 − *C*_{2}90) enabled a stabilized displacement to the yield strength with a raised amount of the overall stored energy density. The degressive stiffness behavior of *C*_{2}45 − *C*_{2}90 was linearized by the serial connection of 2*x C*_{2}45 − *C*_{2}90 (see Fig. 15). In particular, CE 2*x C*_{2}45 − *C*_{2}90 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 Δ*s* of 2*x C*_{1}45 − *C*_{1}45 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/mm^{3}. In a separate scenario, CE *C*_{1}90 − *C*_{2}90 (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 [*M*_{1}*M*_{2}]. Consequently, the rigid and loading base made contact before the yield strength was reached. However, several building blocks (e.g., *C*_{1}90 − *C*_{2}45, *C*_{2}45 − *C*_{1}45, and *C*_{1}90 − *C*_{1}45) showed promising and preferable results with high elastic displacement and large storable energy per material volume. In contrast to CE_{90 deg}, the synthesis of the CE_{45 deg} resulted in only one building block, having a translation of Δ*s* = 4.47 mm, an applied force of *F* = 10.2 N, and a stored energy of 1.63 × 10^{−1} mJ/mm^{3} (based on the profile C_{1}45 − C_{1}45′), 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 CE_{45 deg} building block is described together with a selected set of CE_{0 deg} and CE_{90 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 CE_{0 deg}, CE_{45 deg}, and CE_{90 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 $\gamma \u226545deg$. Some of these PBF building blocks are displayed in Fig. 10(a).

The additive manufactured CE_{90 deg}*C*_{2}45 − *C*_{2}90 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 CE_{90 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 CE_{90 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 [*M*_{1}*M*_{2}] was realized by replacing the rigid base *B*: *C*_{1}45 − *C*_{1}45′ with *A*: *C*_{1}45 − *C*_{1}45′. 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 CE_{90 deg} based on the profile *C*_{2}45 − *C*_{2}90 and the base *B*: *C*_{1}45 − *C*_{1}45′. 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 *C*_{2}45 − *C*_{2}90 and the base *A*: *C*_{1}45 − *C*_{1}45′. 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 $\delta Ori=0deg$, $\delta Ori=45deg$, and $\delta Ori=90deg$ 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 CE_{0 deg} and CE_{45 deg}, the building blocks based on the profile *C*_{1}45 − *C*_{1}45′ 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 CE_{90 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 $\delta Ori=0deg$ to $\delta Ori=90deg$ via the reduction of the build angle from $\gamma \u226545deg$ to $\gamma \u226525deg$. However, the manufacturability of the surfaces up to $\gamma \u226525deg$ has not yet been reported in the literature and has restricted the findings of this paper. Notwithstanding its limitations, a building angle of $\gamma \u226530deg$, as confirmed in the literature [29,38], resulted in an excluded range of 0 – 90 deg within $\delta Ori=20deg\u221230deg$.

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 CE_{90 deg} building blocks could be potentially applied for a customizable robotic spring actuator. Further, the bistable character of CE_{90 deg}*C*_{2}45 − *C*_{2}90 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 $\delta Ori=0deg,\delta Ori=45deg$, and $\delta Ori=90deg$ 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

*θ*_{A}=tangential angle of the curve at starting point

*A*, deg*δ*_{Ori}=orientation of the compliant element, deg

*σ*_{yield}=yield strength, MPa

- Δ
*s*=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 CE

_{90 deg}building block. This 3D building block is an exemplary synthesis with the profile C

_{1}45 − C

_{1}45 and the B-base C

_{1}45 − C

_{1}45′. Using trigonometric expressions for the angles

*α*,

*β*,

*γ*and

*δ*

_{Ori}, the following equations can be obtained:

*l*results to Eq. (7).

*Self-Supported Range for α and β*. Table 3 provides a list of the respective ranges of *α* and *β* assuming $\gamma \u226545deg$ and *δ*_{Ori}. For CE_{0 deg}, the profile beam remains in the self-supported building space for $\alpha =45deg$ and $\alpha =90deg$. It should be regarded that increasing α would result in reduced compliance caused by the shortened profile length *l*. In addition, for $\beta \u226422.5deg$, 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 CE_{90 deg}, and the beam profile is bounded within the conical space for $0deg\u2264\alpha \u226440deg$. Moreover, CE_{45 deg} is self-supported for $\alpha =0deg$ or $\alpha =90deg$. 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 CE_{0 deg}, CE_{45 deg}, and CE_{90 deg} building blocks. The force-displacement characteristics of the CE_{0 deg} and CE_{45 deg} building blocks (based on the profile *C*_{1}45 − *C*_{1}45′) give a linear regression equation with a correlation of 99.75%. The CE_{90 deg} building blocks *C*_{1}90 − *C*_{1}45, *C*_{1}90 − *C*_{2}45, *C*_{2}45 − *C*_{1}45, and *C*_{2}45 − *C*_{2}90 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 *C*_{2}45 − *C*_{2}90 (2*x C*_{2}45 − *C*_{2}90) purely linearizes the degressive curve of a single *C*_{2}45 − *C*_{2}90 and correlates with a linear regression equation to 100%. The degressive curve of CE_{90 deg} is caused by the loading type and boundary condition of the CE_{90 deg} building blocks.

GE Additive 17-4 PH