Additive manufactured continuum mechanisms based on shape-programmable and micro-sized building blocks

ABSTRACT Micro-additive manufacturing techniques have the potential to meet the demand for miniaturised functional components for minimally invasive surgical instruments. These techniques create monolithic, compliant mechanisms with micro-sized free-form structures that can be tailored to patient-specific surgical procedures. The automated design synthesis of the mechanisms using building blocks results in structures that are shape-programmable. This is achieved through an algorithmic-based computational workflow, which automatically converts user-specified 2D and 3D curves into discrete curve segments. The actuated motion of the mechanisms can be designed to move in a specific way, both forwardly and inversely. The mechanisms are manufactured using micro-laser powder bed fusion and hardenable stainless steel 17-4 PH. By carefully selecting the process parameters, it is possible to 3D-print micro-sized features such as a compliant beam thickness of 80 μm and an actuation hole of 100 μm. Both 2D planar curved mechanisms and 3D spatial curved mechanisms have been implemented and experimentally validated.


Introduction
Recent advances in industry and academia envision the design capabilities of micro-additive manufacturing (micro-AM) for the creation of miniaturised parts with small feature sizes on the microscale (Chin et al. 2020;Nagarajan et al. 2019;Vaezi, Seitz, and Yang 2013).The free-form capabilities of micro-AM enable the fabrication of three-dimensional (3D) structures that are more complex, time-and labour-intensive, or not possible to produce using conventional micromanufacturing technologies (Modica, Marrocco, and Fassi 2017).In addition to its free-form potential, micro-AM offers the ability to customize designs to specific functional requirements for applications, such as in the field of minimally invasive surgery (MIS) (Culmone, Smit, and Breedveld 2019).MIS requires surgical instruments for complicated procedures within confined spaces and often demands improved manoeuvrability and miniaturisation.As a result, the instruments have the ability to navigate through multiplecurved paths, such as those with multiple radii in an S-shaped curve (Figure 1).However, existing devices have many functional components that result in high assembly and fabrication efforts, particularly on a miniaturised scale (Ali et al. 2019;Sakes et al. 2018;Swaney et al. 2017).
Miniaturised functional components such as monolithic compliant mechanisms (CMs) can greatly benefit from AM and meet the demands for MIS instruments within clinical applications (e.g.endonasal surgery, neurosurgery, laparoscopic surgery, and ocular surgery), as reviewed in recent publications (Culmone, Smit, and Breedveld 2019;Zanaty et al. 2019;Lussenburg, Sakes, and Breedveld 2021).The continuum mechanism achieves its steerable motion through the elastic deformation of its structures.Its monolithic design reduces the time and effort required for the manual assembly of micro-sized features (Howell and Spencer 2013).These mechanisms can be integrated as a joint or end effector to make the MIS instrument more manoeuvrable, providing surgeons with more freedom in its steerability (Culmone, Smit, and Breedveld 2019;Lussenburg, Sakes, and Breedveld 2021;Jelínek and Breedveld 2015;Culmone et al. 2020).Furthermore, the design and motion of the instruments can be personalised to the surgical procedure, patient anatomy or surgeon's preferences based on pre-operative CT or MRI imaging (Culmone, Smit, and Breedveld 2019;da Veiga et al. 2020;Lussenburg, Sakes, and Breedveld 2021).Pre-operative, customizable mechanisms have been demonstrated using various polymer-based 3D-printing technologies (Culmone et al. 2020;Desai et al. 2019;Henselmans, Smit, and Breedveld 2019;Henselmans et al. 2020;Krieger et al. 2016).However, metals offer higher resistance to fragility, creep, and sterilisation than polymers (Coemert et al. 2017;Laban et al. 2021;Ngo et al. 2018), making them the preferred material for surgical instruments in conventional manufacturing techniques.Consequently, metallic AM techniques (e.g.laser powder bed fusion (LPBF)), are seen as a promising technology for the creation of meso-and macrofeatures (> 100 μm) within 3D-printed mechanisms (Danun et al. 2021;Feng, Hong, and Xie 2020b;Hu et al. 2019b).Micro-LPBF, in particular, is a micro-AM technique based on LPBF technologies that can print micro-sized features of < 100 μm (Chin et al. 2020;Nagarajan et al. 2019;Vaezi, Seitz, and Yang 2013).
Despite the potential of metallic AM, the number of LPBF CMs is limited to date (Lussenburg, Sakes, and Breedveld 2021) due to several domain-specific challenges.One challenge within the AM domain is the requirement for support structures (Calignano 2014;Langelaar 2016;Thomas 2009).Removing support structures is time-consuming and can damage fragile beam structures, especially if they are micro-sized (Chin et al. 2020;Coemert et al. 2017).To reduce the tedious effort of removing support structures, it is necessary to follow the design rules of LPBF, such as maintaining an overhang angle γ ≥ 45°or using proper print orientation (Calignano 2014;Langelaar 2016;Thomas 2009).Consequently, designing AM parts using LPBF can be challenging, even for experienced AM designers.This design effort in the AM domain adds to the design challenges of CMs, such as design synthesis and structural analysis.These two issues lead to a long manual design synthesis phase with multiple trial-and-error approaches and require high computational time and power for structural analysis (Gallego and Herder 2009;Yu et al. 2009;Zhu et al. 2020).A further challenge is the design of CMs with sufficient structural flexibility to enhance their limited range of elastic deformation, which is given by material properties, geometric/topological design, and boundary conditions (Howell and Spencer 2013).
In prior work, a design-for-AM synthesis approach based on building blocks (BBs) was presented to address design and manufacturing challenges for the LPBF of CMs.The BB approach helped to synthesize a CM in a self-supporting manner (Danun et al. 2021).In this work, these BBs are taken as a foundation to further focus on the automated design synthesis of customizable and miniaturised continuum mechanisms.When customizing for functionality, appropriate considerations within design synthesis and structural analysis approaches are crucial, taking into account the strengths and weaknesses of different methods (Huang et al. 2020).Generally, the BB design synthesis favours design programming for user-defined inputs (Ding et al. 2018;Huang et al. 2020).This combines parametric design with programming methods to generate automated 3D BB models (Huang et al. 2020;Yao et al. 2019).The modular and discretized models are synthesized as individual units that, when assembled and actuated, result in the desired morphing S-shaped curved mechanism (Figure 1).Their programmable BB nature also leads to an automated design and assembly of the mechanism (Huang et al. 2020;Yao et al. 2019).
The contribution of this work is to leverage the advantages of self-supported BBs for the automated design synthesis and micro-LPBF of shape-programmable and miniaturised continuum mechanisms.Specifically, this work proposes: (1) A design approach for the local stiffness tailoring of individual BBs (2) Design topologies for automated design programming of customized mechanisms in target curved shapes (3) Manufacturability and micro-LPBF of continuum mechanisms with free-form features on the microscale (4) Experimental validation of the miniaturised case studies Figure 2 provides a section-by-section overview of the contributions of this article.The article begins in Section 2.1 with the geometric design of the BB and its kinematic description, including the analytical modelling of BBs.Based on this, the design approach for tailoring the compliance of the BBs is presented in Section 2.2.Additionally, the automated design synthesis workflow, which includes integrated procedures for programmatically shaping the overall curved shape of the mechanism, is described in Section 2.3.A set of automated, configurable 3D models with in-plane and out-of-plane curved shapes are generated to provide the design topologies (Section 2.3).In Section 3, the manufacturability and micro-LPBF of the mechanism is performed.Finally, in Section 4, six case studies of wire-driven continuum mechanisms are experimentally validated and their design-programmed shapes are compared to their measured results.

BB design synthesis and kinematic modelling
A BB is comprised of four identical beam elements that are arranged in a square-like configuration to accommodate the overhang requirement.These beam elements are angled at γ ≥ 45°with respect to the built plate to ensure minimal use of support structures during manufacturing.Each beam element features a double-curved arc form, with arcs that are tangential and horizontal at the midpoint of the beam, and an inclination angle of α.This angle has been determined through previous study to yield optimal results (Danun et al. 2021).The final design of the BB can be observed in Figure 3(a).The mechanism is constructed by assembling the BBs in series, as depicted in Figure 3(b).Additionally, a hole is integrated into each corner of the BB, which serves as a guide for actuation wires.A total of four wires are used to actuate the structure, with each wire corresponding to one corner.When a load is applied to one of the wires, a force F is generated at the distal tip of the structure, resulting in a bending motion of the steerable distal tip in the direction of the applied force, as illustrated in Figure 3(c).The proximal end of the mechanism is rigidly connected.
The actuated motion of the mechanism can be shaped by defining the mechanical properties of the individual, discretized BBs.The mechanical properties are set by the designer using geometrical dimensions that serve as the basis for the analytic model.These parameters are outlined in Table 1 and shown in Figure A1 and A2 of Appendix A. The geometric design is defined by the length of the beam element (l beam ), the inclination angle (α), the outer size (D), and the height (h i ) of the BB.An Equation (1) that transforms the applied load (F) to the desired curvature radius (k o ) of the BB is used in conjunction with these geometrical parameters to programme the mechanism.The mathematical derivation of this quasi-static model is described in detail in Appendix A. This Equation ( 1) is used in the automated design programming of the mechanism in the following sections.A compensation factor (c) is introduced into the model to analytically consider difficult-to-model phenomena (e.g.process-induced, material-induced, and mathematical assumptions, etc.), which may cause deviation between the as-designed and experimentally tested mechanism behaviour.Furthermore, the trigonometrical projections of the beam elements to the curvature plane are combined into the factor m. The parameter v l considers the ratio between l beam and the effective beam length (l eff ).The parameter l eff is used to tailor the stiffness of a BB, as described in the following section (Section 2.2).

Design approach of tailorable BBs
The curved shape of the steerable mechanism can be programmed locally by adjusting the beam length l beam of the BB (Figure 4a).This allows for the tailoring of the bending stiffness of each beam and the overall BB.The beam length (l beam ) is controlled by defining the dimensions of notch volumes that are integrated between two neighbouring beam elements.These volumes are designed and programmed in a way to set the local size of l beam .This results in an effective beam length (l eff ) defined as the distance between the steep-angled intersection of two neighbouring beams and the actuation hole.Since there are eight beams in two neighbouring BBs, there are also eight notch volumes.These volumes can be seen in Figure 4(b) and are marked in red.By stacking multiple BBs and adding notch volumes to them, the mechanism can be programmed with tailored compliance over its entire mechanism length L. By varying the stiffness of the elements, their bending deformation (Du) or curvature radius (k i ) can be influenced.The size of individual notch volumes (size notch volume ) is therefore defined as a percentage, ranging from size notch volume = 0% to size notch volume = 100%, as can be seen in Figure 4(b).By knowing the sized beam length (l size ) and the effective beam length (l eff ), the size notch volume can be calculated.The more the l eff is reduced by the notch volume, the higher is the bending stiffness of the beam.In contrast, the maximal compliant deformation of a BB is defined by a minimal size notch volume = 0%.Depending on the size of the volume within the structure, different parts can be made more flexible, stiffer, or blocked completely.As can be seen in Equation (2), the tailored k i of each beam element has a cubic dependency on the percentual volume of the k o with a size notch volume = 0%.Further details of the derivation of Equation (2) can be found in Appendix A .
As the size of the eight tailorable notch volumes are the only design parameters that are changed within BBs, all the information needed to fully describe their configuration is condensed into the figured notation (Figure 4c).This notation will be used in subsequent sections for a clear and traceable description of the BB tailoring.For  reference, the BB actuation holes are labelled anticlockwise with letters from A to D. The volumes start from A and extend to corner B and so on.On each beam element, there are two neighbouring volumes, resulting in a total of eight tailored volumes.Eight rectangles represent the size of these eight customizable notch volumes (Figure 4c).A darkened region indicates the actual sized beam length (l size ) of the corresponding notch volume, and the bright region represents the effective beam length (l eff ).The line at the end of the rectangles indicates the starting point of the respective volumes.The inner rectangles within the square-like configuration correspond to the lower notch volume of a BB, whereas the outer rectangles correspond to the upper notch volume.In addition to the visual indication, the notch volume size is separately listed as a percentage ratio of the initial l eff to the l size .The marked blue square at the actuation holes shows the position of the applied force (F) via the actuation wires.

Symmetric and asymmetric mechanism design
Mechanisms with surgical-specific shapes were achieved by combining the design approach discussed in previous sections (Section 2.2) with the introduction of symmetric and asymmetric design topologies.These design topologies enable the mechanism to achieve planar and spatial motions, such as C-curved shape, L-curved shape, in-plane and out-of-plane S-curved shape.The distinction between mechanisms with symmetrical and asymmetrical BBs is shown in Figure 5.A symmetrical BB has constant stiffness properties in the four directions of actuation.The design is a series of BB connections with a uniform configuration of notch volume sizes, and thus it has the same notch volume size in each direction.When a force (F) is applied at the top of a symmetrical segment, the segment bends in that direction with a directly determined curvature k i .Actuating one of the remaining wires results in the same k i .In asymmetrical design structures, the eight individual notch volumes do not have equal length.This dissimilarity results in a direction-dependent bending deformation.In Figure 5 As shown in Figure 5, the symmetric design topologies result in a C-curved and L-curved shape, while the asymmetric configurations enable in-plane and out-ofplane S-curved shapes.Each of these curves has a distinct arrangement of notch volumes.Depending on the notch volume size and configuration of the structure, different parts and shapes can be programmed for the surgical procedure.The C-curved shape consists of identical BBs with the same notch volume size.The bending deformation is uniform throughout the entire length of the mechanism.Figure 5 shows an example structure where each volume has a minimal notch size.The curvature radius is the same for all BBs, resulting in a constant k o .To customize k i over the length of the mechanism, the bending stiffness is varied locally.Such a customized arrangement allows for an L-curved shape as an extension to the C-curved shape.As mentioned earlier, in symmetric configurations, each BB has equal notch volume sizes.However, the L-curved shape has a continuous variation in its notch volumes, enabling a continuous change but still symmetric bending deformation across the mechanism.The structure consists of a more flexible lower section and a stiffer upper section, or vice versa.The stiffer section will deflect less under load compared to the flexible section.The parameter k i can exhibit both continuous and discontinuous changes between the two extremes.As presented in Figure 5, the upper section of the structure is more flexible and thus deforms in a circular manner, while the lower section, with an increased notch volume, remains relatively straight.
An S-curved shape consists of two bends in opposing directions, which is achieved by combining two asymmetrical sections and arranging them in series.The inplane S-curved shape is created by limiting the bending deformation on two opposing sides of the beam structure.The notch volume, which extends from point C to B or D to A, effectively blocks the corresponding beam elements, restricting the deformation of the part to a 2D plane.As a result, no deformation is possible at corner B, while deformation is unrestricted at corners C and D. This analogy holds true for the opposite side, with corners D and A.
An in-plane S-curved shape can be synthesized by combining two sections with this notch volume configuration and rotating them by 180°relative to each other.Actuation is achieved by applying simultaneous loads at holes B and D, causing the two bending sections to deform and produce an in-plane S-curved shape (Figure 5).The out-of-plane S-curved shape functions similarly, but with the upper section of the structure rotated by 90°relative to the lower one, created by applying a load at hole C. The volume arrangements for in-plane and out-of-plane S-curved shapes are shown in detail in Figures A3 and A4 of Appendix B.
Furthermore, the design topologies for customizable curves are presented through multiple case studies in Section 4.

Automated design workflow for shapeprogramming
A computational design workflow, utilising the RhinoPython scripting, has been implemented in the CAD software Rhinoceros 6 (Robert McNeel & Associates, Seattle, USA) to automatically synthesize continuum mechanisms.This workflow comprises of a main Python script that contains the logic, instructions, and controls the flow of the automation process by calling functions and classes from the imported library Rhino-ScriptSyntax and several custom modules.The custom modules are mainly used for two tasks: automated design synthesis of the continuum mechanism and automated design synthesis of the notch volumes by referencing user-specific curves.These synthesized designs are then assembled within the main Python script.Consequently, this workflow is used to programme the mechanism synthesis with symmetric and asymmetric design topologies and tailor the BB locally bending deformation with the analytical model.The workflow is used for both forward and inverse design programming.The forward shape programming of the mechanism is realised by considering Equations ( 1) and ( 2).In the case of inverse design programming, the user specifies a curve with the desired shape of the mechanism.The design algorithm takes this curve as its input and processes it into a 3D configured part.The automated process chain is shown in Figure 6 and can be divided into four general steps.First, the input curve of the user (Figure 6a) is discretized into i-segments (Figure 6b).Each segment curve length corresponds to the height h i of a single BB.The script generates and stacks these BBs in their initial state on top of each other to reach the length corresponding to the input curve length.Thereby, the number of both segments and aggregated BB match each other.Each segment's starting point is denoted with a point P i .At each P i , the k i is calculated, and a tangent line t i is constructed (Figure 6c).The direction of curvature is derived by calculating the angle w i between the tangents of two neighbouring segments.This calculation generates a transformation matrix from one t i to the next t i+1 .If w i is positive, the curve direction bends to the right in the curve reference system.If the value of the angle is negative, it turns to the left.Curved shapes with only positive w i are classified as symmetric C-and L-curved shapes.Curved shapes with positive and negative w i are detected as asymmetric in-plane or out-of-plane Scurved shapes.The distinction between the two types of S-curved shapes is realised by analysing the alignment of the BB in an x-y plane or out of the plane (x-z plane).
After the discretization of the curve, the k i for each segment is passed to the analytical model, and an automated mechanism design is derived in reverse.Equation ( 2) is used to translate k i of the input curve into the size of the volumes.A set of eight notch volume sizes per BB is defined, specifying the tailored stiffness of each BB.For each volume, a percentage value is determined.Before these values are used in the automated design synthesis, the previously detected direction of the input curve is considered for the definition of the design topology.The design algorithm combines the information of notch volume size and curve type and returns an array containing the eight percentages of notch volumes for each BB.The contents of this array correspond to the information in the notation explained in Section 2.2.At this stage, the mechanism is assembled at its initial unconfigured state with the derived information about the required notch volume sizes (highlighted in reddish in Figure 6(d).This assembly is stacked spatially by iterating through all BBs.For every BB, the eight-volume sizes are retrieved from the array.These magnitudes are translated into a volume at its assigned spatial location within the respective BB.This configuration step leads to the final 3D model of the user-defined mechanism, as shown in Figure 6(e).Afterward, the model can be exported as an STL-file for the manufacturing process.

Micro-LPBF of mechanisms
The manufacturing process for micro-AM via LPBF is similar to the conventional process of LPBF.In addition to the well-known powder-specific, laser-specific, and powder-bed-specific parameters, micro-AM requires special attention to the selection of three main factors: smaller powder particles, thinner layer thickness, and smaller laser spot.By downscaling these main factors, the printed feature resolution is increased from common LPBF feature sizes of 0.2-0.4mm to < 0.1 mm.In this work, the micro-AM of miniaturised continuum mechanisms was achieved using a commercial TRUMPF TruPrint 1000 machine.The machine's minimum laser spot diameter of 30 µm and minimum layer thickness of 12 µm were taken into account during the process (TRUMPF 2021).To ensure a homogeneous powder packing density and correlate it with the layer thickness, the material supplier's powder with an average spherical particle size of D 50 = 14-15 μm and its material properties was selected (m4p material solutions GmbH 2023).The microparts were manufactured using precipitation hardened stainless steel powder 17-4 PH (H900).This material is suitable for use in surgical instruments, which supports the clinical use of the continuum mechanism in potential future applications, due to its corrosion resistance, high yield strength properties, good wear resistance, biocompatibility, manufacturability, and affordability (Giganto et al. 2022).Furthermore, the structures were heat-treated to increase the yield strength and ultimate strength, while the Young's modulus remained almost unchanged.This heat treatment enhanced the material properties, meaning the material can withstand higher energy without permanent deformation (Yadollahi et al. 2015).The selected process and material parameters are summarised in Table 2. Additionally, the external forces introduced during the separation process of the miniaturised part from the build plate could damage the delicate mechanism.To mitigate this, the support structures were separated from the build plate using a wire electrical discharge machining (wire-EDM) process.
The design guidelines for micro-LPBF are similar to those of a typical LPBF process.In particular, the overhang angle of γ ≥ 45°applies to meso-, macro-AM, and micro-AM since angle measures are not affected by down-scaling.Therefore, the square-like self-supported BB shape was kept similar to that of macro-BB design.In addition to the overhang design rule, in the specific case of micromechanisms, two features were given iterative and careful consideration in the design and fabrication.These features were its wall thickness and the actuation hole diameter (Figure 7).A wall thickness with an acceptable definition was achieved by printing at least three laser fusion lines along the 80 μm beam element thickness cross-section.To pass a maximal 50 μm actuation wire diameter, holes with twofold sizing of minimal 100 μm were aimed to ensure a wide gap between hole and wire.To realise this 100 μm, an actuation hole diameter of 180 μm was chosen.As a result, the final design considered a tolerance margin of an additional 80 μm.The actuation holes are self-supported by designing the holes in a squared design.

Part validation of case studies
Six mechanisms were fabricated using micro-LPBF to validate the symmetric and asymmetric design topologies.Case studies were conducted based on both forward and inverse shape programming.Figure 8 illustrates six case studies, including both planar and spatial motion.
The compensation factor c within the analytical model (see Equation ( 1)) accounts for mathematical simplifications and inevitable inaccuracies associated with the LPBF process (as described in Section 2.1).Therefore, the C-curved shape C 1 was tested to determine the compensation factor c. The geometric BB dimensional parameters for C 1 (as listed in Table 1) were identical for the remaining forward and inverse case studies (as shown in Figure 8).The only variations were the number of BBs and the integrated notch volume sizes.As seen in Table 1, a compensation factor of c = 2.5 was found for the calculation k o of C 1 .Furthermore, the analytical model used in the computational design workflow established a relationship between the effective beam length (l eff ), the force (F) acting on the structure, and the resulting curvature k i .An experimental setup was used to measure these three values (as described in Appendix C).The force (F) acting on the structure was measured through actuation wires using a tensile testing machine.The parts were fixed to the machine with a mounting system.A reference scale was captured via a ruler to measure the curvature.The measurement setup is depicted in Figure A5 of Appendix C. When the defined F was applied, an image of the structure was taken.The images with known F were then imported into the CAD-tool Rhinoceros 6 to determine the radius of curvature.A curve was fitted through the bent neutral axis of the mechanism, and its radius was measured using the radial dimension tool in Rhinoceros 6.In the case of forward design programming, as shown in Figure 8, the l eff of C 2 was stiffened by 23.5% while taking Equations ( 1) and (2) into account.The notation and resulting tested curves of C 1 and its counterpart C 2 are illustrated in Figures 9 and 10, respectively.
Both mechanisms were analysed and aggregated with 72 BBs.C 1 and C 2 were loaded with a F = 0.3 N and a curvature radius of k Test C1 = 0.1208 1 mm and k Test C2 = 0.0545 1 mm , respectively, were measured.A deviation of 1% for C 1 and 2% for C 2 between the tested and modelled curvature k Model C1 = 0.1201 1 mm and k Model C2 = 0.0537 1 mm , respectively, was observed.Both L-shaped samples, L 1 and L 2 , were inversely synthesized, with the variation of beam element stiffness being derived from user-defined curves.The part had a continuously varying beam element stiffness, transitioning from stiff at the attachment point to flexible at the loaded free end, as indicated by the black arrows in Figure 8.The 72 BBs of L 1 exhibited a continuously changing stiffness from 0% to 35% when loaded with F = 0.6 N, as shown in Figure 11.Similarly, L 2 , which also had 72 BBs, had a symmetrically programmed shape, with the highest k i at the centre and the lowest stiffness at both ends, as indicated by the dashed line in Figure 8.The spectrum of stiffness change for L 2 ranged from 0% to 22.5% when subjected to F = 0.9 N (see Figure 11).Both L-shaped samples displayed a maximum deviation of 0.8 mm from the target curve, corresponding to approximately 5% of the mechanism length of L = 17.3 mm.The curves of the two L-shaped samples are depicted in Figure 12.
Similar to the L-shaped samples, the two S-shaped samples were synthesized from a user-defined curve (shown in Figure 8).The in-plane S-shaped (IPS) sample was composed of a total of 114 BBs and comprised two point-symmetrically mirrored sections, inspired by and corresponding to L 2 , but with an asymmetric design pattern.As described in Section 2.3.1, the lower section IPS-B and the upper section IPS-U of the IPS are related to each other through a 180°rotation (clockwise) and have different stiffness ranges, as illustrated in Figure 13.The side-ways blocking of IPS-B and IPS-U was achieved by reducing l eff through a constant notch volume size of 60%.A continuous variation of the beam element notch volume size from 0% to 40% was used to transform the user-defined curve.
Both IPS-B and IPS-U exhibited symmetric curve characteristics, with the dashed line serving as the axis of symmetry (as seen in Figure 8).They displayed a maximum curvature in the centre and decreasing curvature towards the ends, as indicated by the starting and ending points of the black arrows in Figure 8.The modelled curve of the IPS is depicted in Figure 14.The deviation of the tested curve from the nominally intended curve was measured simultaneously with a F = 2.0 N applied at holes B and D. The maximum deviation from the nominal curve was 0.6 mm, which was nearly 2% of the mechanism length of L = 26.8mm.Because the IPS deflects using two actuation wires, it was possible to actuate both wires simultaneously or individually.Additionally, by controlling the deflection of IPS-B through the tension of one wire, the force acting on IPS-U can be regulated by the actuation of the other wire.
The actuation of the out-of-plane S-shaped (OPS) sample was achieved by placing the upper section OPS-U in a 90°anticlockwise orientation relative to the lower section OPS-B.The OPS comprised 72 BBs.Similar to the IPS, the side-ways blocking of the beam elements was reached with a constant notch volume size of 60%.In contrast to the IPS, the transformation of the user-defined curve was accomplished Figure 12.The modelled blue curve of L 1 (left) exhibited a maximum deviation of 0.8 mm from the experimentally red curve when subjected to a F = 0.6 N. Similarly, when L 2 (right) was loaded with a F = 0.9 N, a maximum deviation of 0.8 mm was observed when comparing the modelled curve to the experimentally loaded curve.
by implementing a continuous change in stiffness between 0% and 50%, as shown in Figure 15.The actuation of the OPS can be attained by applying a F = 1.8 N at hole D using a single actuation wire.The two perpendicular sections were tested separately to measure the deviation from the nominal and tested curves.The deviation of the lower section OPS-B from the modelled curve was 1.1 mm and the deviation of the upper section OPS-U was 2 mm, as viewed in Figure 16.An average deviation of 1.55 mm resulted in nearly 9% of the overall mechanism's length of L = 17.3 mm.

BB design approach and design topologies
Generally, BBs favour programmability in their mechanical characteristics (e.g.stiffness or deformation) (Lei et al. 2019;Lin et al. 2021;Zhang, Guo, and Hu 2021).This programmability was accomplished through the use of synthesized BBs, which were aggregated in a repetitive, serial and parallel manner.In this work, the design approach and the symmetric and asymmetric topologies leveraged this characteristic of BBs to customize the mechanism to a user-specified curved shape.
The design approach for tailoring the stiffness of BBs followed the hierarchical design potential of AM (Yang and Zhao 2015).The local volume variation of notches within the individual beam elements led to a controlled bending deformation of the overall mechanism, enabling the user-specified and actuated movement of the mechanism.The modification of the BB functionality was not defined by the geometry or material of the beam structures.It was tailored locally by the sizing of the notch volumes.As can be seen in Equation (A9), the stiffness of the beam structures is defined by Young's modulus (E), the moment of inertia (I), and the length of the beam structures (l beam ).Furthermore, the thickness (t) has a cubic dependency on the bending stiffness of the beam, whereas the width (w) only has a linear dependency.Therefore, it can be concluded that to tune the stiffness of specific beams within the structure, t could be used to set the significant scale of the stiffness, while w could be used for fine-tuning.However, the modification of t and w directly relates to the minimum feature size (e.g. the  wall thickness) characteristic of the LPBF process machine.Consequently, the chosen design approach supports the applicability and reproducibility of the automated BB programming process and reduces its dependency on common feature limitations of the LPBF process.The definition of stiffness through thickness modification also increases the susceptibility of the system to process-induced defects (e.g.surface roughness).The constant surface roughness may impact the stiffness in different relative proportions.At small wall thicknesses, the surface roughness has a much larger ratio than when a large thickness is chosen.Due to this ratio, the beam length was chosen as a more robust choice for local variation of stiffness (Wei et al. 2021).
The symmetric and asymmetric design patterns carried out the forward or inverse programming of C-, L-, and S-curved mechanisms in combination with the above-mentioned tailoring design approach.curve selections in this work refer to a well-established number of mechanism movements that have been explored among traditional continuum mechanisms reported in the literature (Sakes et al. 2018;Ali et al. 2019;Henselmans et al. 2020;Henselmans, Smit, and Breedveld 2019;Gerboni et al. 2015;Coemert et al. 2020).Similar to reported literature, the design topologies allow the user to select the desired number of degrees of freedom (DOF) (Legrand et al. 2021;Eastwood et al. 2018).For symmetric designs, the number Figure 16.The was subjected to a loading of F = 1.8 N. The deviation between the experimental and modelled curves for OPS-B (left) was 1.1 mm.For OPS-U (right), the deviation was 2.0 mm. of actuation wires is directly related to the number of DOFs.A single wire corresponded to one DOF.Additionally, a series of symmetrical sections can lead to other actuation freedom, but this results in higher complexity in wire routing, which is a widespread limitation for continuum mechanisms (Sakes et al. 2018;Culmone et al. 2020;Henselmans et al. 2020;Gerboni et al. 2015).Especially for miniaturised devices, the routing of micro-sized wires is time-consuming.Therefore, the asymmetric design aimed to reduce the complexity of wire routing and supports a controlled and patient-specific movement of the actuated mechanism.The in-plane S-curved shape required at least two wires to realise a 2-DOF that passed through both sections without any intermediate wire outlet or fixation points (Chitalia et al. 2020;Pacheco et al. 2021;York et al. 2015).The realisation of a spatial mechanism was even achieved with one actuation wire, which is an unprecedented benefit.However, these potentials limit their DOF to specific motion planes, which can be complemented by the rotational motion freedom (roll) of the mechanisms.Another advantage of the asymmetric design is its operational simplicity concerning the controlled and actuated DOF.This ease reduces the chance of undesired surgical movement during the procedure and reinforces a user-specified movement.

Automated computational design workflow
The computational design workflow identified the input curve as a C-, L-, or S-curved shape and generated an output geometry corresponding to the design topology and design approach.This automation enabled the rapid generation of different inverted designs without the need for manual notch volume configuration.
The benefit of inverse synthesized designs becomes particularly evident when large and customized structures are desired.Since each BB features eight notch volumes that can be individually defined, the number of parameters to be set quickly increases.For example, the IPS consisted of 114 BBs, and therefore, featured 456 notch volume parameters.By automating this process, the user can save considerable design time and avoid the need for further expertise for the design in AM CMs.This also supports its applicability for a broader interdisciplinary team of (non-)engineers, such as physicians or other health sector employees.In the case of C 2 , it provided proofs that the computational design workflow enabled a forward and automated design synthesis of the notch volume configuration.The direct definition of the volume notch configuration was limited to the level of C-curved shapes, where the size notch volume was kept symmetrical and constant.
The analytical model implemented within the computational design workflow served as an algorithmic design tool, translating between the curvature radius of the user-defined curve and the bending deformation of the BBs.The relation between stiffness, actuated force, and resulting curvature was validated experimentally through various case studies of continuum mechanisms, provided proofs for the design and manufacturing concept.A maximum deviation of 0.6 and 2.0 mm was found (nearly 9%), including a compensation factor c = 2.5, which compensated for various sources of error (e.g. the robustness of 3D-printed compliant structures due to process-induced errors, material anisotropy, and assumed mathematical simplifications [Feng, Hong, and Xie 2020b;Hu et al. 2019b;Wei et al. 2021]).This diversity of the maximum deviations among the case studies can be attributed to their different actuation aspects, such as the number of actuated wires and the magnitude of applied forces.Additionally, the camerabased experimental setup was favourable for testing in-plane continuum mechanisms, as opposed to more complex out-of-plane motions.In endoscopic mechanisms applications, the camera functionality allows for compensation of deviation between the target and actuated curve, enabling the user to visually compensate for the calculated force until the target location is reached.
Despite this, the integration of a simplified analytical model as a design tool within the computational workflow facilitated the design exploration by synthesizing various automated design variants with low-computational power and time.As this simplified analytical model is not a substitute for a more detailed structural analysis, the explored design variants can be further analysed by comprehensive numerical finite element analysis (FEA) methods.Consequently, in this work, the computational workflow was limited in its ability to provide a more accurate representation of the system's structural behaviour (e.g.local mechanical strains or stresses).Therefore, an extension of the computational workflow with additional FEA tools is an area for future work.Furthermore, the use of FEA might be crucial for identifying the root cause of various sources of aforementioned errors.It should be noted that integrating the FEA tool requires approaches that can be realised with low computational power and time (Huang et al. 2020).

Advancement in micro-LPBF
The micro-sized BBs were manufactured by downsizing the parameters of the scalable LPBF process, such as layer thickness, laser spot size, and powder particle diameter.They had a minimum beam thickness of t = 80 μm and a micro-sized actuation hole diameter of ∅ < 100 μm.However, the fabrication of micro-sized holes oriented horizontally to the printing direction was prone to material sagging-in (Feng et al. 2020a).To compensate for this, a rectangular hole and an additional tolerance margin of 100 μm hole size were designed.This approach resulted in micro-sized actuation holes that can be navigated using wires of 50 μm.Despite the challenges associated with the micro-LPBF process, its design freedom has enabled the creation of miniature and automated designs synthesis of continuum mechanisms.The self-supporting manufacturability and micro-LPBF were demonstrated in six case studies of continuum mechanisms, with a particular focus on intricate free-form features such as microsized actuation holes and thin walls.Geometry-related methods were employed to compensate for defects.However, further research is needed to fully understand the impact of process-induced defects on micro-LPBF techniques, similar to the extensive studies conducted on macro-LPBF techniques (Alghamdi et al. 2020;Dallago et al. 2019;Echeta et al. 2020;Fotovvati and Asadi 2019;Jones et al. 2021;Khurana, Hanks, and Frecker 2018;Snyder et al. 2015).This includes optimisation of the process for different micro-sized actuation hole diameters, and a combination of destructive and non-destructive measurement techniques (e.g.micro-CT) to provide a more comprehensive and detailed understanding of the samples (Feng et al. 2020a;Snyder et al. 2016).Additionally, methods for addressing defects through design-related adjustments (e.g.print orientation or geometric tolerance margins), pre-processing adjustments (e.g.tool-path and slicing), in-processing adjustments (e.g.optimisation of machine process parameters), and post-processing (e.g.micro-sandblasting, micro-EDM) should be studied.It is also worthwhile to consider the potential impact of geometrical distortion on the continuum mechanisms due to process-induced thermal stresses in micro-LPBF.Within this work, postbuild tempering using H900 helped to reduce residual stresses within the part.
To overcome the challenges associated with creating micro-sized actuation holes, alternative technologies could be also included.Wireless actuations, such as using magnetic fields (Shao et al. 2021) or thermal changes in shape-memory alloys (Farber et al. 2019), can lead to further miniaturisation (Charreyron et al. 2021;da Veiga et al. 2020) and improved functionality.Additionally, further optimising micro-LPBF process parameters is crucial for ensuring the mechanisms to function as a steerable instrument for MIS applications such as catheterisation, and for scaling the circumference diameter of the mechanism below 3 mm (Ali et al. 2019).
Furthermore, higher resolution in the process parameters enables a circular, self-supported mechanism design with smaller dimensions of the micro-sized BBs (Figure 17b).The maximal concave radii should be < 1 mm for self-supported circular cross-sections in analogy to horizontal self-supported concave radii < 3 mm for common LPBF (Kamat and Pei 2019;Thomas 2009;Wang et al. 2013).In this work, the contour of the mechanism was focused on a squarelike form to follow up on previous work (Danun et al. 2021) and to satisfy the size requirements for MIS < 4 mm (Figure 17a).A squared-like mechanism design also meets the requirements for a surgical instrument as reported in other studies (Coemert et al. 2020).In addition, it should be noted that the contour of the mechanism plays a role in its range of controllability.A more symmetrical contour has a higher benefit for the number of wire-actuated points, leading to a greater capability of motion in various directions compared to a squared mechanism design with four more controlled corners of actuation (Li et al. 2017)

End-to-end digital process chain
The overall ongoing goal of this research is to develop an end-to-end digital process chain that incorporates patient-specific input parameters from pre-operative imaging data (e.g.CT or MRI) and automatically generates an STL file for rapid manufacturing of the patientspecific instrument.This advancement in the applicable digital process chain is expected to significantly reduce time and costs in the design phase (Bernhard et al. 2022;Krieger et al. 2016), as compared to manual design and traditional manufacturing technique which often bring complexity in assembly and manufacture (da Veiga et al. 2020).The resulting mechanisms can be integrated into actuated robotic systems and handheld surgical instruments and can be tested for performance in both phantom and clinical trials (Eastwood et al. 2020;Hu et al. 2019a).To achieve this goal, it is essential to optimise the lead time and cost of manufacturing patient-specific surgical instruments.Addressing the limitations of a shorter lead time from imaging data to finished part and lower costs in AM are consequently crucial for success (Desai et al. 2019;Martelli et al. 2016).

Conclusion
This work presents the micro-AM of monolithic continuum mechanisms for use in MIS.The mechanisms are hierarchically synthesized from individual BBs or their subcomponents, which are compliant beam elements.By locally defining the beam stiffness using integrated notch volumes, the deformation of each BB can be tailored.This design approach is complemented by symmetric and asymmetric design topologies, making the mechanisms patient-specific and customizable in their curved shapes.The programming of the BBs allows for customization of the mechanisms.An automated computational design workflow is used to synthesize the BBs both forwardly and inversely, and an analytical model is implemented as a design tool to transform the discretized user-defined curve into BB deformation.This results in different curved shapes, including symmetric C-and L-shaped curves and asymmetric inplane and out-of-plane S-shaped curves.Six case studies are fabricated using the micro-LPBF process and the material 17-4 PH (H900), four symmetric and two asymmetric.The miniaturised mechanisms are sized to an outer contour dimension of 3.4 mm by downsizing the process parameters: layer thickness, laser spot, and powder particles.Each micro-sized BB has a beam element wall thickness of 80 µm and 100 µm micro-sized actuation holes for actuation with a steel wire outer diameter of 50 µm.The manufacturability and applicability of the case studies are evaluated with the loading of the mechanisms under calculated forces.The deviation between the measured and designed curves ranges between 0.6 and 2.0 mm, which represents maximally 9% of the overall mechanism's length.Further optimisation of the process parameters is expected to allow further downsizing of the mechanism to ∅ < 3 mm for catheter-based applications.The transition from a self-supported diamondlike structure to a circular shape design is foreseen with additional parameter optimisation, similar to selfsupported diameter < 5-6 mm for common LPBF.An end-to-end demonstration of this established digital process chain from user-defined surgical curve to patient-specific shaped curve in clinical intervention is an ongoing vision.

Notes on contributors
Aschraf N. Danun is a doctoral student and research associate at ETH Zurich, Switzerland.He holds a master's and bachelor's degrees in mechanical engineering from TU Munich, Germany, which he received before joining ETH Zurich in 2018.His doctoral research focuses on the design for additive manufacturing of compliant mechanisms, with a specific emphasis on potential applications in the medical and robotic field.
Remo Elmiger is a graduate of ETH Zurich, Switzerland, where he received both his master's and bachelor's degrees in mechanical engineering.He is currently employed in the private sector and his research interests lie in the areas of additive manufacturing and design automation.
Fabio Leuenberger is currently a master's student at ETH Zurich, Switzerland.He completed his bachelor's degree in mechanical engineering from ETH Zurich in 2021.His research focus encompasses the fields of additive manufacturing, medical engineering, and robotics.
Luca Niederhauser is a recent graduate of ETH Zurich, Switzerland, where he received his master's degree in mechanical engineering in 2023, after completing his bachelor's degree in mechanical engineering at ETH Zurich in 2020.He has a focus on additive manufacturing, compliant mechanism designs, mechanics, and control systems.
Jan Szlauzys obtained his diploma as a building technician from the Technical School Suwalki, Poland.He has been working as a laser technician specialist for LaserJob Rapid.3DGmbH since 2015, where he processes laser technology for metallic additive manufacturing.
Lorin Fasel is a doctoral student and research associate at University of Basel, Switzerland.He graduated from ETH Zurich, Switzerland, with a bachelor's degree in mechanical engineering and master's degree in robotics, systems, and control.Since 2018, he is pursuing his PhD at the BIROMED-Lab, where he is focusing on novel approaches for the development of flexible robotic endoscopes.
Mirko Meboldt is a Full Professor for Product Development and Engineering Design at ETH Zürich, Switzerland since 2012.His research focuses on design for new technologies with a special emphasis on AM and digital value chain.He develops methods and tools for design automation that combine application, design, and process knowledge for the best use of AM in an industrial context.

Figure 1 .
Figure 1.The user-specified S-shaped curve (left) is automatically transformed into a steerable continuum mechanism (right) with design-programmable and micro-sized BBs.A match is used for scale purpose.

Figure 2 .
Figure 2. A schematic overview for the automated design programming and micro-LPBF of the continuum mechanisms.

Figure 3 .
Figure3.A self-supporting BB configuration is composed of four beam elements (a).By aggregating these beam elements in a serial manner, a monolithic continuum mechanism is synthesized (b).This mechanism features integrated holes for the actuation wires (c).
, an asymmetrical volume configuration is illustrated.For clarity, the notch volume is highlighted in red.The beam elements, C-B and D-A of the BB, consist of reduced l eff each, while the remaining beam elements (D-C and B-A) have an unconstrained notch volume (size notch volume = 0%)) combination.This configuration realises the asymmetric actuation behaviour of the BB.Actuation forces at holes C and D cause the beam elements to deform in the loading direction.However, the reduced l eff in D-C and B-A results in axial rigidity when they are actuated at hole A or B.

Figure 4 .
Figure 4.The bending deformation (Du) and curvature radius (k i ) are tailored by sizing the beam length (l beam ) of the BBs using tailorable notch volumes (a) and (b).The size of the notch volume (the red-coloured area) increases or decreases the effective beam length (l eff ).A size notch volume = 0% results in the longest possible effective beam length, while size notch volume = 100% results in a completely rigid beam.Therefore, this tailoring defines the sized beam length (l size ) that is noted as the darkened red region in (c).For traceability purposes, a general notation is used to describe the tailorable BBs.

Figure 5 .
Figure5.The integration of the design approach of BB tailoring with programmable design patterns results in both symmetric and asymmetric mechanism configurations.The C-curved and L-curved mechanisms have symmetric design configurations, while the inplane and out-of-plane S-curved mechanisms have asymmetric configurations.

Figure 6 .
Figure6.The computational design workflow for the automated user-specific shape-programming of the steerable mechanism contains multiple steps.A user-defined curve (a) is discretized in single i-segments corresponding to the BB geometry (b).The radius curvature k i and bending angle w i of each segment (c) are transformed via a design algorithm to corresponding notch volumes (highlighted reddish in (d)).The combination of the aggregated mechanism and the derived notch volumes leads to the inversed programmed shape of the mechanism (d)-(e).

Figure 7 .
Figure7.The design of micro-sized features, such as a wall thickness of t = 80 μm and a actuation hole diameter of ∅ = 100 μm, was achieved by downscaling three main process factors: utilising a smaller laser spot diameter of 30 μm, a thinner layer thickness of 12 μm, and smaller powder particles with an average diameter of 15 μm.Additionally, the common design guidelines of macro-LPBF techniques, such as the overhang angle γ ≥ 45°, were also considered in the design process.

Figure 8 .
Figure 8. Six case studies of mechanisms were conducted to validate their forward and inverse shape programming.Both symmetric and asymmetric design topologies were considered for validation.The starting case was the symmetrically designed C-curved shape C 1 , whose effective beam length was stiffened by 23.5% resulting in C 2 .A continuous stiffening of the BBs based on the given curves of the designer resulted in the L-curved shapes L 1 and L 2 .The continuity is represented by the black arrows, where the start of the arrow denotes the lowest stiffness and its end represents the highest stiffness.The dashed line indicates the symmetrical properties of the structures.The in-plane and out-of-plane asymmetric designed S-curved shapes also show this representation and labelling.

Figure 9 .
Figure 9.In the design of symmetrical C-shaped curves, the C 2 curve was constrained by a 23.5% notch volume size, while the C 1 curve had the maximum flexibility and a minimum notch volume size of 0% BB

Figure 11 .
Figure11.The L-shaped samples, L 1 and L 2 , were inversely transformed by implementing a continuous change in stiffness from 0% to 35% (L 1 ) and 0% to 22.5% (L 2 ).The colour gradient in the figures represents a continuous variation in the notch volume size of the rectangular bars.The force was applied at hole C, represented by the blue-coloured square, for both cases.

Figure 13 .
Figure 13.The upper (IPS-U) and lower (IPS-B) bending sections of the in-plane S-shaped sample are related to each other through a 180°rotation (clockwise, as indicated by the black arrow).The side-ways blocking of both sections was achieved by constraining the l eff of the BBs by 60%.The transformation of the user-defined curve in the IPS mechanism was accomplished by varying the l eff from 0% to 40%.Hole B was used for actuating IPS-B, and hole D was used for actuating IPS-U.

Figure 14 .
Figure14.The experimentally tested curve of the IPS (shown in red) exhibited a maximum deviation of 0.6 mm from the modelled curve (shown in blue) when a F = 2.0 N was applied.

Figure 15 .
Figure 15.The blockage of OPS-B and OPS-U was accomplished through the use of a notch volume size of 60%.The change in k i was conveyed through a user-specified curve by gradually varying the notch volume size from 0% to 50%.The anticlockwise black arrows indicate the 90°relationship between the upper section of OPS-B and the lower section, OPS-U.Both sections were loaded at hole D.

Figure 17 .
Figure17.Similar to meso-and macro-LPBF, the boundaries of self-supported circular cross-sections in micro-LPBF should be investigated for ∅ < 2 mm by further downscaling the micro-BBs.This requires higher resolution in the micro-LPBF process (b).However, sizing the outer contour dimension of the squared mechanism at 3.4 mm due to process-related design restrictions made it suitable as a functional component MIS and benefited from self-supported manufacturing (a).

Table 1 .
In its initial and unconfigured state, the dimensional parameters used to define a BB have a curvature radius of k o .

Table 2 .
The micro-LPBF of the metallic continuum mechanisms was fabricated with the hardened stainless steel 17-4 PH (H900) and utilised the following material and process parameters.