Stiff yet Bendy: Tubular Transmissions for Driving Surgical Robots through Flexible Endoscopes

Abstract

Concentric push-pull robots delivered through flexible endoscopes work best if their laser-cut transmission tubes have high axial stiffness, high torsional stiffness, and low bending stiffness. This paper simultaneously addresses all three output stiffness values in the transmission design problem, explicitly considering axial stiffness, whereas prior work on laser-cut tube design has focused on the bending/torsional stiffness ratio. We demonstrate an inherent trade-off present in existing laser-cut patterns: it is difficult to simultaneously achieve high axial stiffness and low bending stiffness because these properties are very tightly correlated. To break this correlation and design all three stiffness independently, we propose a new type of laser material removal pattern that leverages local stiffness asymmetry ($E{I}_x\ne E{I}_y$) in discrete bending segments separated by segments of solid tube. These discrete asymmetric segments are then rifled down the tube to achieve global stiffness symmetry. We parameterize the design and provide a study of the properties through finite-element analysis. We also consider the effect of interference between the tubes when the discrete segments are not aligned. Results show that our discrete asymmetric segment concept can achieve high axial stiffness and torsional stiffness better than previously suggested laser patterns while maintaining equally low bending stiffness. We also experimentally validated the proposed design’s properties and actuation performance with professionally manufactured prototype Nitinol tubes for use in an endoscopic robot system.

I. Background and Motivation

A. Medical Motivation

Colorectal cancer (CRC) is the second leading cause of cancer-related death in the United States. In 2023, an estimated 153,020 people in America will be diagnosed with CRC, and one third of those diagnosed will die from it [1]. Of the patients diagnosed with CRC, many are identified at the local stage and are great candidates for minimally invasive, outpatient endoscopic surgeries. Despite this, many of these patients are required to undergo invasive colectomies, wherein a length of their colon is removed [2]. This is due to the fact that there are limited experienced colorectal surgeons that have the skill set to manipulate traditional endoscopic tools to perform minimally invasive procedures [3]. Endoscopic submucosal dissection (ESD)—a less invasive procedure—is the preferred method for removing these tumors due to the quick recovery time and outpatient nature of the procedure [4], [5].

The tools needed for an ESD procedure currently include a colonoscope and standard trans-endoscopic instruments that pass through the colonoscope tool channels (e.g. electrosurgery, probes, graspers, etc.). The colonoscope is designed for navigation rather than manipulation, and is thus an unwieldy tool that is unsuitable for making dexterous motions present in lesion resection. Current trans-endoscopic instruments can only be inserted along the tool channel axis or rotated axially. They do not have the ability to actively bend or steer independently from the colonoscope. Poor maneuverability and the inability to reach around large polyps have been identified as fundamental challenges to performing ESD in the colon [6]. These factors result in longer operating times and steep surgeon learning curves, which consequently force many patients per year to undergo more invasive surgeries than necessary.

B. Concentric Push-Pull Robot Steerable Segments

A Concentric Push-Pull Robot (CPPR) steerable sheath structure [7], [8], illustrated in Figure 1(a), consists of two thin-walled tubes made from elastic material whose tip regions have had material selectively removed by laser cutting and are attached to one another at their tips. The material removal is asymmetrically biased so that the effective neutral axis is offset away from the geometric center of the tube [8]. Due to neutral axis offset and fixed together tips, translating the tube bases relative to one another induces bending in the tip. A CPPR steerable tip is thus simple in construction and assembly as there is no need for tendon channels which enables it to be very thin walled and free central lumen for surgical tools, hence makes it well-suited to ESD procedures. As investigated in [7] and [8], the laser pattern in the steerable segment need not be constant, and variable curvature shapes can be designed using length-varying laser patterns.

Figure 1.

(a) A CPPR sheath consists of two laser patterned Nitinol tubes attached at their tips. (b) The transmission section is also made from laser-patterned Nitinol tubing and passes through the working channel of the colonoscope.

C. Transmission Segments

The push-pull motions of the CPPR steerable tip are transmitted from the endoscope’s handle to its tip by a transmission section that extends through the endoscope (Figure 1(b)). This transmission can be laser patterned to increase bending flexibility, such that it does not impede motion of the endoscope by increasing its stiffness. The neutral axes in the symmetrically patterned transmission tubes are aligned so that the push-pull actuation forces do not induce any bending in the transmission portion. This completely decouples steerable segment actuation from the transmission. The plane in which the steerable segment bends can be adjusted by axially rotating the transmission tubes at their bases, endowing a single tube pair with three degrees of freedom (insertion, bending, and rotation). The decoupling of transmission shape from steerable segment shape also enables multiple steerable segments to be combined. For example, in Figure 2, a dexterous system for ESD may contain two arms, each made of two independently actuatable tube pairs. The inner tube pair transmission segment exists inside the outer steerable sheaths segment, decoupling the actuation of each tube pair from the other.

Figure 2.

A bi-manual steerable sheath system. The Outer Sheaths pass through the colonoscope. The Inner Sheaths pass through the Outer Sheaths and hold surgical tools within their inner lumen. Each sheath is composed of a pair of tubes and has 3 degrees of freedom (DOF) (insertion, bending and rotation).

The design of the transmission segments is critical to enabling precise and reliable motion of the steerable tip segments. The transmission portion must have high torsional stiffness $\left(GJ\right)$ in order to transmit rotation with minimal reduction, high axial stiffness $\left(AE\right)$ to drive the push-pull actuation of the tip and overcome the tool channel friction, and yet have low bending stiffness $\left(EI\right)$ in all directions in order to be deployed through curved tool channels without affecting the endoscope’s functionality. To achieve all these properties simultaneously, we propose to modify solid Nitinol tubes by laser cutting patterns of slots into the tube walls.

Choosing a laser cut pattern to optimize the various stiffnesses of a tube is an active area of research with applications in catheter shafts [9] and concentric tube robots [10]. In these applications, engineers are primarily concerned with reducing bending stiffness while maintaining high torsional stiffness. Axial stiffness is desirable, but often less critical, so it is more rarely studied. Furthermore, with existing laser patterning techniques, such as brickwork slot [11], dog bone [9], [12], [13], diamond-hole lattice [14], and auxetic patterns [13], [15], [16], the axial stiffness is usually tightly coupled to the bending stiffness, such that the two move almost in lockstep, which we will illustrate later in this paper. But axial stiffness is a critical property for concentric push-pull robots designed for colorectal ESD. The push-pull actuation forces and translations are carried by transmission tubes that must traverse the curved two-meter length of the colonoscope. To overcome tool-channel friction and reduce hysteresis and elongation during actuation, the transmission tubes need high axial stiffness.

D. Contributions

Thus, in this paper, we seek a way to decouple the axial and bending stiffness, in order to create push-pull transmission tubes with high axial and torsional stiffness and low bending stiffness. We contribute the following ideas and analyses toward this goal:

• A new type of laser-cut pattern based on creating a series of short discrete segments with localized stiffness asymmetry.

• Parameter sweeps of conventional brickwork slot patterns and the proposed discrete pattern using thousands of tests with finite-element analysis.

• An optimization framework for transmission design.

Portions of this work appear in the proceedings of the International Symposium on Medical Robotics [17]. In the current evolved paper, we significantly advance beyond [17] by adding the following contributions:

• Experimental validation of the model-predicted stiffnesses of professionally laser-cut prototype tubes, demonstrating model accuracy and improved stiffness properties over conventional designs.

• Implementation and optimization of the proposed transmission designs for a full bi-manual robotic system for ESD.

• Validation of the proposed transmission tubes’ performance during robot actuation.

II. Conventional and Proposed Design Concepts

A. Conventional Brickwork Designs

As a baseline for comparison, we describe and later analyze the designs made by conventional “brickwork” cut patterns [11]. As shown in Figure 3, the main parameters of a brickwork slot pattern are kerf $k$, pitch $h$, cuts per revolution $n$, and overlap angle $\beta$. This type of design is very common in the catheter shafts industry [18] and is representative of many other designs that all have locally symmetric bending stiffness. Our FEA analysis in the following sections will demonstrate the strong coupling between axial and bending stiffness that characterizes these conventional designs.

Figure 3.

Flattened view of an example of the conventional brickwork design: 2 cuts per revolution. The shaded area indicates cut out sections

B. Proposed Discrete Design

To meet the unique requirements of our bi-manual robot for ESD (high axial and torsional stiffness, low bending stiffness), we propose a new type of design that leverages local asymmetry in bending stiffness. Our proposed cut pattern is pictured in Figure 4 and is parameterized by the number of cuts per revolution $n$, slot kerf $k$, pitch $h$, separation angle $\alpha$, and overlap angle $\beta$. The total length of the bending segment is

$$L_b=(N-1)h+k$$

where $N$ is the number of levels in the bending segment. Successive bending segments are arranged serially along the tube length, separated axially by a length $g$ of unmodified solid tube, and angularly by the rifle angle $\theta$.

Figure 4.

Slotted patterns used for FEA characteristic and optimization studies. The shaded areas indicate cut out sections. Pattern parameters are designed using an unrolled pattern that is wrapped over the tube. A six cuts per revolution pattern forms six overlapped angles from 0° to 180° between two consecutive revolutions: one separation angle $\mathit{\alpha }$ and five overlap angles $\mathit{\beta }$. The blue arrows indicate the bending direction of each asymmetric bending segment.

Each asymmetric bending segment is designed to achieve low bending stiffness in a specific direction and high axial stiffness by introducing the separation angle $\alpha$ which creates two “posts” of solid material 180° apart which bend easily but stiffly resist compression and extension. These solid posts may also improve the torsional stiffness of the segment by providing shear strain resistance at opposing points on the cross section. A downside is that the bending stiffness in the orthogonal direction remains higher, creating asymmetry. Thus, to make the global stiffness behavior symmetric, each successive bending section is rotated by $\theta$ with respect to the previous one, such that approximately symmetric stiffness is achieved in just a few segments. Some solid material length $g$, shown in Figure 4 between the discrete bending segments is necessary to shift the line of action of the axial forces without compromising axial stiffness. Through FEA sweeps of the $g$ parameter, we determined 15% to be a rough heuristic of when the length of $g$ starts to affect the axial stiffness more significantly when the tube is in tension/compression.

A rifle angle of $\theta =90°$ is an intuitive choice in order to achieve global stiffness symmetry with only two successive segments (or 3 successive segments for $\theta =60°,4$ for successive segments $\theta =45°$). However, in preliminary experiments, designs with $\theta =90°$ exhibited preferred bending directions, i.e. snapping behavior, due to the repeatedly aligned bending directions.Thus, we seek a rifle angle $\theta$ so that the next bending axis is as far as possible from all previous bending axes. This approach ensures that over the entire length, the transmission design has non repeating and evenly spaced bending directions in every direction. This suggests that $\theta$ should be an irrational fraction of 180°. The golden ratio $\varphi =\frac{1+\sqrt{5}}{2}$ is wellknown as the “most-irrational” number, in the sense that it is the least well-approximated by an integer ratio with small numerator and denominator [19]. Thus, to create the “least repeating” sequence of discrete bending directions, we choose our rifle angle to be $180°/\varphi \approx 111.24°$, or equivalently the supplementary angle $\theta =68.74°$ (which we call the “golden angle”). This produces a succession of evenly spaced bending axes that never repeat, and for which each successive axis is as far away as possible from any previous axis, thus eliminating any preferred bending directions. We visualize how the global stiffness symmetry would change as we add more bending segments along the length in Figure 5. The plotted symmetry metric is the condition number of the resulting effective global stiffness matrix (i.e. ratio of principal stiffnesses, where 1 is perfectly symmetric), assuming $E{I}_x/E{I}_y=0.1$ in each discrete segment. We can see that symmetry converges just as well when using the golden angle as it does for $\theta =90°$. The later FEA analysis section will demonstrate the de-coupling capability between axial and bending stiffness and a higher torsional stiffness than the conventional brickwork design mentioning below.

Figure 5.

The global stiffness symmetry (ratio of principle stiffnesses) is plotted as a function of number of discrete asymmetric segments for three possible designs with different rifle angles. A symmetric metric of 1 indicates that the principle stiffnesses observed at the tip of the transmission tubes are the same in the x - and y -directions.

III. Analysis

In this section, we will perform several analyses of our proposed laser-cut tube design based on discrete asymmetric segments. The context of all the analysis is planning for a robotic system to be deployed through a two-channel Olympus EVIS EXERA CF-2T160I colonoscope, so our analyses will be performed on tube sizes appropriate for this application. The selected tubes are described in Table I. Each arm is comprised of four tubes (two concentric push-pull pairs). The endoscope has tool-channel diameters of 3.7 mm and 3.2 mm. Radial clearances of 0.1 mm between two adjacent tubes are sufficient to enable smooth push-pull operation with minimal friction while also facilitating a strong weld bond at the tips. The $OD/ID$ (Outer Diameter/Inner Diameter) ratio for these chosen tube sizes is chosen to be 1.08, which is well-within standard hypotube production capabilities. Using the same $OD/ID$ for all tubes is also convenient for analysis because it allows nondimensionalized simulation results to be applied across the entire tube set. In the following sections of this paper, we will use the largest two tubes in the 3.7 mm channel, Tube 1 and Tube 2, for the analysis, manufacturing and experimental validations.

Table I.

Selected Tube Dimensions

	OD (mm)	ID (mm)	Arm 1	Arm 2
Tube 1	3.3	3.04	✔
Tube 2	2.84	2.62	✔	✔
Tube 3	2.42	2.24	✔	✔
Tube 4	2.04	1.88	✔	✔
Tube 5	1.68	1.54		✔

✔: tube is a member of the outer pair.

✔: tube is a member of the outer pair.

A. Geometric Analysis of Achievable Curvature

The achievable curvature of a single laser cut tube is limited by slot closure, which we measure in simulation by extrapolating the FEA solution of a single tube. Furthermore, the achievable curvature of a concentric tube pair is potentially limited by inter-tube contact when the discrete bending segments are not aligned due to tube translation. We can conservatively approximate this effect by considering a worst-case scenario where $\theta =90°$ is used on two concentrically nested discrete transmission tubes and their bending axis are perfectly out of phase as depicted in Figure 6. In this case, aligned bending segments are repeated after every two successive segments, enabling us to analyze the geometry in a plane. Analyzing the geometry using the small-angle approximation then yields a conservative max achievable curvature $u_{max}$ as a function of the design parameters and the clearance between the tubes:

$$u_{max}\approx \frac{8c}{L^2},L=2\left(L_b+g\right)$$ (1)

In our design analysis and optimization below, we use this relationship to drive the parameter $g$ (given values for $N,h,k$, and $c$) in order to make sure that the designs we are analyzing can always achieve a given maximum curvature (such as the curvature of the colonoscope), individually and when paired with another tube.

Figure 6.

An exaggerated view of tube contact occurs between 2 discrete design tubes with the assumption that the bending plane of the inner tube is misaligned with that of the outer.

B. FEA Stiffness Analysis - Parameter Sweep

In the discussions that follow, we nondimensionalize the effective stiffness parameters as $A{E}^{*}=AE/A{E}_0,E{I}^{*}=EI/E{I}_0$, and $G{J}^{*}=GJ/G{J}_0$, where a subscript of zero denotes the corresponding stiffness of the unmodified solid tube. The superscript $*$ then denotes the ratio of that original stiffness value.

We performed a parameter sweep over the design space for brickwork patterns to illustrate the limitations of this conventional design. The parameter ranges included in the sweep were kerf $k$ from 50 to 150 microns, pitch $h$ from 200 to 500 microns, cuts per revolution $n$ from 2 to 6, and overlap angle $\beta$ from −10 to 30 degrees. We uniformly sampled 5 points over each of the above ranges in a nested loop to generate 625 unique designs, and performed FEA tests on Tube 1 to determine the resulting effective $E{I}^{*},E{A}^{*}$, and $G{J}^{*}$ of the brickwork pattern.

We also performed a similar parameter sweep for the proposed discrete design, with kerf $k$ from 50 to 150 microns, pitch $h$ from 200 to 500 microns, separation angle $\alpha$ from 5 to 30 degrees, overlap angle $\beta$ from 5 to 30 degrees, $n=6$, and $N=6$, while $g$ was implemented as a driven parameter to satisfy a maximum curvature of $u_{max}=0.02{\text{mm}}^{-1}$, according to Equation 1 (corresponding to a colonoscope radius of curvature of 50 mm). We again uniformly sampled 5 data points from each design parameter in a nested loop, generating 625 designs, and computed all effective stiffnesses using FEA.

The output stiffnesses generated by our parameter sweeps are depicted in the shaded regions of Figure 7(a) and (b). Figure 7(a) illustrates the strong coupling between $E{I}^{*}$ and $A{E}^{*}$ of the conventional brickwork design. All of the conventional designs lie on the $E{I}^{*}=A{E}^{*}$ line, indicating the percent reductions in $AE$ and $EI$ are always the same. In contrast, the proposed discrete design breaks this strong coupling and expands the $A{E}^{*}/E{I}^{*}$ parameter space significantly, enabling $A{E}^{*}$ values more than 3 times higher than previously possible at low $E{I}^{*}$ values. Figure 7(b) shows that brickwork design results in a wide range of $E{I}^{*}/G{J}^{*}$ ratios, including undesirable ones with low $G{J}^{*}$, but it is difficult to achieve high $G{J}^{*}$ with low $E{I}^{*}$. In contrast, the discrete design space does not contain the undesirable low $G{J}^{*}$ region, and its max $G{J}^{*}$ is larger than the best possible conventional design for any given $E{I}^{*}$. These results illustrate the superiority of the proposed discrete designs in lieu of conventional brickwork designs for concentric push-pull transmission tubes for flexible endoscopic applications. Discrete transmission designs can achieve low bending stiffness while maintaining the high axial and torsional stiffness necessary for push-pull actuation over long tool channels. Figure 8 illustrates the space of $G{J}^{*}$ and $A{E}^{*}$ for brickwork designs and discrete designs, with contours describing the minimum $E{I}^{*}$ required to achieve a given combination of $A{E}^{*}$ and $G{J}^{*}$. This relationship may be helpful in determining practically achievable goals for future designs.

Figure 7.

Comparison of output parameter space between conventional brickwork pattern and discrete pattern. While the conventional brickwork design exhibits a strong coupling between $E{I}^{*}$ and $A{E}^{*}$, the proposed discrete design decouples them and achieves higher $A{E}^{*}$. It also improves $G{J}^{*}$ when comparing designs with similar $E{I}^{*}$.

Figure 8.

Contour plot of the minimum $E{I}^{*}$ required for a given combination of $A{E}^{*}$ and $G{J}^{*}$, and the example optimized stiffnesses for Tube 2. In all cases, for a given $E{I}^{*}$, higher $A{E}^{*}$ and $G{J}^{*}$ can be achieved with the discrete design. These charts can be used to visualize the tradeoffs between all three stiffness parameters as a design aid.

C. Parameter Optimization

To inform the design of each tube in our robotic system, we conducted a preliminary design optimization using Tube 1 of Arm 1 in Table I. By setting $k=0.08,\theta =68.74°$, we let $\alpha ,\beta$, and $h$ be the free variables for optimization. The parameter $g$ was driven by (1) as a function of these variables to ensure compliance with the curvature constraint $u_{max}=0.02{\text{mm}}^{-1}$. Three distinct optimization routines were executed for different combinations of the $N$ and $n$ parameters: $n=4,N=4,n=4,N=6$, and $n=6,N=6$. An optimization was also performed for $\theta =45°,n=4,N=4$ for manufacturing of tubes to validate preferred bending angles.

This optimization approach generally encodes our design goals into a constrained optimization problem expressed as:

$$\begin{gathered} \text{maximize}:f(\mathit{\alpha },\mathit{\beta },\mathit{h})=w_{t}G{J}^{*}+w_{a}A{E}^{*} \\ \text{Subject}\text{to}:E{I}^{*}\le E{I}_{\mathit{\text{max}}}^{*} \end{gathered}$$ (2)

where $w_t$ and $w_a$ are scalar weights on torsional and axial stiffness.

For this preliminary design optimization, we chose $E{I}_{\text{max}}^{*}$ such that the $EI$ of each tube is less than 4% of the colonoscope’s stiffness, which is measured to be $\mathrm{26,100}{Nmm}^2$ [20]. For Tubes 1 of Arm 1, this results in $E{I}_{\text{max}}^{*}=1.03\%$. We select the weights as $w_t=1$ and $w_a=3$ to account for the fact that $G{J}^{*}$ is expected to be about 3 times larger than $A{E}^{*}$ for typical designs. We used a Simulated Annealing algorithm [21] for the optimization process because it is well suited to find a global optimum.

Results of the preliminary optimization of Tube 1 are shown in Table II. All of the discrete designs achieve significantly higher $A{E}^{*}$ than $E{I}^{*}$, and $G{J}^{*}$ is much higher still. Especially, the designs with $N=6,n=6$ achieve an $A{E}^{*}$ more than 2 times higher than $E{I}^{*}$. This is a significant improvement over the optimized conventional design. The optimized stiffnesses all lie on the upper boundaries of the parameter spaces depicted in Figure 7(a) and (b). In terms of practical performance gains, the higher axial stiffness afforded by our discrete design would result in approximately 2 times less axial elongation/compression during push pull actuation. This is a significant factor in the ability of a push-pull robot to operate with high-precision and overcome transmission friction and lag.

Table II.

Optimal Discrete and Brickwork Designs of Tube 1 and theirs corresponding stiffness from Fea and Experiment

		Design 1		Design 2		Design 3		Design 4		Brickwork
Design Variable	$\alpha$ (degs)	−8.5		−7.7		−11.5		−15		5
	$\beta$ (degs)	15.7		12.7		17.8		16		N/A
	$\theta$ (degs)	68.74		68.74		68.74		45		N/A
	gap (mm)	2.02		1.4		2.07		0.865		N/A
	h (mm)	0.21		0.2		0.2		0.21		0.2
	n	6		4		6		4		N/A
	n	4		4		6		4		4
Results		FEA	Exp.	FEA	Exp.	FEA	Experiment	FEA	Experiment	FEA	Experiment
	$EI\left(Nm{m}^2\right)$	1076	1119.6	1035	1095.8	1100	1219.3	1104	1187.8	1003	978.5
	$GJ(Nm{m}^2)$	12748	13566.2	12303	11761.4	10734	11097.5	10643	11114.1	8035	8688.6
	$AE\left(N\right)$	2226	1824	1235	1005.3	2326	1796.1	1019	1099.3	1012	965.4

IV. Single Tube Experimental Validation

The optimized discrete and brickwork designs, found in Section III-C, were used to manufacture prototypes for validation in this section. The experimental results are listed in Table II. The bending stiffness, torsional stiffness, and axial stiffness were experimentally determined through the following.

1). Bending Stiffness:

The bending stiffness properties of the patterned tubes were determined by recording small deflection force-displacement data from a three-point bending setup, as shown in Figure 9(a). The data is calculated with the Euler-Bernoulli beam equation for a simply supported beam with a midpoint-load applied:

$$E{I}_{exp}=\frac{F{L}^3}{48{\delta }_{EI}}$$ (3)

where $E{I}_{\mathit{\text{exp}}}$ is the experimental bending stiffness of testing tubes, F is the applied midpoint force, L is the length of the supported beam, and ${\delta }_{EI}$ is the midpoint displacement of the beam. The tested tubes are supported with rollers at each end. The midpoint force $F$, ranging from 0 to 0.5N in 0.05N increment, was applied with a force gauge (Mark-10 MR03–20). A manual linear actuator is used to record a displacement ${\delta }_{EI}$ at the midpoint along the beam. A linear fit is performed on a graph with $\left(F/{\delta }_{EI}\right)$ as the dependent variable and $\left(L^3/48\right)$ as the independent variable. The slope of the linear fit corresponds to $E{I}_{\mathit{\text{exp}}}$.

Figure 9.

Bending, Torsional, and Axial Experiment Setup used for experiment on the five different designs of the transmission section.

2). Torsional Stiffness:

The torsional rigidity properties of the patterned tubes were determined by recording small deflection torque-displacement data from the torsional experimental setup, as shown in Figure 9(b). The data is calculated with the Hooke’s Law spring equation for a fixed proximal end and torque applied at distal end:

$$G{J}_{exp}=\frac{\tau L}{{\theta }_{GJ}}$$ (4)

where $G{J}_{\mathit{\text{exp}}}$ is the experimental torsional stiffness of testing tubes, $\tau$ is the recorded proximal torque, L is the length of the testing tubes, and ${\theta }_{GJ}$ is the applied angular displacements at the distal end. A manual micro-angular actuator is used to apply angular displacements ${\theta }_{GJ}$, ranging from 1 to 10 degrees in 1 degree increment, at the distal end of the tubes. The according torque at the proximal end was recorded by the Torque gauge (Mark-10 MR50–12). A linear fit is performed on the $\tau /{\theta }_{GJ}$ as dependent variable and $L$ as independent variable. The slope of the linear fit corresponds to $G{J}_{\mathit{\text{exp}}}$.

3). Axial Stiffness:

The axial stiffness properties of the patterned tubes were determined by recording small deflection force-displacement data from the axial stiffness experimental setup, as shown in Figure 9(c). The data is calculated with Hooke’s Law for a spring with fixed proximal end and force applied at distal end:

$$A{E}_{exp}=\frac{FL}{{\delta }_{AE}}$$ (5)

where $A{E}_{\mathit{\text{exp}}}$ is the axial stiffness of testing tube, $F$ is the recorded proximal force, L is the length of the supported beam, and ${\delta }_{AE}$ is the applied displacements of the beam. A manual linear actuator is used to apply a displacement ${\delta }_{AE}$, range from 0 to 0.625mm in 0.125mm increments at the proximal end. The according axial force was recorded by a force gauge (Mark-10 MR03–20). A linear fit is performed on the $F/{\delta }_{AE}$ as dependent variable and $L$ as independent variable. The slope of the linear fit corresponds to $A{E}_{\mathit{\text{exp}}}$.

4). Results:

The experimental results of $A{E}_{\mathit{\text{exp}}},E{I}_{\mathit{\text{exp}}}$, and $G{J}_{\mathit{\text{exp}}}$ of Tube 1 are listed in Table II, and are compared with the FEA results. Compared with the FEA result, the experimental results show an average absolute difference of 8.27% for EI, 17.69% for AE, and 4.92% for GJ. Additionally, the optimized discrete design with the 45° rifle angle qualitatively exhibited preferred bending directions, which is consistent with prior preliminary observations.

V. Implementation and Performance for an Endoscopic Robotic System

In this section we detail the tube design and validate the total transmission stiffness when actuating a steerable sheath tip.

A. Tube Design

The Olympus EVIS EXERA CF-2T160I colonoscope’s tool channels are each approximately 1500 mm long, and tools must pass through three sections with different curvature requirements in each. The entry section of the tool channel is approximately 120 mm long and has a measured fixed curvature of $0.025{\text{mm}}^{-1}$, annotated as Section A in Figure 10. The bending segment at the scope tip is approximately 150 mm long and able to be actuated to curvature as large as $0.05{\text{mm}}^{-1}$, annotated as Section C in Figure 10. The middle section of the scope is significantly longer, 1265 mm, and will experience a smaller curvature limit of $0.02{\text{mm}}^{-1}$, annotated as Section B in Figure 10. To make the transmission as stiff as possible, while still being able to transmit motion smoothly through the three endoscope segments, we employ a stiffer design pattern in the dominant Section B (based on our discrete pattern) with more flexible designs (based on a brickwork pattern) in the shorter Sections A and C.

Figure 10.

The inside view of the colonoscope channel when the Outer and Inner Steerable Sheaths pass through. Section A and C of the colonoscope are short and have greater curvature compare to the longer section $B$.

To find the optimal transmission design for section $B$, we use the process described in Section III-C, on the discrete design, with $k=0.025\text{mm},n=4,N=4$. The variables to be optimized are $\alpha ,\beta ,h$ with $g$ as a driven parameter such that curvature $u_{max}=0.02{\text{mm}}^{-1}$ is achievable, and $E{I}_{max}^{*}$ such that each tube’s absolute $EI$ is less than 5% of the colonoscope’s stiffness. That is $E{I}_{max}^{*}=1.5\%$ for Tube 1 and $E{I}_{max}^{*}=2.9\%$ for Tube 2. The steerable tip sections, annotated as section D in Figure 10 were designed with the asymmetric serpentine pattern design used in [8] (shown in Figure 1) and designed to have a maximum actuatable bending angle of 60° (limited by slot closure contact). The length of each tube is the sum of length of section A, B, C, and D and it ranges from 1550 mm to 1600 mm. The FEA optimized stiffness properties of each section of Tube 1 and Tube 2, as well as for each section of the other six tubes of the ESD robotic system are listed in Table III. Among the eight tubes of Arm 1 and Arm 2, three tube sizes overlap, as shown in Table I. Thus, only five unique tube sizes are listed. Arm 1 consists of tube sizes 1, 2, 3 and 4, while Arm 2 consists of tube sizes 2, 3, 4 and 5. The prototype tubes were custom extruded to our diameter specifications, and the slot patterns were cut with a femtosecond laser (Peiertech, China).

Table III.

Nominal Properties of Transmission Segment and Steerable Segment Designs

	Section	Design Pattern	$\text{AE}\left(\mathit{N}\right)$	$\text{EI}\left({N\mathit{\text{mm}}}^2\right)$	$\text{GJ}\left({N\mathit{\text{mm}}}^2\right)$
Tube 1	A & C	Brickwork	488	405	5456
	B	Discrete	1926	1480	8865
	D	Serpentine	*	185	211
Tube 2	A & C	Brickwork	414	409	2554
	B	Discrete	2121	1551	9245
	D	Serpentine	*	175	156
Tube 3	A & C	Brickwork	533	446	2699
	B	Discrete	2558	1315	4410
	D	Serpentine	*	172	154
Tube 4	A & C	Brickwork	588	360	1704
	B	Discrete	3000	1351	2945
	D	Serpentine	*	178	111
Tube 5	A & C	Brickwork	877	408	1307
	B	Discrete	4306	1289	1869
	D	Serpentine	*	103	89

^*The axial stiffness of the steerable tip is so high relative to its bending stiffness, that we model it as infinitely stiff.

B. Model of Transmission Performance and Calibration

The effective axial stiffness of the transmission segment of both Tube 1 and 2 are used to construct a model of relative tube displacement at the actuators ($q$) required to achieve a give tip bending angle $\theta$. The required actuation force $\tau$ to achieve a given bending angle $\theta$ is calculated using the properties of the steerable tip sections as

$$\tau =\frac{\theta }{d}\left[\frac{E{I}_1}{L}+\frac{E{I}_2}{L-d\theta }\right]$$ (6)

where $d$ is the distance between the stiffness centers of the two tubes [8], $L$ is the length of the steerable tip section of Tube 1, and $E{I}_1$ and $E{I}_2$ are the bending stiffnesses of Tube 1 and Tube 2 in the tip section. Then, we calculate the required amount of relative tube translation at the actuators by adding the transmission elongation induced by the actuation force to the nominal displacement needed at the steerable tip section ($\theta d$):

$$q=\theta d+\tau \left(\frac{1}{k_1}+\frac{1}{k_2}\right)$$ (7)

where $k_1$ and $k_2$ are the total effective axial spring stiffnesses of the transmissions of tubes 1 and 2. To calibrate each total transmission stiffness, we recorded force-displacement data for each tube from the axial stiffness experimental set up, shown in Figure 11. The distal end of the transmission was attached to the force gauge (ATI Mini-40 SI-40–2) and the proximal end was fixed to a manual linear actuator. The force and displacment were recorded at 1 mm increments of actuation. A linear fit was performed on the force/displacement graph. The slope of the linear fit corresponds to $k_i$. The resulting effective axial spring constant of the transmission segment of Tube 1 and Tube 2 were $k_1=0.405N/mm$ and $k_2=0.396N/mm$, respectively.

Figure 11.

Experimental Setup for the validation of actuation distance and bending angle of the Outer Steerable Pair with the attached transmission section.

C. Experimental Setup

To test the transmission model and performance during actuation of the steerable tip, we assembled Tubes 1 and 2 concentrically and welded their tips together using laser spot welding (LaserStar 990 Series iWeld). Figure 11(a) shows the proximal end of Tube 2 of the Outer Steerable Pair attached to the force gauge (ATI Mini-40 SI-40–2) and the proximal end of Tube 1 fixed to a manual linear actuator. Figure 11(b) shows the distal tip position of the prototype was tracked using an electromagnetic (EM) tracker system (NDI Aurora). The middle section of the tube pair, between the distal and proximal ends (not shown), is laid on a flat surface.

D. Results

Figure 12 shows the model predictions and experimental results for the actuation displacement of Tube 1’s proximal end and the corresponding steerable tip angular displacement. The experimental data show a similar curvature/actuation slope as the prediction, indicating that the model accurately captures the general response of the transmission segments to push-pull actuation. The hysteresis observed in the experimental data reflects the effects of internal static friction during the push and pull cycles. Experimentally, the actuation limits due to slot closure was seen to occur around $\pm 40°$ and $\pm 20\text{mm}$ actuation limit, whereas they were designed to have slot closure of $\pm 60°$ and $\pm 25\text{mm}$ actuation limit. This is likely due to a small difference in effective laser kerf during manufacturing, such that the serpentine slots are slightly narrower than they were designed to be, thus reaching closure sooner. This effect can be taken into account and compensated for in future prototypes so that the full designed bending range can be achieved.

Figure 12.

The plot of actuation displacement of Tube 1 and the corresponding distal tip angular displacement. The red colored data represent the model prediction of Tube 1 actuation displacements and the corresponding tip displacement. The blue colored data show the experimental hysteresis data of Tube 1 actuation displacements and the corresponding tip displacement.

VI. Conclusion

In this study, we introduced a new laser-cut pattern that simultaneously considers bending stiffness, torsional stiffness, and explicitly axial stiffness. Using a conventional brickwork design as a benchmark, the discrete design breaks the conventionally strong correlation between axial and bending stiffness. Compared to a brickwork design pattern with similar bending stiffness, the discrete designs exhibited more than three times higher axial stiffness and 25% higher torsional stiffness. The effects of design parameters on performance were explored and summarized in parameter space plots, providing guidelines for future custom designs. Our experimental results validated all three predicted FEA stiffnesses in multiple laser-cut prototype tubes. We also validated the performance of the discrete transmission design within a fully optimized set of tubes designed for an ESD robotic system. We compared the angular displacement at the tip and the transmission actuation displacement observed experimentally with those predicted by a model. The new discrete transmission design can improve transmissions for endoscopic robots, offering high torquability and pushability with reduced bending stiffness. It also shows potential as an alternative to conventional patterns used in other fields, such as catheter shafts, steerable needles, and pre-curved concentric-tube robots, requiring tubes with similar mechanical properties.