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Abstract

Single port access surgery has brought many advantages and new challenges to the field of minimally invasive surgery. In principle, many of these challenges can be addressed through robotics. Continuum robots in particular have seen a surge in interest in recent years for their miniaturisation potential. However, many applications with these robots have been developed for contactless or low-force tasks. This paper tackles several barriers to high-force tasks with continuum robots such as suturing for Open Spina Bifida repair. A novel continuum robot end-effector is proposed. The end-effector is composed of a distal bending segment and hybrid gripper. First, the novel hybrid gripper specifically designed for continuum robots is introduced. The hybrid gripper can function both as a general surgical gripper and as a needle driver while overall keeping the required input force limited. Second, an optimisation of the continuum structure of the distal bending segment is conducted. The objective of this optimisation is to sustain high tip forces – required by the said high-force contact tasks – while keeping deformations acceptable. Third, a quasi-static model that predicts the coupled effect of bending and gripping is derived. This model predicts the deformation of the continuum structure that is induced by the gripping action itself. It could hence be used to compensate for such deformation and establish a stable grip with the otherwise flexible instruments. Finally, experiments are conducted to validate the new designs and the model. The hybrid gripper has a diameter of $3.4$ mm, a length of $15.5$ mm and can hold securely a needle with a $4$ N input force, outputting a $12$ N gripping force. The optimised bending segment is validated for Open Spina Bifida suture: it can bend freely up to 90° and it can support a $1$ N tip load up to a 35° bending angle. The bending and gripper model error varies between 1% and 14.55%, yielding useful results to predict the shape of the bending segment and potentially compensate for the disturbance created by actuating the gripper.

Keywords

Continuum robot single port access surgery hybrid gripper anisotropic stiffness quasi-static friction model bi-lateral notched Nitinol tube

Introduction

Single port access (SPA) surgery is an increasingly attractive field of minimally invasive surgery (MIS). SPA promises reduced post-operative pain, improved cosmetic outcomes, shorter hospital stay and faster recovery,¹ but interventions also become more complex to perform.² Collisions between instruments, obstruction between multiple clinicians, poor triangulation of the operating site and less intuitive motion³ complicate adoption of SPA in clinical practice. Robotic assistance has the potential to address above-mentioned issues.

Continuum surgical robots have increasingly appealed to research groups across the globe⁴ due to their inherent compliance and ease of miniaturisation. However, apart from some exceptions such as Bajo and Simaan,⁵ Smoljkic,⁶ and Donat et al.⁷ most applications of continuum robots have been developed for contactless or low-force tasks, below 0.1 N (e.g. viewing, lasering, ring and peg manipulation). Indeed, transmitting forces through such compliant robot complicates control and reliably doing such remains a challenge. A popular option to limit the impact of external or actuation forces is to use stiffening sheaths.^8–10 These are however complex to manufacture. Establishing large stiffness variations in a miniaturised form factor remains an unsolved problem. Notched bending segments are popular due to their compact size, monolithic composition and hence lower part number and ease of manufacture. By altering the geometry of the notches, the compliance of these continuum robotics can be programmed. But so far – as far as the authors are aware of – no efforts have been done to search for optimal configurations to execute high-force tasks with these notched tubes.

While some interventions do not require active contact such as endoscopy or some catheter navigation tasks, many others do require applying large forces to instruments and targeted tissues. One of these procedures is Open Spina Bifida (OSB) repair. OSB is one of the most common birth defects, affecting $3, 5 : 10000$ births in the USA.¹¹ OSB follows from an incomplete closing of the neural tube; often resulting in severe motor and neuro-developmental impairment. OSB repair requires precise tissue resection, tissue separation and suturing. Given the fragile nature and small dimensions of a foetus, this task is highly challenging. The authors have been developing a novel multi-arm continuum robotic platform to treat OSB in Vandebroek et al.¹² (Figure 1). It consists of an $11$ mm outer-diameter (OD) SPA-robot featuring a macro-micro approach. The macro (proximal) triangulation platform consists of a plurality of concentric tubes with OD ranging from $4.67$ to $3.4$ mm. The micro (distal) sections are fluidic-actuated Nitinol bending segments. These take advantage of the natural lumens of the concentric tubes. McKibben actuators and transmission cables are nested inside the smallest concentric tube.

Figure 1.

Representation of a multi arm robot similar to the design in Vandebroek et al.¹² over its Open Spina Bifida target. The main contributions of this paper here are marked: ① hybrid gripper, ② bending segment pattern optimisation and ③ bending segment kinematic model.

Developing instruments such as grippers, scissors or needle drivers for such miniature continuum robots is challenging. A pragmatic approach would employ commercial $1$ mm OD flexible instruments, for example ‘11,003 KB: Grasping Forceps, flexible, double action jaws, diameter 1mm, length 60cm’(K. Storz, Tubbingen, Germany), in the robot’s working channel. However, such instruments deliver neither the needed output force nor rigidity. Several researchers introduced instruments purposely designed for continuum robots. Zhang et al. proposed a $6$ mm OD sliding-pin needle driver offering high gripping force with limited force input.¹³ Salle et al. proposed several $10$ mm OD motorised and SMA grippers for cardiac surgery.¹⁴ The diameters of these instruments are too large to fit the $3.5$ mm OD required for SPA-OSB.¹² Two small wristed designs of respectively $3.5$ mm and $3$ mm OD are presented in Fujii et al.¹⁵ and Ohdaira et al.¹⁶ but they are only suitable for straight, rigid instruments.

This paper contribution to develop a new continuum robot end-effector for SPA suturing can be divided in three main contributions. First, Section II presents the novel design of a hybrid between gripper and needle-driver. The instrument has been purposely developed for continuum robots. The hybrid design aims to offer both general tissue and object manipulation capability in addition to secure needle holding. It also operates at low actuation forces. This is important as it helps to minimise the impact of gripping force on the continuum robot’s pose and shape. Second, a structural optimisation for high-force contact tasks of flexible bending segments is presented in Section III. This work envisioned task concerns foetal suturing of OSB closure. The developed flexible bending segments rely on anisotropic behaviour: they can apply large forces with little deformation in their transversal plane (yz) while being compliant in their actuation plane (xz); offering a wide bending angle in the latter. An optimisation is conducted to match this stiffness anisotropy to the task. Third, Section IV introduces a detailed quasi-static model of the bending segment. The model takes the perturbation created by the gripper actuation into account. This model can then be utilised to compensate for the influence of the gripper actuation and keep the segment and gripper pose steady while opening or closing the gripper’s jaws. As far as the authors are aware of, this type of compensation model has not yet been reported in literature. The three contributions are experimentally validated in section V before concluding the paper with a discussion on the model, its potential and insights for future work in section VI.

Hybrid gripper design

Instrumentation of continuum robots is difficult for two main reasons. First, since continuum robots are typically considered when miniaturisation is at stake, the instruments themselves also need to be small in size. Second, any force applied on or by the instrument may affect the robot’s pose. Bearing these challenges in mind, two mechanical advantages have been explored. First, the mechanism has been devised to output a large gripping force for a limited input force. Second, the needle is secured using form-closure rather than force closure. The gripper can function as a needle driver and as general surgical forceps. As far as the authors are aware of, this hybrid design is novel in the field of surgical needle drivers. The gripper design specifications are derived next in Section II.A. Section II.B presents the hybrid design in detail.

Design specifications

The different requirements the hybrid gripper design must respect are listed as follows:

S1 Minimal length: the gripper must be as short as possible to keep the moment lever small when external forces are applied to the gripper. This helps minimising additional bending moments and stresses in the bending segment.

S2 Diameter when closed: $< 3.9 mm$ : the gripper must be smaller than the inner diameter of the concentric tube that it fits in. This ensure that the gripper can safely be moved in and out of the larger concentric tube.

S3 Jaw opening angle $\geq 30 °$ : the gripper needs to open its jaw sufficiently wide for griping of tissue or large needles. The $30 °$ opening angle is comparable to manual laparoscopic needle drivers.¹³

S4 General-purpose gripping surface: the gripper must provide the capability to grasp and manipulate various types of tissues and objects. It should therefore have a dented surface like generic MIS forceps.

S5 Needle interchangeability: the gripper must be able to hold needles of various sizes. Suture needles come in different shapes, lengths, curvatures and diameters. Diameters range from 0.04 to 1.7 mm. Needles used for OSB repair range between $0.4$ and $0.6$ mm OD according to our clinicians.

S6 Needle gripping force $\geq 15$ N: the gripper must be able to hold a needle steadily during a suture. Literature suggests required gripping forces ranging from $0.4$ N for micro-surgical interventions¹⁷ to $40$ N for a $15$ mm-diameter laparoscopic robot.¹⁸ To determine the required force level, experiments were conducted to gauge the needed minimum grasping force for suturing with the novel gripper. A static input force was applied to a prototype of the needle driver with a needle in its jaws. The force was increased until the grip force was high enough to pierce through artificial skin made of Ecoflex (Smooth-on, USA). This artificial skin is more resistant to piercing than human skin, which adds a safety factor to the measurement. The minimum input force was found at $4$ N, yielding a $12$ N gripping force for a $20 °$ jaw opening angle (See (11) further for this relation). An extra 25% safety factor was taken, leading to a required $15$ N gripping force.

Design and prototyping

To design a compact gripper that can deliver high gripping forces and secure needles in place while minimising its influence on the robot pose, two mechanical advantage principles are implemented. The first principle is force multiplication. Inspiration for the initial design inspiration came from a classic, symmetrical five bar mechanism using a push-pull actuation. To increase the robustness, strength and ease of manufacturing, the design evolved into a three bar mechanism by rendering one-half of the mechanism rigid. In the three bar version, only one jaw is mobile while the other remains fixed. The fixed jaw can serve the additional purpose of a spatula. The designed three bar mechanism allows to multiply the input force thanks to the lever effect taking place.

The second principle is form closure. To securely hold an object, robotic graspers rely either on force or form closure.¹⁹ Force closure requires applying sufficiently large compression forces to produce friction that holds the desired object steady in the graspers. Form closure on the other hand consists in completely constraining the object geometrically. This typically can take place at lower force levels. Based on this principle, a portion offering a three-points contact was designed at the most proximal part of the gripper jaws. The three-points contact forces the needle in a groove and in the right orientation, perpendicular to the gripper’s longitudinal direction. In fact, this principle has been commercialised (e.g. by Olympus Corporation, Tokyo, Japan) for rigid, straight needle drivers where it is referred to as ‘self-alignment’ or ‘self-righting’ needle drivers.²⁰ The novelty of the design presented here is two-fold: first, utilising this ‘self-alignment’ principle for continuum robots. Second, implementing the ‘self-alignment’ needle-driver together with a traditional gripper. Figure 2(a) shows the resulting design. The following paragraphs verify if this design respects the specifications presented in Section II.A.

S1 The three bar mechanism allows to have a compact and powerful mechanism, minimising the length. The final length of the gripper is $15.5$ mm. At the needle gripping point (where high forces are applied), the length is $12.75 mm$ . The gripper is mounted on the tip of a 40 mm bending segment, contributing to a 25% increase in bending moments.

S2 The gripper’s nominal diameter is $3.4$ mm. However, the mechanism is asymmetric. The three bar mechanism locally increases the gripper radius to $1.9 mm < 3.9 mm / 2$ .

S3 The jaw can reach an opening angle of $32 ° > 30 °$ . In order to reach a wider opening angle while keeping the same force multiplication, the mechanism should be made longer, which would conflict with S1.

S4 The gripping surface has been divided into two zones. The proximal zone is described in S5. The distal zone is composed of a flat surface with small pyramidal-shaped teeth (200 $μ m$ high) to improve grip, for example on soft tissues (Figure 2(a)).

S5 The proximal zone is designed to ensure a three-point contact (Figure 2(b) and (c)). This allows to secure multiple needle sizes (up to a needle diameter of $1$ mm). It also forces the needle in the right orientation (perpendicular to the gripper’s symmetry-plane, as is standard for suturing).

S6 The length of the bars has been maximised so that the closed gripper just fits the outer tube ID; while keeping the gripper’s total length fixed. This means that angles $θ_{1}$ and $θ_{2}$ will increase, as shown in Figure 3. This maximises the output gripping force for a given input pulling force, as will be shown in (11). Similarly, to the micro bending segment presented in Vandebroek et al.,¹² the gripper is actuated via a miniature McKibben muscle and its return movement is provided by a compression spring visible in Figure 2(a). In order to establish a strong gripping force even when the jaws are closed, link © is not straight but bends at an angle $γ$ at the level of the third joint (Figure 3). Without this angle, the gripping force would reach zero when the jaws are closed. Both analytic calculations and a dynamic simulation in Inventor (Autodesk, California, USA) were run with a $10 N$ muscle input force (Figure 4) out of a maximum of $12.5 N$ for the employed McKibben actuator.²¹ The static equilibrium equations linking input and gripping force are derived in the next paragraph.

Figure 2.

(a) Hybrid gripper needle-driver showing the cross-section plane (green), (b) cross-sectional view of the three-points contact for needle locking (0.4 mm OD needle), and (c) same with a 0.8 mm OD needle.

Figure 3.

(a) Link diagram of the gripper mechanism and (b) free-body diagram of the mechanism.

Figure 4.

Simulated and theoretical $F_{g}$ vs jaw angle for a pulling force $F_{m} = 10 N$ .

Static equilibrium equations: Figure 3 shows the link diagram of the gripper mechanism. There are three links: Ⓐ is the main body; Ⓑ is the first bar of length $l_{1}$ ; Ⓒ is the second bar composed of two lengths $l_{2} + l_{3}$ with an angle $γ$ between them. The first joint $J_{1}$ is a prismatic joint combined with a revolute joint between main body Ⓐ and Ⓑ. The second joint $J_{2}$ is a revolute joint between Ⓑ and Ⓒ. The third joint $J_{3}$ is a revolute joint between Ⓒ and the main body. The first and third joint are co-linear with the prismatic joint axis. $F_{m}$ is the muscle input force. $F_{g}$ is the gripping output force. $F_{i, j}$ and $M_{i, j}$ are the resulting internal forces and moments. The following assumptions are used in the upcoming derivation:

friction (sliding and rotational) is negligible;

$F_{g}$ acts perpendicular to the moving jaw at a given distance $l_{4}$ from $J_{3}$ (Figure 3);

the main (action) analysis is done in the gripper’s symmetry (xy)-plane.

Figure 3(b) shows a free-body diagram that helps derive the relation between the input and output forces. From the diagram one can establish the equilibrium of link 1 along the $x$ -axis ( $⇌ x$ ), along the $y$ -axis ( $⇌ y$ ) and the moment equilibrium around $J_{2}$ or $O_{1, 2}$ ( $↻ O_{1, 2}$ ):

{\begin{matrix} ⇌ x : F_{0, 1_{x}} - F_{1, 2_{x}} = 0 \Leftrightarrow F_{0, 1_{x}} = F_{1, 2_{x}} (1) \\ ⇌ y : - F_{m} - F_{1, 2_{y}} = 0 \Leftrightarrow F_{m} = - F_{1, 2_{y}} (2) \\ ↻ O_{1, 2} : M_{1, 2} + F_{0, 1_{x}} l_{1} \cos θ_{1} - F_{m} l_{1} \sin θ_{1} = 0 (3) \end{matrix}

Since revolute joints cannot transmit moments, $M_{1, 2} = 0$ and (3) becomes:

F_{{0, 1}_{x}} = F_{m} \tan θ_{1}

(4)

The equilibrium of link 2 gives:

{\begin{matrix} ⇌ x : F_{2, 1_{x}} + F_{2, 0_{x}} + F_{g} \cos (θ_{2} - γ) = 0 (5) \\ ⇌ y : F_{2, 1_{y}} - F_{2, 0_{y}} - F_{g} \sin (θ_{2} - γ) = 0 (6) \\ ↻ O_{2, 0} : M_{2, 0} + F_{2, 1_{x}} l_{2} \cos θ_{2} - F_{2, 1_{y}} l_{2} \sin θ_{2} - F_{g} l_{4} = 0 (7) \end{matrix}

Following the action-reaction principle, forces on each side of a virtual cut at $J_{2}$ are equal:

{\begin{matrix} F_{1}, 2_{x} = F_{2}, 1_{x} = - F_{m} (8) \\ F_{1}, 2_{y} = - F_{2}, 1_{y} = F_{m} \tan θ_{1} (9) \end{matrix}

Since $J_{3}$ is also revolute, $M_{2, 0} = 0$ . Inserting this together with (8) and (9) into (7) yields:

F_{g} = \frac{F_{m}}{l_{4}} (l_{2} \sin θ_{2} + l_{2} \cos θ_{2} \tan θ_{1})

(10)

For equal lengths $l_{1} = l_{2} = l$ , the triangle formed by the three joints becomes an isosceles triangle making $θ = θ_{1} = θ_{2}$ . In which case (10) reduces to:

F_{g} = \frac{F_{m}}{l_{4}} 2 l \sin θ

(11)

Figure 4 shows the evolution of $F_{g}$ in function of the jaw angle given a fixed $F_{m} = 10 N$ . The blue line represents the theoretical results while the red line shows the result of simulation in Inventor (Autodesk Inc., USA) with the actual gripper. The theoretical and simulated values overlap. It can be seen that the gripping force exceeds $15 N$ in all positions. Note that this realised gripping force will be validated experimentally in Section IV. Some deviation is expected due to friction as well as manufacturing imperfections.

Prototype: Figure 5 shows a prototype that was 3D printed with a Mlab Cusing R (GE Additive, USA). Measurements with a calliper showed 3%–10% increases in some key dimensions. To bring parts back to nominal dimensions, parts were etched in a hydrofluoric acid solution ( $4 % H F - 13 % H N O_{3} - 83 % H_{2} 0$ ). The parts were dipped 1 min before measuring again, until reaching the nominal dimensions. the shortest etch time was $5$ minutes and the longest was $10$ minutes. Finally, holes and slots were cleared out during post-processing with a column drill. A total of $6$ grippers were built following these manufacturing steps.

Figure 5.

Hybrid gripper prototype.

Bending segment structural optimisation

As diameters go down instruments become more flexible. In Vandebroek et al.,¹² 8 requirements for the OSB repair multi-arm robot were established. They were all met except one: the earlier prototype of the bending segment failed to sustain a transversal force of 1 N. This section describes a structural optimisation conducted to maximise the force that can be delivered at the end of a flexible instrument shaft without buckling or reaching superelasticity. Oliver-Butler et al.²² conducted an optimisation for single-sided cut-outs but only optimising the cut-out depth to maximise the bending segment stiffness.

This section first describes the geometry and design parameters of the bending segment. Second, requirements for the optimisation are listed. Third, stresses appearing in the bending segment are analysed analytically and through finite element analysis (FEA). Fourth, the optimisation procedure is elaborated. Fifth, the results are discussed and finally, the results are verified against FEA.

Geometry

In order to optimise the flexibility of the bending segment, a cut-out pattern is proposed. Many different patterns have been explored before, as shown in Table 1 of Legrand et al.²³ In Peirs et al.,²⁴ a $5$ mm OD Nitinol flexible distal tip with asymmetric cut-outs has been developed for endoscopic robot surgery. In Rucker et al.,²⁵ anisotropic bending stiffness is also investigated. The authors developed an anisotropic model and proved that this anisotropic bending stiffness solved the ‘snapping’ problem, which is one of the main hurdles to regular concentric tubes. The design optimised in this paper is similar to Peirs et al.²⁴ with two main differences: the segment in Peirs et al.²⁴ has 2 degrees of freedom (DOFs) versus 1 DOF here and the width of the gap $g$ is minimised here to protect the inner lumen as much as possible. Figure 6(a) shows the cut-out geometry being optimised. Such geometry can e.g. be obtained by laser-cutting the surface of a Nitinol tube. As such, a cylindrical tube of inner and outer radius $R_{i}$ and $R_{o}$ can be transformed into a series of $n$ pairs of parallel flexible beams of width $a$ , height $b$ , beam angle $φ$ (visible on Figure 6(b)) and beam length $L$ ; and $n$ rigid full tube sections of length $L_{R}$ . In this work a right-handed orthonormal coordinate frame is employed whereby the y-axis is collinear with the longitudinal axis of the tube and the xz plane is perpendicular to this axis. Then, at $x = R_{o}$ , one can observe the outer surface of a succession of rigid ‘teeth’ with height $h$ and spaced by gaps $g$ . The gap $g$ will help determine the maximal bending angle of each section. In this configuration the maximum angle will be reached when neighbouring section teeth contact each other. The teeth then act as mechanical stoppers that protect the central beams from excessive stresses. It also stiffens the bending segment once it is fully bent as the contacting teeth create additional friction forces that resist motion in the yz plane. The pattern is periodic with spacing $p = L + L_{R} = h + g$ . The total length of the bending segment is then $L_{T} = np$ . The cut-out depth $c$ defines the fillet radius at the beam extremities. These fillets reduce local stress concentrations at the corners. As long as $c$ is small enough ( $c \leq 0.2 l$ ), its influence on the length of the flexible beam can be ignored.

Table 1.

Maximum stress analytically calculated at the four corners beam corners A,B,C,D. $σ_{VM, th}$ are the analytically (theoretical) Von Mises stresses at the beam corners. $σ_{VM, FEA}$ are the same Von Mises stresses but obtained through FEA for validation.

	A	B	C	D
$σ_{1}$ [MPa]	−44	−44	−44	−44
$σ_{2}$ [MPa]	+117	+117	−117	−117
$σ_{3}$ [MPa]	−1970	+1970	+1970	−1970
$σ_{4}$ [MPa]	−3030	+3030	−3030	+3030
$σ_{VM, th}$ [MPa]	−4927	5073	−1221	899
$σ_{VM, FEA}$ [MPa]	−4658	4926	−1282	742
Deviation [%]	5.5	2.9	4.8	21.1

Figure 6.

(a) Bending segment geometry and cut-out parameters, (b) second moment of area of the beams, equivalent to the subtraction of two portions of a circle, and (c) second moment of area of a portion of a circle.

In order to calculate the bending stresses and the deformations, the second moments of area are necessary. While cutting the notches, the laser remains perpendicular to the tube surface. This means that the beam section takes on the shape of Figure 6(b). If $φ$ is small (as in Vandebroek et al.¹²), the section can be approximated by a rectangle of area $a \times b$ . Otherwise, Figure 6(b) shows that the beam shape is the difference between two circular sections. The second moments of area of a circular section $I_{x, s}$ , $I_{z, s}$ are given by:

{\begin{matrix} I_{x, s} = \frac{R^{4}}{8} (φ + \sin φ) \\ I_{z, s} = \frac{R^{4}}{8} (φ - \sin φ) \end{matrix}

(12)

Taking the radius of each circular sections as $R_{i}$ and $R_{o}$ , the second moments of area of the difference $I_{x, d}$ , $I_{z, d}$ can be calculated as:

{\begin{matrix} I_{x, d} = \frac{R_{o}^{4} - R_{i}^{4}}{8} (φ + \sin φ) \\ I_{z, d} = \frac{R_{o}^{4} - R_{i}^{4}}{8} (φ - \sin φ) \end{matrix}

(13)

As there are two beams in parallel (Figure 6(b)), the total second moments of area $I_{x}$ , $I_{z}$ are:

{\begin{matrix} I_{x} = 2 . I_{x, d} = \frac{R_{o}^{4} - R_{i}^{4}}{4} (φ + \sin φ) \\ I_{z} = 2 . I_{z, d} = \frac{R_{o}^{4} - R_{i}^{4}}{4} (φ - \sin φ) \end{matrix}

(14)

Now that the geometry has been described, design requirements for the bending segment must be developed to impose constraints for the optimisation.

Requirements

Following, four requirements R1 − R4 are taken into consideration for optimising the bending segment:

R1 Tube diameters: A radial clearance is necessary for smooth operation between concentric tubes. This clearance was empirically found at 5% of the larger tube ID. The bending segment OD should therefore be below $3.71$ mm as the tube it nests in has an ID of $3.91$ mm. Also, the ID should exceed $2.2$ mm to leave room for artificial muscles, cables and/or instruments.

R2 Tip force: the bending segment should withstand forces for tissue manipulation and suturing tissues. Forces and torques were measured while performing repetitive sutures on a chicken thigh in Vandebroek et al.¹² Most sutures required less than $1$ N, the largest peak reaching $1.2$ N. Chicken skin being thicker than a foetus skin, $1$ N was set as requirement.

R3 Critical buckling load: in case the gripper is actuated while the bending segment is straight, the critical buckling load

F_{crit} = \frac{π^{2} E I_{z}}{4 L_{T}^{2}}

(15)

must exceed $4$ N as this is the minimum axial force required to securely grip a needle (see S6 in Section II.A).

R4 Bending angle: from the workspace analysis in Vandebroek et al,¹² the required bending angle to reach all points of the OSB defect was identified as $π / 2$ . Note that this $π / 2$ bending range is used for reaching, lifting and resection of tissue. However, at a certain bending angle, suturing will not be possible anymore as the needle orientation does not coincide with the rotation axis and the gripper cannot rotate alone around its own axis at the tip. The maximum bending angle possible to perform a suture was found empirically at $π / 6$ . The required bending angle while sustaining $1$ N is therefore set at $π / 6$ .

Maximum stress identification

From Vandebroek et al.,¹² the transversal load (z direction, Figure 6) that the bending segment undergoes at its tip during a suture through delicate tissue was estimated to be below $1$ N (R2). Based on this load the maximum stress is derived in the following.

(1) Material choice: The chosen material for the bending segment is Nitinol. Its stress-strain behaviour can be approximated as a concatenation of two elastic domains as detailed in Lagoudas.²⁶ First, the material goes through a linear elastic domain up to a first yield point $σ_{y}$ . Beyond this point, a plateau in the stress-strain curve arises where the material behaves in a super-elastic manner. This results in large deformations for only small increments in stress. Such large deformations are not desirable when executing complex motions under large forces. Therefore the segment should be designed such that stresses remain in the linear elastic region (austenitic state). However, the initial yield of Nitinol is still high; situated between $1 - 2 %$ , which is $10$ times the yield of steel. This makes it suitable if large curvatures need to be reached. The Nitinol employed in this work was supplied by Euroflex (Germany). The datasheet provided by Euroflex describes a Young modulus $E = 60$ GPa and a yield point of $σ_{y} = 500$ MPa at 15 $° C$ . It is important to note that Nitinol is temperature sensitive, a variation of a few degrees will observably affect its Young modulus.

By designing the segment such that stresses remain below $σ_{y}$ , simple linear elastic theory and FEA can be used to determine the maximum allowable transversal load that the tube can withstand. In the following sections, the stress is first derived and then later used in the structural optimisation. Afterwards, the theoretical derivations are verified by FEA.

(2) Theoretical stress derivation - torsional analogy: There are $2$ loads acting on the bending segment. First, the load from the McKibben muscle $F_{m}$ . This load is applied in the $y$ direction at a distance $d$ (along the $x$ -axis) from the $y$ -axis. Namely at the location where the pulling cable is anchored to the tip. The muscle force creates both a compressive stress $σ_{1}$ and a bending stress $σ_{2}$ . Second, $F_{t}$ , the load from the tip interaction is applied in the $z$ direction (the direction of the needle tip). This load works at at an arc length along the bending segment $s = L_{T}$ (with $L_{T}$ the total length of the segment). If the bending segment is straight, $F_{t}$ is simply applied in $x = 0; y = L_{T}$ . In this configuration, $F_{t}$ creates a shear stress $τ_{1}$ in the $z$ -direction and a bending stress $σ_{3}$ . If the segment is bent, $F_{t}$ generates an additional a torsional stress $τ_{2}$ .

It is clear that given the remote action of $F_{t}$ , the maximum stress will be found at the most proximal section; where the moment arm is the largest. Therefore, the problem can be simplified by looking at a cross section of the most proximal cut-out. Given the geometry of a single cut-out, the load case can be represented in simplified form as a loading of two parallel beams (depicted in Figure 7). One can understand that the maximum stress occurs when the segment is fully bent. $F_{m}$ will cause maximal stress in this situation. For all loads except for the torsional moment $M_{y}$ , the stress in the $2$ beams can easily be computed. It was not found possible to analytically compute the exact torsional stiffness of the two parallel beams however. Instead, the following analogy is proposed as an analytical approximation.

Figure 7.

Torsional analogy: the most proximal cut-out under a torsional moment $M_{y}$ (left) can be approximated as two clamped parallel beams each undergoing a force $F_{a}$ and a counter-moment $M_{b}$ .

When observing the small deformation behaviour of the parallel beams during torsion of the segment, one can observe that the two beams actually bend almost exclusively in the $x$ direction. The analogy replaces each beam of the cut-out by an equivalent guided beam (Figure 8). For small deformations, thanks to the superposition principle applicable in the first linear elastic region of NiTi, the torsional moment $M_{y}$ (Figure 7) creates a tip displacement $δ_{y}$ but no end rotation $θ_{y}$ (Figure 8). $M_{y}$ can therefore be decomposed into a lateral force $F_{a} = 2 . M_{y} / (R_{o} + R_{i})$ , creating a tip displacement $δ_{a}$ and rotation $θ_{a}$ and a counter bending moment $M_{b}$ that creates an opposite tip displacement $δ_{b}$ and rotation $θ_{b}$ . The counter moment cancels the beams rotation so that $θ_{y} = θ_{a} + θ_{b} = 0$ and $δ_{y} = δ_{a} + δ_{b}$ . Using elastic beam theory allows to find:

{\begin{matrix} M_{b} = \frac{F_{a} l}{2} \\ δ_{y} = \frac{F_{a} l^{3}}{12 EI} \end{matrix}

(16)

Figure 8.

Guided beam analogy to replace $M_{y}$ by $F_{a}$ and $M_{b}$ .

This analogy was validated running an FEA in Inventor (Autodesk, USA). Mesh parameters consisted of an average element size of 4% of the bounding box length, minimum element size of 2% of the average size, a grading factor of $1.5$ and a maximum turn angle of $60 °$ . A finer local mesh control that produces a $0.02$ mm element size is used for the beam surfaces to provide good convergence and avoid corner stresses. The mesh elements are curved and triangular. Testing for $M_{y} = 5$ N.mm, the displacement correlation was above 99% and the yield stress correlation above 97% for the four beam corners where maximum stress is found (A,B,C,D) shown on Figure 7.

Using this analogy, all stresses can now be calculated. Shear stresses ( $τ$ ) can be neglected compared to normal stresses ( $σ$ ) for bending problems as the maximum $σ$ are typically larger than the maximum $τ$ and they occur at the beam surface where $τ = 0$ . The shear stresses generated by $F_{t}$ and $F_{a}$ are therefore ignored. Four components to normal stress $σ$ are identified:

\begin{matrix} σ_{1} = \frac{F_{m}}{A}, σ_{2} = \frac{F_{m} db}{2 I_{z}}, \\ σ_{3} = \frac{F_{t} R_{b} \sin θ R_{o}}{I_{x}}, σ_{4} = \frac{M_{b} a}{2 I_{z}} . \end{matrix}

(17)

With $F_{m}$ compressive stress $σ_{1}$ , $F_{m}$ bending stress $σ_{2}$ , $F_{t}$ bending stress $σ_{3}$ , torsional analogy $M_{b}$ bending stress (created by $F_{t}$ torsional moment) $σ_{4}$ and the bending radius $R_{b}$ .

(3) FEA verification: In order to verify the accuracy of the above analytic derivation, a numerical verification is conducted using the parameters of the previously designed bending segment from Vandebroek et al.¹²: $F_{m} = 7$ N, $F_{t} = 1$ N and $L_{T} = 40$ mm. Table 1 presents the four corner stresses at (17) and the computed Von Mises stress $σ_{VM}$ . Note that the latter is simply the sum of the $σ_{1}, σ_{2}, σ_{3}$ and $σ_{4}$ as all stresses are longitudinal.

The FEA is conducted in Inventor (Autodesk, USA) using the same mesh parameters as earlier. The proximal notched section of the tube is clamped at the base. $F_{m}$ is applied along $y$ with an offset $d = - 1.5 mm$ along $x$ . $F_{t}$ is applied along $z$ where the tip of the fully bent segment would be. Given $L_{T} = 40 mm$ and $θ = 90 °$ , the tip would be at $25.5$ mm both in $y$ and $x$ . From Table 1 one can observe that the deviation between numerically computed corner stresses and the analytically computed values (Figure 9) remain close to 5%, except for D. This is because D has the lowest stress among the four so despite having a smaller absolute error than A, it has a larger relative deviation. Errors are expected as the analytical model does not take into account that the beams are not truly rectangular, also corner concentrations and fillets at the beam extremities are not taken into account in (17).

Figure 9.

FEA verification of the stress analysis. The theoretically calculated stress are shown in the black frame for reference.

These high stress values confirm that the previous segment from Vandebroek et al.¹² that was initially utilised for non-contact tasks is not adequate for suturing and hence needs optimisation because $σ_{VM} > σ_{y} = 500 MPa$ . It is important to note that in reality the material would not experience stresses as high as 5000 MPa but would rather go into superelasticity. These beams would therefore deform until the mechanical stoppers are reached with only a small increase of stress above 500 MPa.

Structural optimisation

To optimise the structure, the design problem is now formulated as a non-linear constrained optimisation problem. The objective of the optimisation consists in finding the configuration that allows the largest bending angle $θ_{T}$ with $F_{T} = 1 N$ applied at the tip while keeping the maximum stress below $σ_{y}$ . Table 2 summarises the employed optimisation parameters. The fixed parameters are the geometric parameters. They come from Vandebroek et al.,¹² except for $L_{R}$ . On one hand, $L_{R}$ should be minimised as it reduces the length available for bending. On the other hand, $L_{R}$ should still be large enough to prevent overly large stress concentrations. The limit for $L_{R}$ was found to be $0.15 mm$ using FEA while keeping other parameters within the limits established in Table 2.

Table 2.

Optimisation parameters.

Fixed param.	$F_{t}$ [N]	$L_{T}$ [mm]	$L_{R}$ [mm]	d [mm]	E [GPa]
Values	1	40	0.15	1.5	63
Variables	$R_{i}$	n	a[mm]	b[mm]	$θ$
Initial guess	1.25	50	0.2	0.4	$\frac{π}{3}$
Limits	[1;2]	[1;100]	[0.1;2]	[0.1;0.75]	$[0.1; \frac{π}{2}]$
Optim. values 1	1.1	28	0.69	0.75	0.73
Optim. values 2	1.333	28	0.82	0.445	0.62

The initial list of optimisation variables includes $R_{i}$ , $n$ , $p$ , $l$ , $a$ , $b$ , $g$ , $α$ (with $α$ the bending angle of each section). However, some of these variables depend on each other or on the fixed parameters. They form the following set of dependencies along with other variables explicitly determined with equality constraints:

\begin{array}{l} R_{o} = R_{i} + b \\ p = L_{T} / n \\ L = p - L_{R} \\ α = θ_{T} / n \\ φ = a / R_{o} \\ g = (R_{o} - c - a / 2) \tan α \\ R_{b} = L_{T} / θ \\ I_{x} = \frac{R_{o}^{4} - R_{i}^{4}}{4} (φ + \sin φ) \\ I_{z} = \frac{R_{o}^{4} - R_{i}^{4}}{4} (φ - \sin φ) \\ F_{m} = \frac{α E I_{z}}{d l} \\ M_{t} = F_{t} R_{b} (1 - \cos θ) \\ F_{a} = M_{t} / d \\ M_{b} = \frac{M_{t} l}{2 d} \end{array}

(18)

Taking into account these relationships reduces the number of independent variables to four: $(R_{i}, n, a, b)$ . Next, there are seven inequality constraints: first, the four corner stresses from Table 1 must remain below $σ_{y} = 500$ MPa. Second, $R_{o}$ must remain below $3.7 / 2 = 1.85 mm$ as specified in R2. Third, the maximally required muscle force $F_{m}$ must remain below $2 / 3$ of the muscle capacity = $40$ N as a safety factor. Finally, the critical buckling load $F_{crit}$ (15) must exceed $4$ N as elaborated in R3. The optimisation problem can therefore be written as:

\begin{matrix} max_{R_{i}, n, a, b} θ_{T} \\ s . t . | σ_{VM, A} | \leq 500, | σ_{VM, B} | \leq 500 \\ | σ_{VM, C} | \leq 500, | σ_{VM, D} | \leq 500 \\ R_{o} \leq 1.85 mm, F_{crit} \geq 4 N, F_{m} \leq 40 N \end{matrix}

(19)

In order to start the optimisation, an initial guess and limits have to be given to the variables $(R_{i}, n, a, b)$ . The limits shown in Table 2 come from manufacturability and geometrical constraints. Ten combinations of initial guesses were tested and they all converged towards the same solution.

Results

Two different non-linear constrained optimisation solvers were used in MATLAB (The MathWorks, USA) in order to confront the solvers convergence: APM from http://apmonitor.com/ and fmincon. Both solvers converged towards the same optimal values shown in Table 2.

The first optimisation yielded a bending angle $θ_{T} = 0.73$ . This means that $\frac{0.73}{π / 2} 100 = 46.5 %$ of the bending workspace can sustain the $1$ N tip load, respecting R4. The required muscle force to reach this angle is $F_{m} = 35$ N.

In order to build a prototype and experimentally validate this optimisation, these initial results were used to select a Nitinol tube available in stock at Euroflex (Germany) with parameters close to Optim. values 1 (as custom production is costly). A few tubes were identified. Their dimensions were used as inputs in the optimisation algorithm. The properties of the tube that provided the highest $θ_{T}$ is described in Table 2, Optim. values 2. The final geometry can reach $θ_{T} = 0.62$ with $F_{m} = 30$ N while sustaining the $1$ N load, still respecting R4. On Figure 10 the bending segment from Vandebroek et al.¹² and the optimised design are shown next to each other. The two segments have the same bending range but the stiffened design of the new version allows it to sustain a $1$ N tip force over a $35.5$ bending. The previous design could only sustain the force up to about $10 °$ , which is a 255% improvement.

Figure 10.

Previous design of Vandebroek et al.¹² (left) versus new optimised design (right).

FEA verification

Running the simulation in Inventor (Autodesk, USA) using $F_{m} = 30$ N, $F_{t} = 1$ N and $L_{T} = 40$ mm yields Figure 11. Mesh parameters consisted of an average element size of $0.03$ , minimum element size of $0.2$ , a grading factor of $1.5$ and a maximum turn angle of $60 °$ . A local mesh control of $0.025$ mm element size was selected for the beam surfaces. The stresses in corners A and B are exceeding $σ_{y}$ while stresses in corner C and D are lower. One can see that these higher values are concentrated in a very small area of corner stress, meaning that the beam is overall not going under superelasticity. Since the bending segment needed to be stiffer than the previous design, the beams are wider. This increases the error inherent to the rectangular beam approximation used in the analytical derivation. Curved beams are more flexible than rectangular beams (for identical $a$ and $b$ parameters). This explains why that the segment is fully bent as shown on Figure 12. It is therefore normal that the stresses exceed $σ_{y}$ as the limit bending range for sustaining a 1 N transversal tip force was also exceeded. Adjusting $F_{m}$ to reach the desired bending angle drops the corner stresses back below their limits.

Figure 11.

FEA optimisation verification stresses - $F_{m} = 30$ N, $F_{t} = 1$ N. Some small corner stress concentrations exceed the stress limit.

Figure 12.

FEA optimisation verification displacement - $F_{m} = 30$ N, $F_{t} = 1$ N. The notch is fully bent.

Kinematic model and control

The following model takes inspiration from the developments by Legrand et al.²³ and Swaney et al.²⁷ for unilateral notches. It describes the kinematics of a notched Nitinol tube manufactured by laser-cutting $n$ bilateral symmetric notches. The geometry of the bending segment and its notches follows the description in Section III.A. Despite the fact that bending only needs to take place in one direction for OSB treatment based on the here employed macro-micro configuration, a bilateral pattern was selected because, for an equivalent bending stiffness, it provides a higher torsional stiffness than a unilateral design. A bilateral segment can therefore support higher transversal tip loads. For the following analysis, the most distal notched section corresponds to $j = 1$ , $j \in [1, n]$ whereas $j = n$ corresponds to the most proximal notch. A first cable is attached at the extremity of notch section 1. Deflection is obtained by pulling this so-called ‘bending cable’. Once tension is released, the elastic property of the segment brings it back to its straight configuration. The hybrid gripper described in Section II is mounted at the tip of the bending segment. A second cable, the so-called ‘gripper cable’, actuates the gripper when pulled. The following model implements both force contributions incrementally as detailed below.

Modelling approach and hypotheses

First, it is assumed that the segment is initially in a straight configuration and that the input tension $T_{bend}$ on the bending cable is equal to the force that the cable applies at the tip of the segment $F_{app}$ (Figure 13).

Figure 13.

Single cut-out section kinematics. Only half of the bending segment is represented thanks to the YZ plane symmetry.

Next, the segment is considered in a deformed state. The bending cable now generates normal and friction forces on the inner wall of the bending segment. These forces reduce the amplitude of the tip force but also create additional moments. These updated forces and moments yield a different segment shape; which in turn modifies again the different components. To find an equilibrium, this problem is solved iteratively until the change in bending angle between two successive iterations is found negligible. The limit is fixed at $< 0.15 °$ as this correspond to a spatial displacement of 0.05 mm, or 5% of the required resolution for the robot (1 mm¹²).

Once this equilibrium is obtained, the gripper at the tip of the bending segment is actuated by pulling the gripper cable. This action creates additional tip, friction and normal forces. The updated problem is again solved iteratively, now taking into accounts forces and moments from both the bending cable and the gripper cable. The computational method is summarised in Algorithm and at the end of this section. The kinematic derivations of this section are based on the following hypotheses

H1 The non-linear behaviour of Nitinol can be ignored because the stress does not exceed $σ_{y} = 500$ MPa.

H2 The deformation of the teeth can be neglected due to the large difference in stiffness. This was also demonstrated in Legrand et al.²³ for unilateral teeth.

H3 The friction between tension wire and backbone can be modelled using the capstan equation.

H4 The lever at which the bending cable acts to create the bending moment is given by $d = R_{i} - R_{c}$ with $R_{c}$ the radius of the bending cable.

Deformation model of a Nitinol notched backbone – Frictionless model

For space and clarity considerations, the following elaboration is done with respect to Figure 13 which shows only the right half (cut along the $yz$ plane) of an otherwise symmetric bilateral notch. The gap $g$ , and therefore the maximal bending, are also exaggerated. Each notched section $j$ can be divided in two parts: a flexible beam and a rigid ‘tooth’. For $j = 1$ at the distal end of the bending segment, the bending cable exerts a load $F_{app}$ . This force creates a bending moment $M$ along the flexible beam portion. This moment is then transmitted along the entirety of the backbone, going through repetitions of rigid and flexible sections. For future kinematic derivations, five local frames are defined per section: $(x_{0}, y_{0})$ at the base of the flexible beam, on the neutral axis; $(x_{1}, y_{1})$ at the top of the flexible beam, on the neutral axis; $(x_{2}, y_{2})$ at the base of the rigid tooth, on the inner radius; $(x_{3}, y_{3})$ at the top of the rigid tooth, on the inner radius; $(x_{4}, y_{4})$ in the middle of the rigid tooth, on the inner radius. First, the deformation of a single beam section is studied. Second, the limit case, where the deformation is so large that the adjacent teeth come into contact, is detailed. Third, each section deformation is concatenated to reconstruct the complete shape of the bending segment.

The frictionless model is detailed first as the bending segment is straight in its initial configuration, meaning there are no friction forces. As soon as the segment starts deforming, this changes and this is described in Subsection IV.C.

(1) Deformation of a beam section: The bending cable is attached at the wall of the tube. The cable then runs along the inner wall of the tube, at a distance $R_{i}$ as shown on Figure 13. According to H4, the lever at which $F_{app}$ acts to create the bending moment $M$ is the distance $d = R_{i} - R_{c}$ with $R_{c}$ the radius of the bending cable:

M = F_{app} . d

(20)

This moment is transmitted over the entire bending segment. Classic Euler-Bernoulli beam theory can then be applied (H1) to obtain the bending stresses at a distance ‘x’ from the neutral axis:

σ = \frac{M . x}{I}

(21)

Additionally, the relationship between bending moment and curvature is

M = \frac{E . I}{ρ}

(22)

With $I = I_{z} = \frac{R_{o}^{4} - R_{i}^{4}}{4} . (ϕ - \sin ϕ)$ as derived above. The bending angle $θ (s)$ as a function of the beam arc length $s$ , varying from $s = 0$ to $s = L$ , can be found as:

θ (s) = \frac{M . s}{E . I}

(23)

The following geometric relations can then be used to obtain the Cartesian coordinates of any point situated on the neutral axis of the deflected beam at a given arc length $s$ from the base of the section:

{\begin{matrix} \frac{dx}{ds} = \sin (θ (s)) \\ \frac{dy}{ds} = \cos (θ (s)) \end{matrix}

(24)

To find the positions $x_{1}$ and $y_{1}$ of the distal end of the beam (Figure 13), the previous expression can be integrated over the entire beam length ( $s = L$ ), yielding:

{\begin{matrix} x_{1} = \frac{- EI}{M} \cos (\frac{ML}{EI}) + \frac{EI}{M} = \frac{- L}{θ} \cos (θ) + \frac{L}{θ} \\ y_{1} = \frac{EI}{M} \sin (\frac{ML}{EI}) = \frac{L}{θ} \sin (θ) \end{matrix}

(25)

And the transformation matrix that expresses the position and orientation of frame 1 with respect to frame 0 of an arbitrary section j can be found as

T_{1, j}^{0, j} = [\begin{matrix} \cos (θ_{j}) & \sin (θ_{j}) & 0 & x_{1, j} \\ - \sin (θ_{j}) & \cos (θ_{j}) & 0 & y_{1, j} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(26)

With $θ_{j} = θ_{j} (L)$ . The rotation matrix takes this representation because $θ$ is negative for clockwise rotation, so $\cos (- θ) = \cos (θ)$ and $\sin (- θ) = - \sin (θ)$ . Since the rigid teeth do not undergo any deformation, the bottom inner wall position on the tooth $(x_{2, j}, y_{2, j})$ can be obtained by adding a translation term to (26):

[\begin{matrix} x_{2, j} \\ y_{2, j} \\ 0 \\ 1 \end{matrix}] = T_{1, j}^{0, j} [\begin{matrix} R_{i} \\ - t \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} R_{i} . \cos (θ_{j}) - t . \sin (θ_{j}) + x_{1, j} \\ - R_{i} . \sin (θ_{j}) - t . \cos (θ_{j}) + y_{1, j} \\ 0 \\ 1 \end{matrix}]

(27)

With $t = \frac{L - g}{2}$ (Figure 13). In a similar fashion, the inner wall top position on the tooth $(x_{3, j}, y_{3, j})$ can be found

[\begin{matrix} x_{3, j} \\ y_{3, j} \\ 0 \\ 1 \end{matrix}] = T_{1, j}^{0, j} [\begin{matrix} R_{i} \\ h + t \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} R_{i} . \cos (θ_{j}) + (h + t) . \sin (θ_{j}) + x_{1, j} \\ - R_{i} . \sin (θ_{j}) + (h + t) . \cos (θ_{j}) + y_{1, j} \\ 0 \\ 1 \end{matrix}]

(28)

Finally, the middle of the tooth inner wall $(x_{4, j}, y_{4, j})$

[\begin{matrix} x_{4, j} \\ y_{4, j} \\ 0 \\ 1 \end{matrix}] = T_{1, j}^{0, j} [\begin{matrix} R_{i} \\ h / 2 \\ 0 \\ 1 \end{matrix}] = [\begin{matrix} R_{i} . \cos (θ_{j}) + (h / 2) . \sin (θ_{j}) + x_{1, j} \\ - R_{i} . \sin (θ_{j}) + (h / 2) . \cos (θ_{j}) + y_{1, j} \\ 0 \\ 1 \end{matrix}]

(29)

(2) Deformations in case of teeth contact: If the moment $M$ is sufficiently large, the bending angle reaches $θ_{\max}$ where the teeth make contact (Figure 14). $θ_{\max}$ , and therefore the limit torque $M_{limit}$ , can be found with a trigonometric approximation: one can observe on Figure 14 that the teeth entering contact form an isosceles triangle (black dots). The base of the triangle can therefore also be approximated as $g$ . The sides of equal lengths of the triangle measure $R_{o} - c - a / 2$ . Cutting the triangle in half along its height yields two rectangle triangles, allowing to calculate:

\begin{matrix} θ_{\max} / 2 = \overset{- 1}{\sin} (\frac{g / 2}{R_{o} - c - a / 2}) \\ \Leftrightarrow θ_{\max} = 2 \overset{- 1}{\sin} (\frac{g}{2 (R_{o} - c - a / 2)}) \end{matrix}

(30)

Figure 14.

Teeth contact when the moment reaches or exceeds $M_{limit}$ .

then inserting (30) in (23) gives:

M_{limit} = \frac{EI}{L} . θ_{\max} = \frac{EI}{L} . \overset{- 1}{\sin} (\frac{g}{R_{o} - c - a / 2})

(31)

The bending radius can then be found by forming two rectangle triangles with intersecting $x_{0}$ and $x_{1}$ axes as shown on Figure 14 (red dashes) by determining $ψ = π / 2 - θ_{\max}$

R_{b, j} = \frac{y_{1, j}}{\cos (π / 2 - θ_{\max})} = \frac{y_{1, j}}{\sin (θ_{\max})}

(32)

or from the arc length formula:

R_{b} = \frac{L}{θ}

(33)

(3) Deformation of the entire backbone: In order to obtain the overall shape of the Nitinol bending segment undergoing a force $F_{app}$ at the most distal tooth, each section can be concatenated using transformation matrices linking the frames ${0, j - 1}$ and ${0, j}$ :

T_{0, j - 1}^{0, j} = [\begin{matrix} \cos (θ_{i}) & \sin (θ_{i}) & 0 & x_{1, j} + h_{i} . \sin (θ_{i}) \\ - \sin (θ_{i}) & \cos (θ_{i}) & 0 & y_{1, j} + h_{i} . \cos (θ_{i}) \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(34)

The world reference frame corresponds to $(x_{0, n}, y_{0, n})$ visible in Figure 15. The overall shape of the bending segment can be derived by identifying the origins of frames 0–4 for each section and then expressing them in the world reference frame. The origin of frame ${0, j}$ , $j < n$ , is found as:

O_{0, j}^{0, n} = Π_{m = j - 1}^{1} (T_{0, m}^{0, m + 1}) . [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}]

(35)

Figure 15.

Deformed pose of the bending segment.

The origin of frame ${1, j}$ :

O_{1, j}^{0, n} = Π_{m = j - 1}^{1} (T_{0, m}^{0, m + 1}) . [\begin{matrix} x_{1, j} \\ y_{1, j} \\ 0 \\ 1 \end{matrix}]

(36)

The origin of frame ${2, j}$ :

O_{2, j}^{0, n} = Π_{m = j - 1}^{1} (T_{0, m}^{0, m + 1}) . T_{1, j}^{0, j} . [\begin{matrix} R_{i} \\ - t \\ 0 \\ 1 \end{matrix}]

(37)

The origin of frame ${3, j}$ :

O_{3, j}^{0, n} = Π_{m = j - 1}^{1} (T_{0, m}^{0, m + 1}) . T_{1, j}^{0, j} . [\begin{matrix} R_{i} \\ h_{j} + t \\ 0 \\ 1 \end{matrix}]

(38)

and the origin of frame ${4, j}$ :

O_{4, j}^{0, n} = Π_{m = j - 1}^{1} (T_{0, m}^{0, m + 1}) . T_{1, j}^{0, j} . [\begin{matrix} R_{i} \\ h_{j} / 2 \\ 0 \\ 1 \end{matrix}]

(39)

The total bending angle at a specific notch $j$ is simply:

Θ_{j} = \sum_{i = n}^{j} θ_{i}

(40)

and the total bending angle of the entire segment is:

Θ = Θ_{1} = \sum_{i = n}^{1} θ_{i}

(41)

Deformation model of each Nitinol notched section - Friction model

(1) Friction between the backbone and the cable - Capstan equation: Now that the segment starts bending, $F_{app}$ will actually decreases due to the friction forces that arise between the cable and the inner wall of the backbone. Also force components appear that act normally $N_{j}$ and tangentially $F_{f_{j}}$ to the inner surface of each section. Approximating each of these inner wall contacts as a circular arc of arc angle $β_{j}$ and radius $R_{f, j} = R_{b, j} - R_{i}$ (Figure 16(b)), with $R_{b, j}$ obtained from (33), the Capstan equation can be used. The Capstan equation estimates the difference between input tension $T_{i}$ and output tension $T_{o}$ along the segment caused by the combination of friction and normal forces as:

T_{i} = T_{o} . e^{μ β}

(42)

Figure 16.

Capstan based friction model: (a) illustration of the Capstan equation and (b) Capstan approximation applied to the notched backbone.

with $μ$ the coefficient of friction and $β$ the wrap angle as shown in Figure 16(a). Using the bending radius of a particular beam section, the wrap angle $β_{j}$ can be calculated between the two points $O_{2, j}$ and $O_{3, j}$ (Figure 16(a)) as:

\sin (β_{j} / 2) = \frac{\sqrt{{(x_{3, j} - x_{2, j})}^{2} + {(y_{3, j} - y_{2, j})}^{2}}}{2 R_{f, j}}

(43)

And since $β_{j}$ is small, a further approximation $\sin β_{j} ≃ β_{j}$ gives:

β_{j} = \frac{\sqrt{{(x_{3, j} - x_{2, j})}^{2} + {(y_{3, j} - y_{2, j})}^{2}}}{R_{f, j}}

(44)

Then, starting from the bottom section $n$ with a known input tension $T_{n} = T_{bend}$ , the output tension force in the cable can be calculated section by section in order the find the final applied $T_{1} = F_{app}$ , so that (42) becomes:

T_{j} = T_{j - 1} e^{μ β}

(45)

The normal force can then be calculated for each section as:

N_{j} = (T_{j} + T_{j - 1}) . \sin (β_{j} / 2) ≃ (T_{j} + T_{j - 1}) . \frac{β_{j}}{2}

(46)

and the friction force as:

F_{f_{j}} = μ . N_{j}

(47)

(2) Free-body diagram computation of the bending moment: Now that extra forces come into play, the bending moment will not be constant anymore along the backbone. In order to compute the bending moment and bending angle of each section, a virtual section is made at each origin $O_{0, j}$ along the $x_{0, j}$ -axis as shown on Figure 17. Note that $F_{grip}$ is already represented on the diagram but is ignored as $F_{grip} = 0$ currently. In the following, a free-body diagram analysis is performed at the first (most distal) and the second section. The subsequent sections will follow the derivations from the second section.

Figure 17.

Free-body diagram of the bending and gripping forces.

The distances from section to section will be used as levers for the calculation of reaction moments. Viewed in the local $O_{0, j}$ frame of each section, the levers are calculated as:

leve r_{j} = [\begin{matrix} x_{1, j} + h_{j} \sin θ_{j} \\ y_{1, j} + h_{j} \cos θ_{j} \\ 1 \end{matrix}]

(48)

(a) First section cut: First, let’s calculate the equilibrium at point $O_{0, 1}$ . The first equation describes the force equilibrium along the $x_{0, 1}$ axis; the second along the $y_{0, 1}$ axis; the third the moment equilibrium around $O_{0, 1}$ :

\begin{matrix} ⇌ x_{0, 1} : F_{x, 1} = (F_{app} + F_{f_{1}}) \sin θ_{1} - N_{1} \cos θ_{1} \\ ⇌ y_{0, 1} : F_{y, 1} = (F_{app} + F_{f_{1}}) \cos θ_{1} + N_{1} \sin θ_{1} \\ ↻ O_{0, 1} M_{r, 1} = F_{app} (d \cos θ_{1} - y_{3, 1} \sin θ_{1}) \\ + (N_{1} \cos θ_{1} - F_{f_{1}} \sin θ_{1}) y_{4, 1} + (F_{f_{1}} \cos θ_{1} + N_{1} \sin θ_{1}) x_{4, 1} \end{matrix}

(49)

Remember that $(x_{4, j}, y_{4, j})$ are each expressed in their local frame $O_{0, j}$ in (29). These forces and moment now allow calculating the new bending angle of section 1

\begin{matrix} θ_{b, 1} = \frac{L}{EI} [F_{app} d + (N_{1} \cos θ_{1} - F_{f_{1}} \sin θ_{1}) \frac{y_{4, 1}}{2} \\ + (F_{f_{1}} \cos θ_{1} + N_{1} \sin θ_{1}) x_{4, 1}] \end{matrix}

(50)

(b) Second and subsequent section cuts: Moving on to the next segment, equilibrium around $O_{0, 2}$ is expressed as

\begin{matrix} ⇌ x_{0, 2} : F_{x, 2} = F_{x, 1} \cos θ_{2} + F_{y, 1} \sin θ_{2} + F_{f_{2}} \sin θ_{2} - N_{2} \cos θ_{2} \\ ⇌ y_{0, 2} : F_{y, 2} = F_{y, 1} \cos θ_{2} - F_{x, 1} \sin θ_{2} + F_{f_{2}} \cos θ_{2} + N_{2} \sin θ_{2} \\ ↻ O_{0, 2} M_{r, 2} = M_{r, 1} + (F_{y, 1} \cos θ_{2} - F_{x, 1} \sin θ_{2}) leve r_{x, 2} \\ - (F_{x, 1} \cos θ_{2} + F_{y, 1} \sin θ_{2}) leve r_{y, 2} + (F_{f_{2}} \cos θ_{2} + N_{2} \sin θ_{2}) x_{4, 2} \\ + (N_{2} \cos θ_{2} - F_{f_{2}} \sin θ_{2}) y_{4, 2} \end{matrix}

(51)

and the new bending angle of section 2 becomes:

\begin{array}{l} θ_{b, 2} = \frac{L}{E I} [M_{r, 1} + (F_{y, 1} \cos θ_{2} - F_{x, 1} \sin θ_{2}) l e v e r_{x, 2} . \\ - (F_{x, 1} \cos θ_{2} + F_{y, 1} \sin θ_{2}) \frac{l e v e r_{y, 2}}{2} \\ + (F_{f_{2}} \cos θ_{2} + N_{2} \sin θ_{2}) x_{4, 2} + (N_{2} \cos θ_{2} - F_{f_{2}} \sin θ_{2}) \frac{y_{4, 2}}{2}] \end{array}

(52)

The following sections follow the same form, yielding

\begin{array}{l} θ_{b, j} = \frac{L}{E I} [M_{r, j - 1} + (F_{y, j - 1} \cos θ_{j} - F_{x, j - 1} \sin θ_{j}) l e v e r_{x, j} . \\ - (F_{x, j - 1} \cos θ_{j} + F_{y, j - 1} \sin θ_{j}) \frac{l e v e r_{y, j}}{2} \\ + (F_{f_{j}} \cos θ_{j} + N_{j} \sin θ_{j}) x_{4, j} + (N_{j} \cos θ_{j} - F_{f_{j}} \sin θ_{j}) \frac{y_{4, j}}{2}] \end{array}

(53)

(c) Deformation of the entire backbone: Equations and transformation matrices (34) to (41) can be used to reconstruct the entire deformed shape of the backbone, using the newly calculated $θ_{j}$ .

Gripper contribution

Now that the bending segment has reached the position desired by the surgeon, the surgeon can activate the gripper. Hereto, the gripper cable is put under tension $T_{grip}$ by an actuator. The gripper will close, but additional forces will also appear on the segment which - complicating matters – will cause a deviation from the originally intended position. The impact of gripper actuation requires a separate analysis. While $F_{app}$ acts on the outer radius of the bending segment, the gripper force $F_{grip}$ is applied at the anchor point $(x_{A}, y_{A})$ located at the centre of the tube’s top section, visible in Figure 18. As a result a different path will be followed by the gripper cable. This has two implications: first, the differing cable path will determine the angle at which $F_{grip}$ acts; second, it will determine which sections and inner surfaces are in contact with the gripper cable and hence which sections will undergo friction and normal forces from the gripper cable. This calls for a new free-body diagram analysis to be conducted. The shape of the entire bending segment needs to be recomputed. This analysis departs thus from the equilibrium configuration that was obtained before initiating the gripper actuation.

(1) Gripper force angle identification: When tensioning the gripper cable attached at the anchor point, the cable, shown in green in Figure 18, will naturally take the shortest path. If there is no straight path between the anchor point and the actuator (i.e. the bending segment is sufficiently bent), the cable will establish contact with the inner wall of the tube. This contact will be tangential to the inner wall up to an arc length $s_{T}$ and location $(x_{T}, y_{T})$ where the cable loses contact again and stretches to the distal anchor point in a straight line. This boundary depends on the shape of the segment and the gripping force. Figure 18 defines an angle $ξ$ expressed relative to the world frame’s y-axis. This angle describes the orientation of the straight portion of the gripping cable and hence also how $F_{grip}$ engages at the anchor point. If the line between anchor point and the location where the gripper cable is pulled at the base of the segment does not intersect with the inner wall, the gripper cable does not make contact with the inner wall and is straight throughout its entire path. $ξ$ is then simply $\arctan (x_{A} / y_{A})$ and $F_{grip} = T_{grip}$ .

Figure 18.

Routing of the gripper cable (green), definition of $(x_{A}, y_{A})$ , $(x_{T}, y_{T})$ and angles.

In order to find $(x_{T}, y_{T})$ , first the equation of the inner wall curve $c_{iw}$ needs to be defined. In section II.D, a circle approximation was used since the moment was constant over the entire backbone. Due to the gripper and cables forces, this is no longer the case. Here, a power approximation is proposed instead:

c_{iw} (x) = a x^{b} + c

(54)

From the previous argumentation, the line connecting $(x_{T}, y_{T})$ and $(x_{A}, y_{A})$ has to be tangent to $c_{iw}$ . The slope of (54) can be expressed by differentiating (54) to $x$ as:

m_{1} = c'_{iw} (x) = ab x^{b - 1}

(55)

The slope can also be written as the slope between the attachment and the tangent contact point:

m_{2} = \frac{y_{T} - y_{A}}{x_{T} - x_{A}} = \frac{{ax}_{T}^{b} + c - y_{A}}{x_{T} - x_{A}}

(56)

Now, equating (55) and (56) as $m_{1} = m_{2}$

m_{1} = m_{2} \Leftrightarrow {abx}_{T}^{b - 1} = \frac{{ax}_{T}^{b} + c - y_{A}}{x_{T} - x_{A}}

(57)

one can solve this numerically to find $x_{T}$ as, before the inclusion of the gripping force, all the other variables are known. Finally one finds:

α = \arctan (\frac{x_{A} - x_{T}}{y_{A} - y_{T}})

(58)

The same approach can be followed at the base of the segment where the cable leaves the inner wall to return towards the central axis of the tube (assuming the gripper cable is attached to a centrally positioned actuator in the proximal tube portion).

(2) Friction between gripper cable and backbone: Once the cable path is identified, the beam sections that are in contact with the gripper cable can be identified. Equations (42) to (47) can be used again for these sections with $T_{n} = T_{grip}$ in order to find $T_{1} = F_{grip}$ . For the sections where the gripper cable contacts the inner wall, the friction and normal forces will be the sum of the influence of both cables as they act in the same direction and at the same point ( $x_{4}, y_{4})$ .

(3) Free-body diagram analysis: As introduced on Figure 18, $α$ is the angle between the world frame’s $y$ -axis ( $y_{0, n}$ , coinciding with the neutral axis at the segment base) and the direction of $F_{grip}$ . Figure 17 introduces a new angle $γ$ which is the angle between the local $y_{0, 1}$ axis and the direction of $F_{grip}$ . Therefore we have:

γ = Θ_{1} - α

(59)

In order to include the influence of the gripper in the new free-body diagram analysis, only the free-body analysis of the first section must be revisited to deal with the inclusion of $F_{grip}$ . The second and following sections will have the same form as (51) to (53), if replacing $N_{j}$ and $F_{f_{j}}$ with

{\begin{matrix} N_{j} = N_{b, j} + N_{g, j} \\ F_{f_{j}} = F_{f, b_{j}} + F_{f, g_{j}} \end{matrix}

(60)

(a) First section cut

\begin{matrix} ⇌ x_{0, 1} : F_{x, 1} = (F_{app} + F_{f_{1}}) \sin θ_{1} - N_{1} \cos θ_{1} - F_{grip} \sin γ \\ ⇌ y_{0, 1} : F_{y, 1} = (F_{app} + F_{f_{1}}) \cos θ_{1} + N_{1} \sin θ_{1} + F_{grip} \cos γ \\ ↻ O_{0, 1} M_{r, 1} = F_{app} (d \cos θ_{1} - y_{3, 1} \sin θ_{1}) \\ + (N_{1} \cos θ_{1} - F_{f_{1}} \sin θ_{1}) y_{4, 1} + (F_{f_{1}} \cos θ_{1} + N_{1} \sin θ_{1}) x_{4, 1} \\ + F_{grip} (y_{A} \sin γ + x_{A} \cos γ) \end{matrix}

(61)

These forces and moment now allow us to calculate the new bending angle of section 1

\begin{array}{l} θ_{b, 1} = \frac{L}{E I} [F_{a p p} d + (N_{1} \cos θ_{1} - F_{f_{1}} \sin θ_{1}) \frac{y_{4, 1}}{2} \\ + (F_{f_{1}} \cos θ_{1} + N_{1} \sin θ_{1}) x_{4, 1} + F_{g r i p} \sin γ \frac{y_{A}}{2} + F_{g r i p} \cos γ x_{A}] \end{array}

(62)

(b) Deformation of the entire backbone: The same equations and transformation matrices as in section IV.B can be used to reconstruct the entire deformed shape of the backbone, using the newly calculated $θ_{j}$ .

In summary, Algorithm 1 presents the computational method used to find the final bending segment shape in MATLAB (The MathWorks, USA).

Algorithm 1: Calculating initial deformed shape.
Input : $T_{bend}$ $T_{bend} = F_{app}$ ; $M = F_{app} d$ ; for $j = 1 : n$ do if $M_{j} > M_{limit}$ then $M_{j} = M_{limit}$ (31); else $M_{j} = M$ ; end $θ_{j} = \frac{M_{j} L}{EI}$ ; $(x_{1, j}; y_{1, j})$ (25); $O_{0, j}^{0, n}$ (35); end $Θ = \sum_{i = n}^{1} θ_{i}$ ; Output: $O_{0, j}^{0, n} (T_{bend}); Θ$

Once the initial deformed shape is obtained, Algorithm 2 loops equations taking into account normal and friction forces until finding an equilibrium, with $T_{grip} = 0$ initially.

Algorithm 2: Friction & normal forces equilibrium, Gripper contribution.
Input: $T_{bend}$ ; $T_{grip}$ ; $O_{0, j}^{0, n}$ ; $Θ$ $δ_{θ} = 1$ ; while $\| δ_{θ} \| \geq 0.001$ do $T_{n} = T_{bend}$ ; $Θ_{prev} = Θ$ ; for $j = n : - 1 : 1$ do $β_{j}$ : (44); $T_{j}$ ; $N_{j}$ ; $F_{f_{j}}$ : (45) → (47); end $F_{app} = T_{1}$ ; for $j = 1 : n$ do $FD B_{j}$ : (48) → (53); $(x_{1, j}; y_{1, j})$ (25); $O_{0, j}^{0, n}$ (35); end $Θ = \sum_{i = n}^{1} θ_{i}$ ; $δ_{θ} = Θ - Θ_{prev}$ ; end $F_{t} = F_{ap p_{1}} e^{μ_{s} \sum_{j = 1}^{n} β_{j}}$ Output: $O_{0, j}^{0, n} (T_{bend}, Θ); Θ$

Algorithm 2: Friction & normal forces equilibrium, Gripper contribution.

Input:

T_{bend}

;

T_{grip}

;

O_{0, j}^{0, n}

;

Θ

δ_{θ} = 1

;
while

| δ_{θ} | \geq 0.001

T_{n} = T_{bend}

;

Θ_{prev} = Θ

;
for

j = n : - 1 : 1

β_{j}

: (44);

T_{j}

;

N_{j}

;

F_{f_{j}}

: (45) → (47);
end

F_{app} = T_{1}

;
for

j = 1 : n

FD B_{j}

: (48) → (53);

(x_{1, j}; y_{1, j})

(25);

O_{0, j}^{0, n}

(35);
end

Θ = \sum_{i = n}^{1} θ_{i}

;

δ_{θ} = Θ - Θ_{prev}

;
end

F_{t} = F_{ap p_{1}} e^{μ_{s} \sum_{j = 1}^{n} β_{j}}

Output:

O_{0, j}^{0, n} (T_{bend}, Θ); Θ

Once an equilibrium is obtained, Algorithm 2 can be run again to add the influence of the gripper with $T_{grip} > 0$ ; looping once more with all forces combined in order to find the final equilibrium shape of the bending segment.

Experimental validation

Gripping force validation

To verify the gripping force (Figure 4 and (11)), the experimental setup of Figure 19 is built. A fixed force is applied by a 5 mm McKibben muscle (DMSP-5, Festo, Germany) to the gripper. The muscle is attached to a load cell to measure the force it generates. An Ultra High Molecular Weight Polyethylene (UHMWPE) rope of 0.2 mm diameter from Caperlan (Decathlon, France) is attached perpendicularly to the gripper moving jaw. A load cell at the base of the rope allows measuring the force generated by the gripper. The rope loops 180° around a steel eyelet fixed at the end of this second load cell in order to easily adjust the tension in the rope. The rope can then be pulled until the desired jaw opening angle is reached. The force required to open the jaw can be measured by this second load cell. However, the forces measured by this load cell needs to be corrected by two factors: first, the tension applied on the free end of the rope is higher than the tension applied on the gripper due to the Capstan equation. The first factor is therefore $f_{1} = e^{μ . π}$ with $μ = 0.175$ the coefficient of friction between steel and UHMWPE (one finds a range of $μ = [0.15 - 0.2]$ in suppliers catalogues). The second factor follows from the fact that the reaction force measured at the load cell is the sum of tension forces exerted by both rope ends, so $f_{2} = 2$ . Therefore the true measured gripping force is:

F_{grip} = \frac{F_{measured}}{f_{1} . f_{2}}

(63)

Figure 19.

(a) Experimental validation setup of the gripping force. (b) Zoomed-in view of the gripper during the experiment.

Figure 20 shows the theoretical gripping forces obtained with various input forces and jaw angles. The ‘x’ marks show experimental data points. The experimental values are slightly below theoretical ones due to frictional effects neglected in the mechanism static equations (1) to (11). The experimental results do not cover the whole spectrum of forces due to limitations of the gripper material strength, but sufficiently cover the intended force spectrum.

Figure 20.

Theoretical (lines) and experimental (x) gripping force vs jaw angle and input force.

Optimisation validation

In order to confirm the optimisation of the bending segment, a $1$ N transversal load is applied at the tip of the bending segment (Figure 21). The bending angle is then increased by incrementing $T_{bend}$ with steps of $2$ N. For each angle the corresponding tip deflection (due to the transversal load) in the XZ-plane is measured on calibrated paper. Five repetitions of this experiment are plotted in Figure 22. The figure shows that the segment starts deflecting along the direction of the transversal load. This happens twice at $35 °$ and 3 times at a $40 °$ bending angle. Overall this corresponds well to the angle of $35.5 °$ predicted in Section III.B. For illustration purpose a hysteretic recovery behaviour - upon return to the straight configuration (in rep 1) is shown as well, demonstrating the superelastic behaviour that took place.

Figure 21.

Tip deflection under 1 N load (a) $0 °$ bending showing the reference measurement line (green), (b) $40 °$ bending, marking the beginning of the tip deflection along the z axis, and (c) $75 °$ bending, showing a large deflection along the z axis.

Figure 22.

Tip deflection vs bending angle with 1 N tip load.

Bending segment model

In order to verify the bending model of Section IV, the segment is mounted onto the setup shown in Figure 23. Two markers are placed at the segment’s proximal and distal extremities in order to track the bending angle, similar to the dynamic characterisation in.¹² The gripper is mounted at the tip of the bending segment. Two cables are connected to a $5$ mm McKibben muscle (DMSP-5, Festo, Germany) each, which are in turn connected to a load cell each. The first muscle actuates the gripper cable. The second actuates the bending cable.

Figure 23.

Experimental validation setup for the bending segment.

Figure 24 is obtained by varying the bending force from $0$ to $40$ N and back to $0$ N by increments of $4$ N. The bending angle is then measured via image processing of the markers via the MATLAB routine imfindcircles. This manipulation was repeated 10 times.

Figure 24.

Experimental validation of the bending segment model. Tip bending angle vs input force.

The 10 loading and unloading phases are decently superposed, showing good experiment repeatability. Some hysteresis can be observed as the loading and unloading phases do not take the same path. Both phases take somewhat of an ‘S’-shape typical of hysteresis. The mean between all phases is plotted as a green dotted line. The bending segment model prediction is plotted as a blue dashed line. A decent correlation between the average of loading and unloading phases (green dotted line) and the model (blue dashed line) can be observed. The two lines start and end at the same points but model takes less of an accentuated S-shape and seems more linear. This is probably because the material hysteresis was not taken into account in the model.

Gripper influence model

Now that the position model has been validated, the addition of the gripper must also be validated. In order to verify the model, 3 bending forces $T_{bend}$ and 3 gripping forces $T_{grip}$ have been applied and measured. This was repeated 10 times. Figure 25 shows the resulting characterisation.

Figure 25.

Experimental validation of the gripper influence on the bending segment deflection angle.

Figure 25 is presented in the form of a box plot for each of the 10 measurements at each $T_{bend}$ and $T_{grip}$ . Each horizontal line represents the median value of each box. Table 3 compares these experimental values to those predicted by the model. Similarly to the position model (Figure 24), the model is more accurate at the extremities of the range, while the error is largest around the middle region. The relative error fluctuate between 1.04% and 14.55%, with a mean error of 8.23%.

Table 3.

Gripper influence on the bending angle.

$F_{bend} [N] - F_{grip} [N]$	Exp. $(+ Δ) [°]$	Model $(+ Δ) [°]$	Abs. $ε [°]$	Rel. $ε [%]$	Abs. $Δ ε [°]$	Rel. $Δ ε [%]$
12-0	18.30	16.94	−1.36	−8.03
12-3	20.82 (+2.52)	20.18 (+3.24)	−0.64	−3.15	0.72	22.36
12-6	23.69 (+5.39)	23.45 (+6.51)	−0.24	−1.04	1.12	17.13
12-9	26.49 (+8.19)	27 (+10.06)	0.51	1.89	1.87	18.59
24-0	32.09	36.04	3.95	10.95
24-3	34.7 (+2.61)	40.61 (+4.57)	5.91	14.55	1.96	42.94
24-6	38.67 (+6.58)	44.49 (+8.45)	5.82	13.08	1.87	22.18
24-9	42.56 (+10.47)	49.23 (+13.19)	6.67	13.55	2.72	20.67
36-0	51.46	54.86	3.4	6.19
36-3	55.04 (+3.58)	60.56 (+5.7)	5.52	9.12	2.12	37.3
36-6	59.95 (+8.49)	65.04 (+10.17)	5.09	7.83	1.69	16.65
36-9	65.12 (+13.66)	69.07 (+14.21)	3.95	5.72	0.55	3.89

Discussion

The position model validated on Figure 24 reportedly shows a good correlation with the average of the loading and unloading curves. The model is based on the loading phase only but this good correlation means that the model could be used as a decent approximation for both loading and unloading phases. With a mean error of 8.23%, the combined bending and gripping model could be useful to counteract the influence of the gripper: by utilising the model prediction, the gripper influence would be reduced by 25% in the worst case and 96% in the best case; with an average reduction of 68%. In order to run this compensation in real time, a look-up table should be implemented as the model takes about 3 s to calculate the bending segment shape and the gripper influence.

Some important limitations must be considered working with Nitinol. This alloy is very capricious to work with: the slightest heat treatment can affect dramatically its properties, rendering the supplier material characteristics erroneous. In addition, even when one is above the austenite finish temperature, the Young’s modulus remains very dependent of the ambient temperature. The experiments of Section VI.B were run several times throughout the year and various meteorological conditions and the maximum deflection obtained for an identical input force (Figure 24) varied from $58 °$ during hot days to $80 °$ during cold days. This is an almost 40% variation. Therefore, the model accuracy could be significantly improved by measuring material properties and conducting experiments at the temperature of application; in this case the human body temperature. This is why Figure 24 shows the lower end of the spectrum: the experiment ran during the hottest day was selected.

Conclusions and future work

This paper first proposed a novel hybrid gripper. The gripper is capable of functioning both as regular surgical forceps and as a needle driver. The design and kinematics were presented in detail. All the design specifications (S1-S6) were met. Second, a structural optimisation of a flexible Nitinol bending segment was explored in order to allow this segment to withstand the force exerted by the needle driver. A model based on two parallel beams was proposed to describe the impact of torsion induced by transversal puncture forces. The model was used in the structural optimisation yielding a bending segment that satisfies all the requirements R1-R4 for OSB suturing. All the derivations and solutions were subsequently validated using FEA. Third, a kinematic model was derived to predict the combined effect of bending and gripping loads acting at the tip of the gripping segment. The model takes into account the friction and normal forces both from bending and gripping cables that act on the inner wall of the bending segment; effectively altering the overall shape of the segment. The model iteratively computes the bending segment shape until an equilibrium is found. Finally the gripper design, the bending segment optimisation and the kinematic model were verified experimentally. This paper contains fundamental contributions that will help precise execution of high-force surgical tasks and manipulations in continuum robotics.

This paper leaves room for future work. In this work only the loading scenario is modelled. The model could thus be expanded by also accounting for the unloading case. A further expansion could look at the scenario where operation takes place in the superelastic region of the Nitinol (i.e. if the gap $g$ was wider and the segment could bend further). The temperature sensitivity of Nitinol could also be taken into account into the model. Having a FEA software capable of modelling Nitinol would also make the design process more robust. Finally, the influence of the gripper upon the bending has been modelled, but not yet incorporated in the control. In principle the gripper model could be used to compensate for the extra bending induced by the gripper. For example, while the gripper muscle contracts to close the jaws, the bending muscle could be relaxed at the same time to maintain the overall position.

Footnotes

Acknowledgements

This research was funded in part by the Wellcome Trust [WT101957]. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. The second part of the funding is a personal Strategic-Basic research Fellowship of the Research Foundation – Flanders (1S96518N). The author gratitude goes to Pr. Van Hooreweder and Ir. Meyers from the MaPS group (KU Leuven) for 3D printing several iterations of gripper prototypes.

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Funding from Wellcome Trust [WT101957] and a personal SB research grant from the FWO – Flanders (1S96518N).

ORCID iDs

Tom Vandebroek

Tom Vercauteren

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Design and modelling of an anisotropic continuum robot end-effector for single-port access surgery suturing

Abstract

Keywords

Introduction

Hybrid gripper design

Design specifications

Design and prototyping

Bending segment structural optimisation

Geometry

Requirements

Maximum stress identification

Structural optimisation

Results

FEA verification

Kinematic model and control

Modelling approach and hypotheses

Deformation model of a Nitinol notched backbone – Frictionless model

Deformation model of each Nitinol notched section - Friction model

Gripper contribution

Experimental validation

Gripping force validation

Optimisation validation

Bending segment model

Gripper influence model

Discussion

Conclusions and future work

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iDs

References