This application relates to the field of magnetically actuated shape-forming surgical continuum manipulators, methods of manufacture and methods of operation thereof.

BACKGROUND

Surgical continuum manipulators (“CMs”) have been used to assist with and enable surgical procedures in the form of catheters and endoscopes for at least the last 120 years. Traditional continuum manipulators rely on body rigidity to transmit forces and torques from proximal to distal ends. This approach relies on operator skill, offers limited accuracy or dexterity and the process itself can cause tissue trauma.

These limitations may be mitigated with the use of soft robotic manipulators which are primarily fabricated from elastomeric materials. Such robotic manipulators may be fluid driven, tendon driven, made from shape memory alloy or electroactive polymer, or magnetically actuated.

Tip driven magnetically actuated CMs wherein the tip of the device is magnetically driven have been demonstrated to increase control and reduce trauma during the negotiation of anatomical convolutions. Example are described in:

S Jeon, AK Hoshiar, K Kim, S Lee, E Kim, S Lee, J-y Kim, BJ Nelson, H-J Cha, B-J Yi and H Choi, “A Magnetically Controlled Soft Microrobot Steering a Guidewire in a Three- Dimensional Phantom Vascular Network”, Soft Robotics, vol 6, no 1 , pp54-68, Oct 2018 https://doi.org/10.1089/soro.2018.0019, and Y Kim, GA Parada, S Liu and X Zhao, “Ferromagnetic soft continuum robots”, Science Robotics, vol 4, no 33, p.eaax7239,2019.

These systems, however, can only assume the body shape of their respective conduit via anatomical interactions. The highly convoluted geometries and millimetre scale workspaces make this a challenging area of research and magnetic actuation has its own attendant complexities regarding the modelling and simulation of long, slender and potentially unstable elastomers.

It is therefore an object of the present invention to provide an improved magnetically actuated shape-forming surgical continuum manipulator. BRIEF SUMMARY OF THE DISCLOSURE

The invention is defined in the appended claims. According to a first aspect of the invention, there is provided a magnetic shape-forming surgical continuum manipulator (“CM”) comprising an elastomeric base material and a plurality of magnetic elements, the plurality of magnetic elements being located at a plurality of points along a length of the CM and each magnetic element having a predetermined magnetic profile, whereby the shape of the CM can be magnetically manipulated substantially along said length by the application of an external magnetic field and, optionally, a magnetic field gradient.

In an embodiment, the plurality of magnetic elements comprises magnetic particles dispersed in the elastomeric base material. The magnetic particles may be dispersed at different concentrations and/or have different magnetic profiles along said length.

In another embodiment, the plurality of magnetic elements comprises multiple spaced permanent magnets embedded in the elastomeric base material.

In an embodiment, the shape-forming surgical continuum manipulator further comprises a lumen along said length providing a working channel therethrough. Optical fibres for laser ablation, for example, could be provided and operated via said lumen.

Preferably, the magnetic shape-forming surgical continuum manipulator has an external diameter of less than 2mm.

In an embodiment, the magnetic shape-forming surgical continuum manipulator further comprises one or more sensors.

In an embodiment, the elastomeric base material has an anisotropic elasticity distribution which can improve bending performance of the CM by reducing torsion.

The magnetic shape-forming surgical continuum manipulator may further comprise a reinforcing element having higher stiffness than said elastomeric based material. The reinforcing element may comprise a helical element.

According to a second aspect of the invention, there is provided a method of manufacturing a magnetic shape-forming surgical continuum manipulator according to any of the preceding paragraphs comprising the steps of: a. Combining said magnetic elements with the elastomeric material by dispersing or embedding said magnetic elements therein; and b. Magnetizing said magnetic elements to create said predetermined magnetic profile.

In an embodiment, the combining step comprises extruding said elastomeric material. Alternatively, the combining step comprises moulding said elastomeric material in a shaped tray.

The combining step may be performed before, after or during said magnetizing step.

According to a third aspect of the invention, there is provided a method of controlling a magnetic shape-forming surgical continuum manipulator according to any of the preceding paragraphs comprising the steps of: a. applying an external magnetic field to the CM; b. allowing the CM to adopt a shape along the length thereof as a result of manipulating said external magnetic field.

“Manipulating” said external magnetic field may simply mean switching the field on or off, and/or may mean applying a magnetic field gradient.

In an embodiment, the method further comprises the step of pulling the CM to a new location as a result of the application and/or manipulation of said external magnetic field. Preferably, a pulling force is applied along the length of the CM.

In an embodiment, in step b, the CM adopts a stiffened shape in order to provide a working channel via said lumen. Alternatively, in step b, the CM adopts a dynamically changing shape dependent on said manipulation of the external magnetic field.

In an embodiment, said external magnetic field is applied by dual arm collaborative magnetic manipulation, electromagnetic coils or magnetic resonance imaging (MRI).

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be more particularly described, by way of example only, with reference to the accompanying drawings in which: Figure 1 is a side view of a CM according to an embodiment of the invention;

Figure 2 is a side view of a CM according to another embodiment of the invention;

Figures 3A- 3D and 4 illustrate manufacturing methods for prototype CMs according to an aspect of the invention;

Figure 5 is a schematic representation of dual arm control of a CM; and

Figure 6 illustrates an extrusion and magnetisation method for prototype CMs.

Figures 7A - 7D show a fabrication process for a CM section with a helical reinforcement element.

Figures 8A and 8B show a CM without a helical reinforcement element and a CM with a helical reinforcement element.

DETAILED DESCRIPTION

Throughout the description and claims of this specification, the term “continuum manipulator” or “CM” is intended to refer to a surgical continuum manipulator, tentacle or robotic manipulator having an elongate shape which can be manipulated. The definition extends to prototypes of any of the above, including those prototypes which have no surgical function.

Throughout the description and claims of this specification, the term “shape forming” is intended to refer to the property of a CM whereby its shape, in particular its curvature, can be selected, controlled or manipulated along part or all of its length.

The “proximal” end of a CM means the tail end of the CM, the end nearest the point of origin and nearest the clinician.

The “distal” end of a CM means the leading end of the CM, the end furthest from the point of origin and furthest from the clinician. Throughout the description and claims of this specification, the words “comprise” and “contain” and variations of the words, for example “comprising” and “comprises”, means “including but not limited to”, and is not intended to (and does not) exclude other moieties, additives, components, integers or steps.

Throughout the description and claims of this specification, the singular encompasses the plural unless the context otherwise requires. In particular, where the indefinite article is used, the specification is to be understood as contemplating plurality as well as singularity, unless the context requires otherwise.

Features, integers, characteristics, compounds, chemical moieties or groups described in conjunction with a particular aspect, embodiment or example of the invention are to be understood to be applicable to any other aspect, embodiment or example described herein unless incompatible therewith.

Referring to Figure 1 , a “multi-segment” magnetic shape-forming continuum manipulator (“CM”) is shown comprising an elastomeric base material 1 and a plurality of magnetic elements 2 embedded therein. The elastomeric base material 1 is a silicone elastomer such as Ecoflex™ 00-30. The magnetic elements 2 are equispaced permanent magnets made, for example, from silicone elastomer doped with neodymium-iron-boron (NDFeB) microparticles with an average diameter of 5μm. The permanent magnets can each have their own individual magnetisation direction. Onboard sensors such as Hall effect sensors and IMUs (inertial measurement units) may be integrated or co-located with the permanent magnets so that they are spaced along the CM.

In an alternative embodiment shown in Figure 2, a “single segment” CM comprises an elastomeric base material 1 doped throughout with a plurality of magnetic elements in the form of a plurality of magnetic particles. Optionally there may be regions 2’ having a greater concentration of magnetic particles. In manufacturing a prototype, particles of NdFeB were added to a prepolymer in a 1 :1 ratio by weight equating to a volumetric ratio of 0.88:0.12 (Ecoflex™:NdFeB). The composite was mixed and degassed in a high vacuum mixer (ARV- 310, THINKYMIXER, Japan) at 1400rpm, 20.0kPa for 90 seconds and then injected onto a straight cylindrical mould of diameter d=1 ,5mm and length 20mm and left to cure. The mould contained a centrally aligned 0.25mm diameter Nitinol needle running for 10mm of its length. This needle remained embedded in the polymer and was used to suspend and constrain the prototype during testing. Once the polymer had cured, the prototype was subjected to a uniform field of 46.44 KGauss (4.644 T) (ASC IM-10-30, ASC Scientific, USA) orthogonal to the CM prototype’s principle (longitudinal) axis.

Referring to Figures 3A-3D, a prototype multi-segment CM was manufactured as follows. An unmagnetized elastomer doped with NdFeB was injected into a mould around a centrally aligned needle 3 (Fig 3A). Once cured, the doped elastomer was divided into three identical 7mm segments 2 which were axially separated by 14mm, still on the needle (Fig 3B). Alternatively the doped elastomer was removed from the needle 3 and divided into segments which were then replaced, axially spaced, on a needle of slightly greater diameter so that the axial positioning could be more easily maintained owing to the tighter friction fit.

The needle-mounted segments 2 were then placed in a second mould 4 and an undoped silicone elastomer base material 1 (Ecoflex™ 00-30) was injected around them (Fig 3C). Upon curing of the polymer, the needle 3 was removed save for the final 10mm which remained embedded to act as a mechanical constraint during experiments on the prototype.

The total length of the multi-segment prototype was 52mm (Fig 3D). From bottom to top this can be broken down as 10mm of unconstrained length followed by 42mm of constrained length. In the Figures, the undoped elastomer appears white and the doped segments comprising the magnetic elements 2 appear black. The dimensional accuracy of the fabricated CM prototypes was assessed through image analysis software (LAZ, EZZ, Leica, Germany), calibrated against a known reference length with images obtained using a digital light microscope (DMS300, Leica, Germany). The magnetic element segments 2 had lengths (Mean+/-SD) of 7.4+/-0.43mm and diameters 1.9+/-0.03mm. Specific values and dimensions mentioned above are given by way of example only and are not intended to limit the scope of the appended claims.

Instead of using a needle 3 to maintain the desired axial spacing of the segments 2, this could instead be achieved by the design of the mould shape per se and/or the use of radial pins or other alignment features to hold the segments in place as the elastomer base is moulded around them.

Using the above described method, a CM is manufactured by pre-preparing the magnetic elements 2 and then moulding the undoped elastomer 1 around the magnetic elements 2.

An alternative is to combine the elastomer with sequentially inserted magnetic elements as illustrated in Figure 4. The doped elastomer magnetic segments 2 are prepared in the same way as described above in relation to Figures 3A and 3B and then removed from any supporting needle 3. A magnetic segment 2 is inserted into a mould 4 and then pushed down into the mould by the injection therein of undoped elastomer 1 . Sequential alternate injection of elastomer and insertion of magnetic elements creates a CM with a desired distribution and spacing of magnetic elements 2 in an elastomeric base material 1. Once fully cured, the CM is removed from the mould 4.

Instead of combining the undoped elastomer with the magnetic elements in one of the methods as described above, a further alternative is to extrude undoped elastomer simultaneously with doped elastomer as illustrated in Figure 6. Figure 6 shows a vacuum based extrusion system in which P(t) is applied to selectively extract liquid elastomer (either doped or undoped) from two reservoirs into a tube-shaped mould 4. Doped elastomer in liquid form is provided in reservoir 5. Undoped elastomer in liquid form is provided in reservoir 6. Doped or undoped elastomer can be drawn alternately into the mould 4, or a mixture can be simultaneously drawn from both reservoirs 5, 6, in order to create a desired concentration of doped particles along the length of the CM. Such a continuous distribution may be homogenous or may vary in concentration along the length of the CM.

The apparatus 8 provides localised curing of the CM for example using locally-applied heat or UV from curing apparatus 7. At least part of the apparatus 8 is rotatable about the longitudinal axis of the mould 4 (i.e. in the direction indicated by the arrow 9 in Figure 6) so that heat/UV can be applied as desired.

The mould 4 may move through the apparatus 8, or the apparatus 8 may be linearly translated with respect to the mould 4.

The mould may comprise PVA so that it can easily be removed from the cured CM by dissolving the mould in water.

When manufacturing the “single segment” CM of Figure 2, instead of injecting the composite into a mould as described above, an alternative method is to use a mould formed from a sacrificial gelatin. A cavity of desired shape is formed in a sacrificial gelatin and then the composite (the elastomer and magnetic particle mix) is injected into the sacrificial gelatin mould which supports the composite while it cures. Once cured, a magnetizing step (described below) is performed, after which the sacrificial gelatin mould can be removed by dissolving in hot water, leaving the single segment CM ready for use.

A magnetising step is employed to magnetise the magnetic elements of the CM prior to use in a clinical situation. The CM may be housed in a magnetizing tray (Fig 3D) and exposed to for example a 46.44 KGauss (4.644 T) saturating field. The geometry of the magnetizing tray may be determined by the solution to the inverse static problem for the CM, the solution being generated by a neural network based on a predefined desired shape for the CM.

It is possible to perform the magnetising step of magnetising the doped segments/magnetic elements either before or after the moulding/extrusion step combining the elastomer and doped segments together. As illustrated in Figure 6, it is also possible to perform the magnetising step simultaneously with extrusion using magnetising coil 10.

The elastomer may be moulded or extruded around a removable rod or needle which, when removed, leaves a lumen that can be used as a working channel.

The result is a CM having multiple magnetic elements arranged along its length i.e. not only at its distal tip as is conventionally known. Application of an external magnetic field and optionally a magnetic field gradient means the CM can be driven along a predetermined path by forces applied along its length so that it can be guided carefully through the desired path rather than pushed from the proximal end or pulled from the distal tip. The soft elastomer minimises trauma to surrounding tissues.

The CM may have a generally circular cross-sectional shape although other cross-sectional shapes are possible.

The diverse range of magnetic fields that will be applied to the CM could potentially lead to instability resulting from the CM twisting about its longitudinal axis in search for the minimum energy pose. Adaptive dynamic control of the applied magnetic fields could potentially be used to counteract this instability but this is impractical for real life applications due to the challenges of monitoring and sensing within the human body. An alternative solution is for the CM to have an anisotropic elasticity distribution by reinforcing the elastomer with higher stiffness fibres in order to restrict torsion whilst still permitting bending.

The CM may thus be provided with a helical reinforcing element. The helical reinforcing element 20 may be in the form of a single helix or a double helix (i.e. a pair of helices comprising one left handed helix and one right handed helix).

Steps for forming a CM with helical reinforcing element 20 are shown in Figure 7. As shown in Figure 7A, the helical reinforcing element 20 is made from extruded PLA (polylactide) fibre of diameter 0.4mm (+ or - 0.02mm) wound around a 3D printed cylindrical form 11 featuring the desired helical groove. The fibre is wound around the form 11 and secured before being subjected to a heat cycle peaking at 60°C for 30 minutes. The heat treatment enables the fibre to retain the desired helical shape after removal from the form. This can be repeated for both left and right-handed helices with a helix angle of 0 _H = 85°.

As shown in Figure 7B, a clockwise helix is secured in a first cylindrical mould 12A., Cylindrical inserts 14 are provided at intervals to create cavities at predefined desired angles for the magnetic elements 2 to be inserted later. The elastomeric base material is injected into the mould around the clockwise helix and cured.

One removed from the first mould 12A, the cured structure is placed within a second, anticlockwise helix and the inserts 14 are removed so that magnetic elements 2 (permanent magnets) can be placed in the resulting cavities. Next, as shown in Figure 7C, the structure is placed in a second mould 12B so that additional elastomeric base material can be injected to secure the magnetic elements and anti-clockwise helix in place. The completed reinforced CM 20 is shown in Figure 7D, removed from the second mould.

Figure 8A shows a CM 20A without helical reinforcement. When subjected to a magnetic field, the unreinforced CM has a mean twist of 145° ± 12° (where 180° would indicate a complete reversal of the permanent magnets) and mean bend is just 6° ± 5°. Figure 8B shows a CM 20B with helical reinforcement. In the reinforced CM 20B mean twist is reduced to 49° ±5 ° and, due to preservation of magnetic energy, mean bend increases to 40° ±7°. In other words, when unreinforced 20A and reinforced 20B CMs are subjected to identical magnetic fields, the unreinforced CM 20A twists, without producing a sufficient bend angle, whereas the reinforced CM 20B twists significantly less, and produces the required bend angle.

The magnetic elements are magnetised, before clinical use, with a magnetic profile that can be actuated during clinical use in order to determine the shape of the CM. The CM may be designed to have a specific predetermined shape that can be “switched on” by the external magnetic field when the CM has reached its destination. Alternatively, the CM may be designed with a specific insertion profile that can be dynamically controlled by the external magnetic field and a magnetic field gradient so that each segment moves in a “follow my leader” fashion to avoid obstructions and to follow a desired path during insertion.

Independent control of the magnetic elements enables the CM to adopt a shape along its length that can be selected for the specific clinical application and indeed the anatomical structures of a specific patient. This enables the CM to adopt a shape conforming to tortuous curvilinear trajectories without exerting significant pressure on surrounding tissues. Control along the length of the CM provides the ability to stiffen part(s) of the CM to accomplish specific surgical tasks that need structural rigidity.

The magnetic elements can have homogenous magnetisation i.e. identical magnetisation for each element, tuneable magnetisation i.e. where the magnetisation can be changed dynamically, or heterogenous magnetisation i.e. where each element has a different magnetisation profile.

In order to actuate the shape-forming aspects of the CM, an external magnetic field and, optionally, a magnetic field gradient is applied. Magnetic fields offer the possibility of manipulating the CM from afar and with penetrate human tissues without inflicting any harm on the patient. Magnetic control of a CM avoids the need for tendons or other internal actuation mechanisms thus facilitating miniaturisation and body flexibility of the CM.

The external magnetic fields and magnetic field gradients can be either uniform in the entire workspace or position-variant. This gives the following example combinations:

• Homogenous magnetisation, uniform fields and position-variant field gradients;

• Homogenous magnetisation and position-variant fields generated by permanent magnets;

• Homogenous magnetisation and position-variant fields generated by electromagnets;

• Tuneable magnetisation and uniform fields;

• Heterogeneous magnetisation and uniform fields;

• Heterogeneous magnetisation, position-variant fields and position-variant field gradients.

The external magnetic fields and magnetic field gradients can be provided by any one of a number of different techniques, for example: electromagnetic coils, MRI (magnetic resonance imaging) or multiple arm collaborative magnetic manipulation. Use of dual arm manipulation is schematically illustrated in Figure 5 but more than two arms could be used.

Reducing the volume of the magnetic elements of the CM in order to facilitate miniaturisation leads to a loss of magnetic wrench for a given field. However this can be directly compensated for through appropriate dimensioning of the external magnetic actuation system. Specifically, more force/torque can be achieved by using more powerful actuation systems without a direct increase in the CM’s dimensions.

For completeness, the complete content of the priority document of the present application is reproduced below and forms part of the description of the present application. Claims follow thereafter.

A learnt approach for the design of magnetically actuated shape forming soft tentacle robots

Peter Lloyd ¹ , Ali Kafash Hoshiar ² , Tomas da Veiga ¹ , Aleks Attanasio ¹ , Nils Marahrens ¹ , James H. Chandler ¹ and Pietro Valdastri ¹

Abstract — Soft continuum robots have the potential to revolutionize minimally invasive surgery. The challenges for such robots are universal; functioning within sensitive, unstructured and convoluted environments which are inconsistent between patients. As such, there exists an open design problem for robots of this genre. Research currently exists relating to the design considerations of on-board actuated soft robots such as fluid and tendon driven manipulators. Magnetically reactive robots, however, exhibit off-board actuation and consequently demonstrate far greater potential for miniaturization and dexterity. In this paper we present a soft, magnetically actuated, shape forming, high aspect ratio ‘tentacle-like’ robot. To overcome the aforementioned design challenges we also propose a novel design methodology based on a Neural Network trained using Finite Element Simulations. We demonstrate how our design approach generates static, two-dimensional tentacle profiles under homogeneous actuation based on predefined, desired deformations. To demonstrate our learnt approach, we fabricate Fig. 1. A sample application of the magnetically actuated tentacle - neurovascular catheter navigation. The target shape is derived from a preand actuate candidate tentacles of 2mm diameter and 42mm operative Magnetic Resonance Image of the brain. Under specific electrolength producing shape profiles within 5% mean absolute magnetic actuation, the desired shape is assumed. percentage error of simulations. With this proof of concept, we make the first step towards showing how tentacles with bespoke methods of actuation are fluid driven [4], tendon driven [5], magnetic profiles may be designed and manufactured to suit specific anatomical constraints. shape memory alloy [6], electroactive polymer [7] and

Index Terms — Modeling, Control, and Learning for Soft magnetic [8] systems. For the non-magnetic ’on-board’ Robots; Soft Robot Materials and Design; Surgical Robotics: actuated systems a fundamental trade-off will always exist Steerable Catheters/Needles. between dexterity and miniaturisation potential; for each additional degree of freedom (DoF) controlled within the

I. INTRODUCTION manipulator, a Anther driveline (e.g. a fluid channel or

Continuum Manipulators (CMs) have been used to assist tendon) must be added. This limitation does not apply to with and enable surgical procedures in the form of catheters magnetic actuation and, consequentially, the magnetic and endoscopes for at least the last 120 years [1], Traditional approach exhibits far greater potential for miniaturisation and

CMs rely on body rigidity to transmit forces and torques therefore surgical application than rival methods. from proximal to distal ends. This approach relies on operaMagnetically actuated tip driven systems [9] [10] have been tor skill, offers limited accuracy or dexterity and the process demonstrated to increase control and reduce trauma [11], [12] itself can cause tissue trauma [2], These limitations may be during the negotiation of anatomical convolutions. Recently a mitigated with the use of soft robotic manipulators which are number of works [13], [14], [8] have demon- strated the primarily fabricated from elastomeric materials and actuated efficacy and miniaturisation potential of such catheters. These through a wide range of methods as detailed in [3], Common systems, however, can only assume the body shape of their respective conduit via anatomical interaction. A soft

Research reported in this article was supportedby the Royal Society, by the Engineering and Physical Sciences Research Council (EPSRC) under continuum robot equipped with lull body-shape control would grant number EP/R045291/1, and by the European Research Council (ERC) possess the potential to assume a predefined shape without under the European Union’s Horizon 2020 research and innovation relying on these forces. We would consider such a robot, programme (grant agreement No 818045). Any opinions, findings and conclusions, or recommendations expressed in this article are those of the with numerous wrenches acting along its length, to be ‘ fully authors and do not necessarily reflect the views of the Royal Society, EPSRC, shape forming’ in contrast to the conventional tip driven or the ERC.

¹ manipulators. The magnetic shape control demon- strated in Peter Lloyd, Tomas da Veiga, Aleks Attanasio, Nils Marahrens, James H. Chandler and Pietro Valdastri are with the Storm Lab, School of Electronic [15], [16] and [17] could, in principle, be exploited to provide and Electrical Engineering, University of Leeds, UK, men9prl, eltjfdv, elaat, a safe railroad to a predefined working location. This would elnma, J.H.Chandler, p.valdastri @leeds.ac.uk)

² enable improvements in safety, procedure time and patient Ali Kafash Hoshiar is with the School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK, comfort; this concept is shown schematically in Fig. 1. For ^• fa.kafashhoshiar@essex.ac.uk) real-world applications, the highly convoluted geometries and millimetre scale workspaces encountered Network (ANN). The output of the trained ANN represents make magnetic actuation a promising, if challenging, avenue the solution to the inverse statics of our soft robot - when we of research. Magnetic actuation, however, comes with its own refer to forward and inverse statics, we are referring to the soft attendant complexities regarding the modelling and serial robot equivalent of kinematics in conventional hard simulation of long, slender and therefore potentially unstable, robotics. The difference being the requirement, for a soft magnetically active elastomers. Henceforth, we refer to our robot, of forces to maintain static equilibrium. This solution slender, shape forming, soft robots as magnetic tentacles. in turn informs the design of our experimental prototypes. The

Shape forming CMs (as opposed to tip-driven CMs) results produced by the ANN are validated for three presented in the literature exhibit a variety of modelling demonstrative shapes in both the underlying FEM and, after methods. Most prominently we observe the Cosserat rod [18], fabrication, in our experimental setup. the constant curvature [19] and the rigid-link [20] models. II. DESIGN APPROACH Each model represents a level of approximation and attendant computational intensity deemed appropriate for its Machine learning techniques which are driven by real- particular application. There also exists the set of world experimental data can minimise or even bypass magnetically actuated shape forming materials [21], [22], modelling assumptions. To train such networks, learning via [16] which have heavily influenced this work. These tend to demonstration [24], or input from a randomized feed employ a full continuum mechanics model via commercial (sometimes referred to as motor babble) [25] offer valid FEM packages as, generally, they are not considering closed methods. However, in the absence of high-volume, reliable loop control applications. The subset at the intersection of sensory data and the ability to rapidly prototype test samples these two groups (where our work resides) is the magnetically these real-world approaches are unfeasible. For our design actuated rectangular cross-section shape forming continuum methodology we therefore employ a FEM as the source of robots appearing in [17] and [23], These use a numerically training data for a fully connected ANN. intensive Fourier representation of manipulator shape to solve Utilizing FEM to simulate the interaction between their statics and dynamics. magnetic and mechanical forces has been successfully

The concept of a fully flexible, shape forming tentacle demonstrated in previous work [21] and been successfully robot fabricated from a magnetically active elastomer, orders applied to CMs [8], In addition, the use of ANNs as surrogates of magnitude softer than present day catheters, represents a of FEMs has also been reported; for example in [26] and [27] step change in the evolution of CM design. These tentacles convolutional neural networks are used to fully recreate FE can be pre-programmed to assume the profile of the conduit through which they are designed to pass. This relies on prior knowledge of the route in question - which may be derived from pre-operative imaging - and a methodology to translate this pathway into a magnetization profile. The contribution presented here offers a first step towards this goal coupled with a design methodology to overcome the inherent complications of magnetically active elastomers.

The two discrete functions of the tentacle can be defined as; quasi-static shape forming and dynamic shape forming. The first, quasi-static role, is to stiffen into a pre-defined shape upon arrival at a specific location such as the tumour at thebaseof the skull illustrated in Fig. 1. This stiffening would provide a safe and robust working channel for the delivery of treatment and the evacuation of tissue whilst also permitting the increased force required for cutting or ablation. The second, dynamic role, would incorporate shape forming Fig. 2. The workflow through the study. A simplified single segment FEM is during navigation to that same working location and would be built to experimentally verify the magnetic and material properties of the tentacle. These properties are applied in a multi -segment FEM and this model driven by an in-homogeneous and transient magnetic field. is replicated by a Neural Network to solve the inverse statics of the The work presented here considers the quasi- static case under Continuum Manipulator. This result is reconciled against the original FEM a planar, homogeneous and time invariant actuating field. and experimental prototypes.

TABLE I

The first contribution of this work is to present the fundamental concept, including the fabrication process, of our fully soft, shape forming tentacle robots. The second contribution is our learnt approach to the two-dimensional design of these tentacles, actuated in a time-invariant homogeneous field. We employ a Finite Element Model (FEM) as the source of training data for a fully connected Artificial Neural create a database for training validation and testing of the ANN; (4) the ANN is trained; (5) the ANN generalises for Fig. 3. (A) The undefomied position under a zero actuating field, the tentacle predefined, novel tentacle shapes; and (6) these shapes are is magnetized in the X direction. (B) The deformed position under an verified in the multi-segment FEM, fabricated, and actuating torque T generated by the field, S, applied orthogonally to the experimentally evaluated under a homogeneous magnetic magnetization m. The deformation gradient tensor Finfluences the effective direction of magnetization. An exploded infinitesimal volume shows the field. Cauchy stress tensor s in two dimensions.

III. MODELLING APPROACH stress components contributing to bending moment. From Euler-Bemoulli beam theory we can thus say that:

A. Constitutive Model m

For the purposes of this study, we assume the elastomer in consideration is homogeneous, isotropic and, for the range of strains experienced herein, incompressible. Assuming a where d is beam deflection, E is elastic modulus, L is length quasi-static state and entirely elastic deformation, under the and d is diameter. As in [29], given invariant geometric and Lagrangian description in [28], the deformation gradient material properties, (3 ) offers a linear correlation between d tensor F is the partial derivative of the deformed position with and B _t . To simulate the system, FEMs were constructed under respect to the relaxed position. the plane strain assumption in two dimensions in COMSOL

For the case of a magnetic material, torque when placed multiphysics suite v5.4 (COMSOL AB, Stock- holm, wholly within a homogeneous magnetic field, may be defined Sweden). The model employed solid mechanics and as the cross product of magnetic moment (m) and the electromagnetics modules connected via the Maxwell surface surrounding field of flux (B). This may be considered as the stress tensor utilizing Newton-Raphson iterative convergence mechanical work of the magnetic torque performed to align within the MUltifrontal Massively Parallel sparse direct the magnetic dipole moments [21], Defining magnetization Solver (MUMPS) option. as M = —we can say that the Cauchy stress tensor for Due to the highly non-linear nature of the Maxwell equa magnetic effects is: tions, the manual mesh optimization process results in local mesh concentrations around the edges of the magnetized segments (1) (visible in Fig. 4). These segment edges were constrained to 50 nodes per 7mm length and 25 nodes per and with reference to the quasi-static assumption we can also 1.5mm diameter. A maximum element growth rate of 1.2 was state that, for the full Cauchy stress tensor: applied throughout. The air domain in which the tentacle is (2) suspended is 250mm x 250mm with a zero gradient boundary condition applied around its periphery. This domain has a where f _g is the body force vector per unit volume; equal to maximum element growth rate of 1.1 and a maximum element gravity as magnetic body forces are included in the stress size of 20mm. For the single segment cantilever case, the tensor. These two equations can be solved for appropriate resultant model was comprised of 27, 000 free triangular (two- boundary conditions to give the deformation gradient tensor dimensional) nodes. and thus the deflection of our tentacle due to magnetic actuation. C. The Multi-Segment Model

The multi-segment ‘tentacle’ structure was represented

B. The Single Segment Cantilever Beam Model with three discrete segments of magnetically doped elas¬

Using the planar cantilever beam of uniform magnetizatomer, 7.0mm long by 1.5mm diameter, embedded withinthe tion, depicted in Fig. 3, as our example, it is possible to magnetically umeactive silicone, 42mm in length by 2mm in compare experimental data with an analytic model and our diameter. The magnetization direction of each segment can be FEM. This permits verification of our assumptions before we independently controlled within the two-dimensional plane. extend the FEM beyond the reach of any analytical solution. To generate a geometrically accurate simulation, the resulting In the simple shear stress model, the symmetrical mechanical FEM, as shown in Fig. 4, was discretized using 238,000 free stress components net to zero leaving only the magnetic triangular (two-dimensional) nodes subject to

poses which are quantified by deflections in x and y at three Points of Interest down the length of the tentacle. necessary to systematically remove duplicate results; those vectors residing in the red shaded region of Fig. 6, from each the same constraints as the single segment mesh (Section III- of the three actuating segments. The resulting lookup table B). reduces from 13,824 systematically produced entries to 2298

IV. THE ARTIFICIAL NEURAL NETWORK unique solutions each with 6 dimensions of input data (two-

The appeal of Machine Learning (ML) for our problem is dimensional magnetization vectors at three Pol) and a its ability to generalize for previously unseen scenarios corresponding 6 dimensions of output data (two-dimensional from sample data thus forecasting future outcomes in the deflection vectors at the same three Pol). These planar absence of a constitutive model. This renders ML an ideal tool deflection vectors are simplified to scalars, representing the for solving contrived inverse static problems such as those lateral x-displacement of the Pol on the actuated manipulator occurring in hyper-redundant, elastomeric CMs. to the unactuated centreline. Under the assumption of negligible tentacle extension (under 5mT actuation a

A. The Dataset maximum stretch of 0.1 % is ob served in the numerical model)

In order to train an ANN, we need large quantities of data. it can be assumed that y-deflection is purely a function of x- In real world applications, as described previously, this can deflection be produced via motor babble. To generate ‘virtual motor i.e. only one DoF exists per segment. This dataset is split 70% babble’, we parametrically swept each of the three for training, 15% for validation and 15% for testing. magnetization input variables. For each magnetized segment B. The Learning Network of the tentacle the modulus of the magnetization vector remains unchanged, the only variable is the direction of The architecture of the ANN employed to replicate the magnetization. As such the input to be swept can be FEM is a fully connected neural network with an output represented by three unit vectors representing magnetization regression layer. The effect of variations in number of neurons direction. and number of hidden layers was assessed based on runtime

An incremental rotation of the magnetization vector of and validation accuracy and the final arrangement emerged ^ _j -of a revolution (24 possible values for each of 3 input containing 6 hidden layers of 20 neurons each. After iterating variables) produces 24* = 13, 824 sequential entries. These for 60 epochs of Levenberg-Marquardt backpropa- gation of actuating variables were fed as inputs to the pre-assembled error employing sigmoid activation functions this FEMand runona 3.2GHz, 32GB, 16 core Intel Xeon Gold processor in a total run time of 38 hours. This process generated a set of deflection vectors at each of three Points of interest (Poi) corresponding to the magnetization vectors at those same Poi. To represent, as a point in space, the global position of a segment of non-zero volume, the centre of mass of each of the three magnetically active segments was chosen as the Pol. An example selection of resultant deformed tentacles is shown in Fig. 5.

As illustrated in Fig. 6, for each segment, the magnetization angle produces a deflection. This deflection reaches some maximum, beyond this point further increases in Fig. 6. Schematic of the positive and negative inflection angles. For a fixed applied field ‘B’, and fixed initial pose a maximum deflection is achievable magnetization angle, θ, will begin to reduce deflection. This by applying a magnetization ‘M’ at an angle 0max(or 9max)· The single phenomenon produces a non-unique relationship between segment tentacle is shown in grey with the unactuated pose in the centre of inputs and outputs - kinematic redundancy - something which the image and the maximum and minimum deflections also shown. For magnetization angles greater than 9max the resultant deflection will drop, is anathema to a Neural Network. It is therefore rendering results produced by magnetizations in the red shaded region repetitious. arrangement gave a 6.3% mean absolute percentage error obtained using a digital light microscope (DMS300, Leica, (MAPE) at the validation phase. Germany). Values of segment lengths (Mean+/-SD) were found to be 7.4+/-0.43mm, and segment diameters were

V. FABRICATION 1.9+/-0.03mm. A. The Single Segment This arrangement was housed in a magnetizing tray (Fig.

The single segment was fabricated from Ecoflex 00-30 7D) and exposed to the same 46.44 KGauss saturating field embedded with neodymium-iron-boron (NdFeB) which was employed to magnetize the single segment. The microparticles with an average diameter of 5 /μm (MQFP-B+, geometry of the magnetizing tray was driven by the solution Magnequench GmbH, Germany). Particles of NdFeB were to the inverse static problem for the soft robot. This solution added to the prepolymer in a 1 : 1 ratio by weight equating to a was generated by the Neural Network based on pre-defined volu- metric ratio of 0.88:0.12 (Ecoflex:NdFeB). The desired deflections. composite was mixed and degassed in a high vacuum mixer VI. EXPERIMENTAL EVALUATION (ARV- 310, THINKYMIXER, Japan) at 1400 rpm, 20.0 kPa for 90 seconds and then injected onto a straight cylindrical A. The Single Segment mold of diameter d= 1.5mm and length 20mm and left to The single segment was hung vertically downwards on its cure. The mold contained a centrally aligned 0.25mm embedded Nitinol needle in the centre of a Helmholtz coil diameter Nitinol needle running for 10mm of its length. This (DXHClO-200, Dexing Magnet Tech. Co., Ltd, Xiamen, needle remained embedded in the polymer and was used to China). As in Fig. 8, 10mm of this 20mm section was suspend and constrain the specimen during testing. Once the constrained by the Nitinol needle and 10mm was free to polymer had cured, the specimen was subjected to a uniform deform. The Helmholtz coil was arranged so as to produce a field of vertically aligned homogeneous magnetic field. The current

46.44 KGauss (4.644 T) (ASC IM-10-30, ASC Scientific, through the coils was ramped from -lOmT to lOmT in 2mT USA) orthogonal to the tentacle’s principle axis. increments to produce a piece-wise increasing actuating field

11. The Multi-Segment Tentacle orthogonal to the undeformed magnetization of the testpiece.

For the multi-segment arrangement, the unmagnetized Images of the specimen were taken on a Nikon D5500 DSLR doped elastomer (Fig. 7A) was divided into three identical with an AF-S NIKKOR 18 -55 mm lens at each field strength 7mm segments (Fig. 7B). Each segment was subsequently and were post-processed in GIMP 2.10 prior to analysis. The embedded, concentrically, at 14mm intervals (in the maximum deflection was measured at the centre of the longitudinal direction) into an undoped silicone host (Ecoflex distal end of the specimen in both the numerical and 00 experimental analyses. The experiment was repeated three

30) (Fig. 7C). A centrally aligned 0.25mm Nitinol needle times, the first iteration of which is shown in Fig. 8 and also supports the full length of the structure during fabrication. in the supporting video. Upon curing this needle is removed save for the final 10mm This analysis was performed to verify the mechanical and which remains embedded to act as the mechanical constraint magnetic properties of the doped elastomer. The Elastic modduring experimentation. The total length of the multiulus of the doped and undoped silicone was measured to be segment tentacle (Fig. 7D) is 52mm. From bottom to top this 91 kPa and 69 kPa respectively. These values were obtained can be broken down as 10mm of constrained length followed using a uniaxial load tester (MultiTest 2.5 -xt, Mecmesin, UK) by 42mm of unconstrained length. Undoped segments appear operating up to a maximum of 100% strain. The Pois- son’s white and doped segments appear black. The dimensional ratio was set to 0.5 for both elastomers representing the accuracy of the fabricated tentacle samples was assessed assumption of incompressibility. The remanent flux density through image analysis software (LAZ EZ, Leica, Germany), calibrated against a known reference length with images

Fig. 7. . Fabrication process of a multi-segment tentacle. (A) Injection molding of a continuous, magnetizable tentacle of diameter 1.5mm. (B) Once cured, the elastomer is cut into 7mm segments and positioned along the Nitinol needle at 14mm centres. (C) The needle is placed in a second mold of diameter 2mm and injected with plain silicone. (D) After curing, demolding and needle removal, the tentacle is placed in a 3D printed magnetizing tray. Here the three trays (i), (ii) and (iii) correspond to the scenarios A, B and C shown in Section VII Results.

of the doped elastomer was calculated to be 107 mT which the three candidate profiles the ANN replicated the FEM with reconciles with comparable works in [17] and [21], a MAPE of 4.4%. This reconciles with the validation error of 6.3% shown in Section IV-B.

B. The Multi-Segment Tentacle Table II shows absolute x-axis deflection and, in brackets,

As with the single segment arrangement the multi-segment x-axis deflection as a percentage of maximum. The absolute tentacle was hung in the vertically aligned homogeneous field deflection reveals discrepancies in magnitude between of the Flelmholtz coil. For the multi-segment model, a field experimental and numerical results. From the images in Fig. of 5mT was applied (10mT actuating fields appear in the 10, it can be observed that the experimental prototypes are supporting video). Images of the tentacles were taken on the deforming into a comparable shape to those requested (if not same camera as the single segment arrangement and again by a comparable magnitude). As such, proportional processed in GIMP 2.10. After superimposition of the deflections are included to give a comparison of shape only. undeformed and deformed images from the experimental This normalizing process has been included purely for setup, deflections were measured in the x-plane from centre representational reasons; to enable the visual shape to centre of each of the three magnetized segments. Due to the comparison available in Fig. 10 to be quantified. The rigid tubular formwork used during the fabrication process, existence of this adjustment is a recognition of two key unactuated tentacle deformations are minimal (MAPE limitations of this study. The first relates to the unmodelled 0.27mm or 0.6% of robot length, when subject to gravity). three-dimensional effects unavoidably embedded in the Furthermore, the zero line for deflection measurements is multi-segment experiment. To accurately rephcate the two- taken from the unactuated tentacle position, thus mitigating dimensional assumption made in the numerical model we the worst of any residual unactuated deformation. would need to magnetize our specimens in a perfectly planar and twist-free fashion. This is unachievable and the

VII. RESULTS consequence of any unwanted axial rotation introduced here

The analytical result derived in Section III using (3) is is a loss of magnetic moment in shown in Fig. 9 and can be seen to be comparable to TABLE II experimental and numerical results in the small strain region. A COMPARISON OF X-AXIS DEFLECTION FOR THE INPUT TO THE As deflection increases, it can be observed that the linear NEURAL NETWORK, THE OUTPUT OF THE FEM AND THE analytical model becomes increasingly inaccurate, quickly EXPERIMENTAL READINGS. DEFLECTIONS ARE ALSO SHOWN, IN reaching a point where it can no longer be said to represent BRACKETS, AS PERCENTAGE OF MAXIMUM. the behavior of the tentacle at all.

The single-segment FEM results, also shown in Fig. 9, reflect the experimental results with a MAPE of 14.9% up to the maximum field strengths of ±10mT. This result effectively verifies the material and magnetic properties of the elastomer in the numerical model and provides the requisite confidence in the FEM to extend the simulation up to the multi-segment tentacle.

As shown in Fig. 10, the multi-segment system was tested using three different x-axis desired deflections. The trained ANN translated these idealized outcomes into magnetization vectors based on its learnt weights. These magnetization vectors were input into the original FEM for validation. For

Fig. 10. Sample experimental results shown against numerical results for three predefined scenarios of the full tentacle. The magnetization vectors are the output of the trained Neural Network using desired deflections as input. The experimental result on the right is shown with B=0 and 5mT. The graphical result on the left shows both experimental and numerical outcomes. (A) Desired deflections (top to bottom) of dc = [2 6 5] (mm). (B) dc = [-1 -2 1] (mm). (C) dc = [1 0 -5] (mm). the plane being considered. The second limitation is the non- automated fabrication and magnetization process of the multiof possible outcomes. segment arrangement. As discussed earlier, the tentacle does Beyond the accuracy of the learner, any significant further not exactly achieve its intended dimensions, exhibits some errors can be attributed to the modelling assumptions entering unactuated deformation and, additionally, will not be the FEM and to limitations in the manual fabrication and magnetized in exactly the intended directions. magnetization process. These will be reduced with further ex¬

Both of these factors are significant areas for future work perimentation, fabrication process refinement, the inclusion discussed in Section VIII. Furthermore, despite the single of more sophisticated elasticity and magnetic field models in segment reconciliation, errors may persist in the measurement the FEM and inclusion of the third spatial dimension. In any and calculation of material and magnetic properties. future application of the proposed method, we would also aim Notwithstanding this, after normalization the shape profiles to integrate an appropriate sensing technique (e.g. [30], [31], give results with less than 10% MAPE in any one of the [32], [33]) to identify and balance the discrepancy between scenarios tested and an average across all scenarios of 5.2%. model and reality. This could inform a specific magnetic field More work will however be required to understand the controller which, in turn, would adjust the field to achieve the discrepancy observed in the magnitude of deflection. desired shape.

To give a broad basis for comparison of shape forming As well as the above improvements this work can now be accuracy (after the adjustment for magnitude has been made) extended for more convoluted, real-world trajectoriesderived this result can be represented as a mean error across all three from pre-operative imaging. This will be enabled by a greater scenarios along the length of the manipulator of 1% of total variety of input variables (including geometric) to the lookup robot length. This same metric compares to the rigid link table. In addition to this, stiffening under actuating fields is model error in [20] of 1.5%. Similarly, mean tip error across known to occur for magnetic elastomers [34], harnessing this all three scenarios is 1% of total robot length. This compares property to improve the capabilities of magnetic tentacles will to the Cosserat rod model error in [18] of 1.7%. form an interesting area of future research, ft should also be

VIII. CONCLUSIONS noted that the current technique for generating magnetic fields is limited to homogeneous field generation within a

Once trained, the ANN produces a reliable replica of the centimetre scale workspace and therefore unsuitable for FEM and is capable of producing forward and inverse static clinical application. For future feasibility, and also to address results in real-time with less than 5% MAPE. This system of the issue of dynamic shape forming for navigation, there are design, of course, is not limited to magnetically actuated CMs a variety of potential methods of field generation available and could well be generalized to other applications. Once that operate on a much larger scale [12], [35] and with more duplicate results have been removed from the training data DoF [36], [37], [10], (Section IV-A), the ANN can provide a useful surrogate of the Looking further ahead, with improved sensing technology, numerical model subject to one further caveat; the operational we may be able to eliminate the FE model altogether and workspace of the robot. Without external assistance the ANN, train the ANN from incoming sensory data thus fully unlike the FEM, has no indication of which deformations he eliminating modelling error and further harnessing the within or outside of the physical scope of the system. This enormous potential of Machine Learning. With this work we requires additional restrictions to the neural network, have begun to demonstrate the potential of our magnetic connecting desired deflections to the lookup table shape forming tentacles and their scope to, in future works, conform to specific anatomical constraints.