The First Interlaced Continuum Robot, Devised to Intrinsically Follow the Leader

Flexible probes that are safely deployed to hard-to-reach targets while avoiding critical structures are strategic in several high-impact application fields, including the biomedical sector and the sector of inspections at large. A critical problem for these tools is the best approach for deploying an entire tool body, not only its tip, on a sought trajectory. A probe that achieves this deployment is considered to follow the leader (or to achieve follow-the-leader deployment) because its body sections follow the track traced by its tip. Follow-the-leader deployment through cavities is complicated due to a lack of external supports. Currently, no definitive implementation for a probe that is intrinsically able to follow the leader, i.e., without relying on external supports, has been achieved. In this paper, we present a completely new device, namely the first interlaced continuum robot, devised to intrinsically follow the leader. We developed the interlaced configuration by pursuing a conceptual approach irrespective of application-specific constraints and assuming two flexible tools with controllable stiffness. We questioned the possibility of solving the previously mentioned deployment problem by harnessing probe symmetry during the design process. This study examines the entire development of the novel interlaced probe: model-based conceptual design, detailed design and prototyping, and preliminary experimental assessment. Our probe can build a track with a radius of curvature that is as small as twice the probe diameter, which enables it to outperform state-of-the-art tools that are aimed at follow-the-leader deployment. Despite the limitations that are inherently associated with its original character, this study provides a prototypical approach to the design of interlaced continuum systems and demonstrates the first interlaced continuum probe, which is intrinsically able to follow the leader.

We then introduced a mathematical model in order to obtain a design value for the probe span ℓ (indicated in Figure 1A). In more detail, we studied the statics of an idealized probe section to be deployed so as to build a circular track portion with curvature (we considered a quasi-static deployment, as most likely for common applications). For any fixed value of , the model aimed at determining the span l = l( ) that allows the probe section to build the sought track portion, within a chosen relative tolerance .
Such a tolerance is properly defined in the sequel, yet let us anticipate that we adopted = 5 %, so that all the fixed working parameters (namely , , , , and ) are now stated. Therefore, formally speaking, we computed l = l( ; , , , , , ), (1) for selected values of . Then, we targeted = (2 ) −1 as design curvature: we aimed at achieving a radius of curvature as low as twice the probe diameter during the deployment. This is very challenging; indeed, it is beyond the capabilities of relevant state-of-the-art systems [5][6][7]. Hence, we used the developed model for determining the design value of the probe span, as follows: ℓ ≅ l((2 ) −1 ; , , , , , ).
In the sequel we detail the procedure that we used for obtaining l as in Eq. 1, and we show exemplificative results including those providing ℓ according to Eq. 2.
Let us finally remark that the proposed model is not tied to a specific characteristic size of the probe: the working parameters can be varied in light of model objectives and constraints (the parameters set is a matter of design). Furthermore, let us anticipate that, as a byproduct of the model, we also obtained a reference value for the rod-clamping force to be achieved through the shape-lockers. This is briefly reported at the end of the present document.
[Modeling approach and working assumptions] We addressed the following basic problem: how to build a circular track portion with curvature by deploying an idealized probe section like the one sketched in Figure 2. The section at hand bends in a plane that contains two of the six deployment rods, for simplicity, and its shape is defined starting from the centerline (dashed in the figure), namely a circular arc with curvature and length ℓ. (With minor abuse of notation, we use the same symbol for the centerline length and for the design value of the probe span.) In particular, the deployment rods are defined through circular arcs that subtend the same angle = ℓ as the centerline. Moreover, the proximal/distal disks (introduced to lock the LCR shape) have negligible thickness and they are perpendicular to the centerline. Furthermore, we neglect the connecting wires because their contribution to the probe bending stiffness is negligible compared to the one of the rods (by construction), and therefore they are not relevant for the subsequent equilibrium problem. Finally, based on the relative position of the continuum robots (CRs) with respect to the bending plane, the shortest arc either belongs to the leader (LCR), as in Figure 2A, or to the follower (FCR), as in Figure 2B. The former configuration is labelled "FP" and the latter "UP", for ease of presentation. while denotes the track curvature. The shortest circular arc either belongs (A) to the leader or (B) to the follower. Track building was addressed by a two-step simplified approach: we firstly determined the load-free pose of the leader (locked, green in the figure) that approximates the circular shape. We then applied the torque associated with the bended rods of the follower (unlocked, blue in the figure), and we studied the resulting deviations (linear and angular ) from the circular shape. For a chosen we obtained the maximum ℓ that permits to approximate the circular shape within a chosen accuracy. Span determination was based on this approach, which holds when also reversing the roles of the CRs, by symmetry. (The B-subfigure is included in Figure 1D of the main text, for ease of presentation.) We tackled the track-building problem by a simplified approach, namely by studying two (static) equilibrium problems associated with the following deployment steps: The unlocked LCR is firstly deployed so as to finally approach the pursued circular shape; then, its shape is locked. Upon successful completion of this step, the equilibrium configuration of the LCR suitably approximates the sought circular track.] We firstly determined the load-free equilibrium pose of the LCR, by minimizing the distance of the distal disk center from the endpoint of the pursued circular centerline, and the angular deviation from the angle that defines the pursued pose of the distal disk. We determined the LCR rod lengths as part of the solution (as for an inversekinematics problem);  [Step#2: The unlocked FCR is then deployed over the locked LCR. Upon successful completion of this step (i.e. provided that the LCR section is stiff enough), the resulting probe section still suitably approximates the pursed circular track.] We then studied the equilibrium pose of the LCR (with the rod lengths determined at Step#1) by accounting for the contact with the FCR. Also in this case we minimized and , and we regarded the resulting pose as a suitable approximation of the pursued track when where denotes the aforementioned non-dimensional tolerance.
In more detail, at Step#2 we assumed that each of the FCR rods only acted on the LCR through a bending torque T = T(κ) depending on the assumed curvature. This is consistent with the constant-curvature assumption of the rods (also for large displacements/deformations [8]), as far as the resulting equilibrium pose of the probe section is close enough to the pursued circular shape. In other words:


When the condition in Eq. 3 holds: the locked (and loaded) LCR section is reasonably close to the pursued circular shape; the FCR rods (as guided by the LCR) are correspondingly close to the associated circular arcs of the circular shape; the mechanical actions exchanged by LCR and FCR can be reasonably described through pure bending torques, as discussed above. In this case, therefore, the model solution is physically-representative (for the statics at hand), and the considered section manages to build the track;  When the condition in Eq. 3 is violated, the considered section cannot be deployed so as to build the pursued track. In this case, the model solution at Step#2 does not accurately describe the actual statics, in particular because the FCR rods substantially differ from the assumed circular arcs, and therefore the mechanical actions exchanged by LCR and FCR are no longer described by pure bending torques. Indeed, the actual pose of the probe section must be determined by solving the coupled problem (LCR and FCR): the two-step approach is no longer suitable. However, we are not interested in static solutions that violate Eq. 3: we are only interested in determining the upper value of ℓ for which Eq. 3 holds. Hence, the proposed two-step approach is commensurate with the model objectives, and can be effectively used for determining the probe span; Indeed, the main idea behind the proposed two-step approach is that deviations from the circular shape can be contained, provided that ℓ is low-enough. In fact, for any fixed values of the torque applied by the unlocked CR only depends on (as far as the rods are circular), yet the stiffness of the locked CR decreases when increasing ℓ.
In general, one might get close to the circular shape at the end of Step#1, yet by approaching unstable configurations of the rods. In that condition, however, the subsequent loading at Step#2 most likely causes the loaded LCR to substantially depart from the sought track. Hence, the proposed modeling approach also detects unstable probe configurations, as exemplified by the results below.
Furthermore, let us remark that, once determined the probe span as discussed above, the conditions for proper track-building are expected to be met at the subsequent deployment step as well, thanks to the symmetry between LCR and FCR in our interlaced concept. Indeed, when building the subsequent track portion, the (unlocked) LCR can properly catch up with the (locked) FCR over the previously-built track portion, by symmetry. Of course, we are simplifying the scenario; for instance, we are not considering the stability of the cumulatively-built track, which can be perturbed during the deployment. However, the way we address track-building by considering a single probe section should be noticed: it is intrinsically based on the symmetry underpinning the interlaced concept, according to which LCR and FCR essentially play identical roles to implement a truly alternating method [9].
Finally, we supposed the deployment rods to be elastic (we assumed the same elastic and cross-sectional properties for all the rods), and we described them as simple Cosserat rods to allow for large displacements/deformations [8]. Moreover, we exploited the momentum balance of the distal disk to couple the mechanical actions of the rods (and thus the CRs at Step#2). Furthermore, we used the pose of the disks (both the distal and the proximal one) to introduce proper boundary conditions at the rods endpoints (ideal clamping assumption, like in [10]). In addition, we neglected gravity, to simplify the derivation and because it is expected to play a minor role for many probe poses (simple estimates support this point, see e.g. [11]). Let us also remark that we deliberately introduced a small set of model parameters in order to contain the uncertainties, and to obtain results effectively usable for probe development.
[Model formulation and solution strategy] The shape of the i-th rod ( = 1,2,3; subscripts are hereafter understood) can be obtained by integrating the following ordinary differential equations [8]: to be complemented with proper boundary conditions. Symbols ∈ ℝ 3 and ∈ (3) respectively denote the rod position and material orientation, which provide a body (i.e. material-attached) frame for each value of the arc length ∈ [0, ℓ ] (ℓ is the rod length). Furthermore, ∈ ℝ 3 and ∈ ℝ 3 indicate the internal force and torque, respectively. Moreover, ∈ ℝ 3 and ∈ ℝ 3 denote kinematic variables, expressed in the body frame, respectively associated with the linear and the angular rate of change of the body frame itself. In particular, is related to extension/shear, while is related to bending/torsion. In addition, the prime operator represents differentiation with respect to the arc length, while (•) ∧ (hat operator) denotes the classical mapping from ℝ 3 to (3), which is the Lie algebra of (3). Let us observe that we expressly neglect body forces and torques in the rod momentum balances (Eq. 6-7), consistently with the considered loading conditions. Furthermore, from Eq. 6 we immediately obtain that is constant along the i-th rod, so that we can omit the formal dependence on the corresponding arc length (besides neglecting Eq. 6 itself).
Then, for an elastic rod the kinematic variables are linked to the internal force and torque by the following constitutive relations (that provide the closure for Eq. 4-7): where the starred quantities refer to the stress-free reference configuration of the rods. Moreover, = ( , , ) and Let us now consider the LCR distal disk; its momentum balances read: where denotes the torque due to contact with the FCR rods. Based on the two-step strategy introduced above, is null at Step#1, while it has a known expression at Step#2 (reported below). It should be noticed that Eq. 10-11 provide a set of boundary conditions for the rods, which couple the internal forces and torques at the endpoints (the reference point for the angular momentum being immaterial).
Let us then consider the geometric boundary conditions for the rods. To the purpose, let denote the relative position of the i-th rod's endpoint with respect to the center of a disk (either proximal or distal, being the same), as represented in the disk-attached frame. Moreover, let indicate the rotation of the distal disk, and the position of its center; furthermore, let and represent the corresponding quantities for the proximal disk. Then, the following compatibility conditions straightforwardly stem from the assumed perfect clamping: ( ( (0) )) ∨ = 0.
In particular, Eq. 13 and Eq. 15 state that the rod orientation at the corresponding endpoints is the same as the one of the disks (the matrix logarithm and the (•) ∨ operator, which is the inverse of (•) ∧ , permit to use a compact notation, as e.g. in [10]).
We are now in a position to simplify the formulation, by exploiting the assumed planar deformation of the idealized probe section. To the purpose, let us assume that each rod deforms on a plane parallel to the plane. Moreover, two of the three rods, say the ones with subscript 2 and 3, undergo the same deformation, thanks to the symmetry with respect to the = 0 bending plane. Hence:


We only solved Eq. 4 for the components and , with = 1,2 (four equations). Moreover, we recast Eq. 5 in terms of a rotation angle around the -axis (two equations). Furthermore, given the internal force components 1 and 1 in rod#1, Eq. 10 immediately provides 2 = 3 = − 1 /2 and 2 = 3 = − 1 /2. Hence, we treated 1 and 1 as two unknown constants, tentatively assigned to integrate the differential problem yet to be determined as part of the solution process. In addition, Eq. 7 only provides a non-null component along the -axis, so that we considered (two equations); similarly, also appeared through the -component only. Moreover, we consistently considered the relevant components of the closure relations, i.e. Eq. 8-9;  As regards the proximal boundary conditions, we directly imposed the ones regarding , and (six values), as provided by Eq. 12-13. Moreover, we treated the proximal values of , say 1 0 and 2 0 as two additional unknown constants, tentatively assigned to integrate the differential problem yet to be determined as part of the solution process;  At the distal end, based on Eq. 11 and Eq. 14-15, we defined the following residuals (let us omit the arc length specification for conciseness): to be minimized as part of the solution process (see below). The symbols [•] and [•] in Eq. 17-18 denote the projection on the corresponding coordinate axis, while , and clearly represent the position and the orientation of the distal disk. For convenience, however, we defined the latter quantities with reference to the pursued circular shape (see Figure 2). In particular, once introduced the coordinates ( , ) of the distal endpoint of the centerline, and by recalling the angle , we introduced the following deviations: to be determined through the minimization process (see below). The expressions of and in terms of ℓ and can be easily derived (see e.g. [12]); they are omitted for brevity.
In light of the above formulation, let us finally detail the solution strategy: We determined the LCR equilibrium pose by minimizing the seven residuals defined in Eq. [16][17][18][19]; in particular, we chose their sum as cost function. To the purpose, we selected the following seven optimization variables: 1 , 1 , 1 0 , 2 0 , ℓ 1 , ℓ 2 and ; We performed the same minimization as for Step#1, yet with the following optimization variables: 1 , 1 , 1 0 , 2 0 , , and (in particular, we adopted the rod lengths obtained at the previous step).
In more detail:  We did not allow for linear deviations of distal disk position at Step#1 in order to approach the sought circular shape as closely as possible (i.e. consistently with the fixed model parameters). Conversely, at Step#2 we expressly introduced and , not to introduce unphysical constraints on the solution.
Indeed, from Step#2 we obtained the values of and ≡ ( 2 + 2 ) 1/2 necessary for assessing track-building capabilities according to Eq. 3;


At Step#2 we assumed to be given by the following expression [8]: where is the length of the j-th FCR rod ( = 1,2,3) in the assumed circular shape (see Figure 2), so that −1 represents its curvature. For brevity, we omit the expression of (it can be easily obtained, see e.g. [12]), as well as the simplification of Eq. 23 enabled by symmetry;  Once recast the problem in non-dimensional form, we numerically integrated the Cosserat differential equations in order to evaluate the cost function. To the purpose, we exploited a classical Dormand-Prince scheme available in Matlab (The Mathworks, Natick, MA, USA), with a tight relative tolerance (10 −8 ). Cost minimization was performed through Matlab as well, in particular by exploiting a quasi-Newton method (we assumed the optimization landscape to be locally smooth);  For any fixed values of we solved a sequence of equilibrium problems, by increasing the centerline length. This way, we could exploit a previously obtained solution to guess the subsequent one (both at Step#1 and Step#2), with clear benefits on convergence and computational cost. This strategy also benefited from the fact that we formulated some of the residuals in terms of variations: as far as the probe achieves track-building, the deviations from the circular shape are small and the algorithm can easily converge by initially assuming null deviations. In addition, we could easily detect unstable rod configurations: stable configuration were incrementally explored up to e.g. a bifurcation (after which the solution was no longer of interest, as discussed above);


In summary, once fixed , we obtained and for increasing values of the centerline length, and we determined the limit value l( ) based on the condition in Eq. 3. Finally, we obtained the design value for the probe span by specifically considering = (2 ) −1 , i.e. by picking l((2 ) −1 ).

[Model results]
Exemplificative results obtained through the proposed modeling strategy are reported in Figure 3 and Figure 4. In particular, in Figure 3 the FP configuration permits to approximate the pursued circular shape, while the UP configuration exhibits rod instabilities and substantially departs from the sought circular track. This is consistent with physical intuition: it is more challenging to achieve a chosen curvature by only pushing with one rod (instead of two). Indeed, we introduced the labels FP and UP to remind of favorable and unfavorable pushing, respectively, for the LCR rods. Yet the careful reader will spot that the strong symmetry underlying our interlaced probe concept should not introduce any bias that favors one rod configuration with respect to the other. And this is indeed the case, since at each deployment move the FP configuration of the LCR corresponds to the UP configuration of the FCR, and vice-versa. Therefore, from Figure 3 we obtain a unique information: we conclude that the span ℓ = 2.5 is too long for accurately building a track with radius of curvature equal to 3 . (and by considering the UP rod configuration). In more detail, Figure 4 shows that accurate track-building can be achieved by choosing ℓ = , while ℓ = 2 leads to instability. Moreover, Figure 5 shows the trend of the computed deviations, namely / and , versus ℓ/ , for selected values of the curvature. In particular, we considered = 1/5, in light of the deployment capabilities stated in [5], as well as the target value = 1/2, to outperform the state of the art. For completeness, we report the trends for both the UP and the FP configuration, the former systematically being the limiting one, as expected. Moreover, we cut the y-axis in the figure in correspondence of the chosen tolerance, namely = 5 %, to enhance readability. Based on the results in Figure 5, by choosing a probe span ℓ/ ≅ 1.5 we should be able to achieve the sought performance, i.e. building a track with a radius of curvature as low as twice the probe diameter. This is the main result of the proposed model: we then addressed the detailed design of the probe by adopting ℓ/ = 1.5. Based on these results, ℓ = 1.5 should permit to achieve the pursued track-building performance, within the chosen tolerance = 5 %. (The trends associated with the UP configuration are reported in Figure 1D of the main text, for ease of presentation.) Let us observe that in Figure 5 we also show the trends associated with = 1, just for reference.
Indeed, this very high curvature might be achieved e.g. by miniaturized flexible tool tips [13], whereas it is extremely challenging for deployable probes. We did not address this very high curvature at the early developmental stage, because practical implementation constraints (primarily those associated with the nonnegligible thickness of the shape-lockers) were likely to hamper the effective achievement of the sought performance. Let us also mention that the computational cost of the curves in Figure 5 was very contained: for any fixed and ℓ, it took a few seconds to order of a minute to compute the deviations (elapsed time, on a common desktop PC). Furthermore, those trends in Figure 5 that are associated with the UP configuration are also reported in the main text ( Figure 1D), for ease of presentation.
Let us finally remark that, as a byproduct of the developed model, we also obtained a reference value for the rod-clamping force to be achieved through the shape-lockers. In more detail, we firstly computed the maximum tangential force (absolute value) occurring at the endpoints of the LCR rods. For the chosen conditions ( = 1/2, ℓ/ = 1.5) we obtained ≅ 5 N. We then introduced the following inequalities (the approximation symbol reminds that we are dealing with simple estimates): where denotes the clamping force (absolute value) on a deployment rod, and indicates the static friction coefficient between the rod and the clamping interface on the shape-locker. In more detail, the condition in Eq. 24 applies to the rod specifically subjected to the extremal value , and it is clearly more stringent than the condition provided by Eq. 25. Differently, the latter condition can be introduced by considering the three rods at once, namely by adding their static friction inequalities (and by assuming that the tangential forces on the two rods that are less loaded sum to , consistently with the equilibrium conditions for the distal disk introduced above). By considering, for instance, a clamping interface made of aluminum, so that ≈ 0.45 [14][15][16], one obtains ≳ 11.1 N from Eq. 24 and ≳ 7.4 N from Eq. 25.
Consequently, one should address the detailed design of the shape-lockers by targeting a clamping force around 10 N. We actually considered this reference value since we chose aluminum for the clamping interface, as detailed in the main text.