A TDCR is usually composed of a flexible backbone along which tendons are guided using spacer disks, these last ones being fixed to the backbone. It is actuated by changing the tendons’ length and applying tendon tensions, which cause a reversible deformation of the backbone. This actuation strategy is referred to as extrinsic, since the actuators used to deform the backbone are not embedded within the manipulator’s body. Extrinsic actuation allows for small diameter to length ratios of the robot and reduces the reflective inertia of the manipulator. TDCR are usually designed with several actuatable segments, by routing additional tendons along the backbone. These tendons are attached at specific intermediate disks fixed on the backbone, delimiting the so-called segments, and routed to the robot base where the actuators are located. The design of most of them uses two or three segments. As a result, they are good candidates for a small-scale robot with high motion capabilities, maneuverability, and interesting FTL deployment properties due to their high number of actuated dof. However, a high number of segments comes at the expense of complex tendon routing, high friction between the tendons, and a bulky actuation unit. Improving the robot motion capabilities can be performed by enhancing the capability of FTL deployment and the workspace.

Regarding FTL deployment, TDCRs have been considered to follow paths composed of sections with planar constant curvature (Palmer et al., 2014; Kang et al., 2016; Neumann and Burgner-Kahrs, 2016; Amanov et al., 2019; Gao et al., 2019). A path can be followed by deploying the TDCR along the desired path and modifying the tendon tensions, which subsequently change the curvature of the segments. To improve accuracy during deployment, a TDCR with extensible segments, which is depicted in top left image in Figure 1, is proposed by Nguyen and Burgner-Kahrs (2015) and further developed by Amanov et al. (2019).

However, due to the fact that existing TDCRs rely on straight tendon routing, only paths composed of sections with planar and constant curvature can be followed. Therefore, to facilitate general path-following, a TDCR design would require more dofs in task space (Amanov et al., 2019). As a consequence, following spatial tortuous paths requires at least two segments and for a more complex spatial path significantly more segments are required. Aside from the challenges caused by using high number of segments, e.g. complex tendon routing and increased friction, FTL deployment accuracy of standard TDCR decreases at the transition between two sections of the path as shown by Amanov et al. (2019).

As an interesting side-effect, the use of extensible segments enhances the workspace and position reduncancy of a TDCR. Existing works in the literature mainly focused on investigating the workspace improvement, i.e. which end-effector positions can be reached by the robot end-effector using the actuated dofs (Blessing and Walker, 2004; Walker et al., 2006; Amanov et al., 2019). A workspace volume increase of 22.5% compared to a TDCR with constant segment length was obtained by Amanov et al. (2019). The positive impact of the segments extensibility on the workspace is also clearly demonstrated by Li et al. (2017). Position redundancy, i.e. the capability of changing the robot configuration while staying at a desired end-effector position, was less considered but is also an important performance metric. It indicates the range of achievable robot tip orientations for the considered tip position, which is critical for manipulation (Simaan et al., 2009) and inspection (Goldman et al., 2013) tasks. The redundancy can also be leveraged for respecting anatomical constraints with the robot body (Gao et al., 2019)), or for minimizing the potential energy stored in the backbone (Moll and Kavraki, 2004). It was demonstrated by Wu et al. (2017) that the position redundancy is strongly impacted by the number of actuated dofs. Considering extensible segments improves also the position redundancy distribution at the workspace center Amanov et al. (2019).

The workspace and steerability of TDCR can also be enhanced by using non-straight tendon routing. In particular, helical tendon routing, which is depicted in the bottom left image in Figure 1, has been used to obtain TDCR configurations, which are not possible with straight tendon routing. As a result, a larger variety of tip orientations can be achieved in the workspace (Gerboni et al., 2015) or different position can be reached, while keeping the same orientation (Rucker and Webster, 2011). Further, using an additional helical tendon to existing straight tendon routing can lead to a fourfold increase of workspace volume and modification of workspace shape as demonstrated by Starke et al. (2017). However, the prototypes presented in Gerboni et al. (2015) and Starke et al. (2017) combine fixed helical and straight tendon routing and may lead to complex tendon routing, especially if several segments are stacked. A similar approach is the configurable tendon routing in (Barreiros et al., 2019), where non-straight tendon routing can be realized. However, the chosen fixed tendon routing is a design parameter and cannot be controlled during the operation of the TDCR.

To conclude, the current design paradigm, i.e. stacking segments with fixed tendon routing, has reached its limit for improving motion capabilities while achieving small scale manipulators. Although different non-straight tendon routings are considered to achieve larger workspaces and, therefore, higher motion capabilities, current tendon routings are fixed and only used as design parameters. We propose to exploit the tendon routing as an additional dof in order to induce torsion in the backbone, thus exploiting all significant deformation modes of a TDCR segment. It is our hypothesis that taking advantage of the fourth dof has a significant impact on the motion capabilities and, therefore, on the FTL deployment as well as position and orientation capabilities of a TDCR.

In this paper, we propose a design of TDCR segment which leverages advantages of both extensibility and variable helical routing, cf. Figure 1. We present our progress in the development of a TDCR with four dof per segment. We introduce the design and evaluate its merits by several assessments in simulation and with real prototypes. In particular, the contributions of this paper include:

• A mechanism able to continuously change the tendon routing from straight to helical routing and capable of FTL deployment along a spatial tortuous path using a simple deployment strategy.

The paper is structured as follows. We first introduce the conceptual design of TDCR in Section 2. We then propose a static model of the robot in Section 3. The FTL deployment capabilities are studied in Section 4, where the deployment strategy and the area-based error are described. The position and orientation capabilities are studied in Section 5. The results are discussed in Section 6 before concluding.

We introduce a categorization of spacer disks into four types. The categories are based on the most general spacer disk, referred to as FSD (floating spacer disk), which can translate along and rotate about the TDCR’s backbone.

Type-0: A spacer disk which can neither rotate nor translate along the backbone.

The first type of spacer disk, i.e., type-0, is fixed to the backbone and is a widely used spacer disk for TDCRs (Rucker and Webster, 2011; Mishra et al., 2017; Starke et al., 2017; Li et al., 2020). End disks and disks attached to the base fall into the category of type-0 FSD. Type I FSD can be found in the literature (Blessing and Walker, 2004; Kang et al., 2016; Amanov et al., 2017; Lastinger et al., 2019; Visentin et al., 2019), where the rotational passive dof is constrained due to its design. The position of each disk can be depended on the repulsion forces between them in order to distribute the disks along the backbone. Adding an additional rotational passive dof leads to an FSD with two dof, i.e., type-III FSD. An example is given by Nguyen and Burgner-Kahrs (2015). However, the design by Nguyen and Burgner-Kahrs (2015) does not utilize the rotational passive dof. Another FSD with rotational passive dof is given by type-II FSD. Since type-II FSD and type-III FSD are coupled with tendons, the resulting orientation of each of them is influenced by the adjacent FSD of any type. To conclude, type-II FSD have not been realized thus far and type-III FSD have not been fully exploited.

Here, we present the general idea of our proposition of TDCR segment with four actuated dof, i.e. bending in two directions, extension and twist, obtained using extrinsic actuation. A schematic design is depicted in Figure 2I. The proposed segment design is partially based on the extensible segment TDCR design described in our previous works by Amanov et al. (2019). It uses flexible, slender and non compressible tubes as the backbone, which shear deformations are typically negligible. Our segment design, called FAS (Fully Actuated Segment), allows then to actuate all significant deformation modes of the backbone. Furthermore, connections to concentric tube continuum robots (CTCR) are made. We kindly refer to Gilbert et al. (2016) and Mahoney et al. (2016) as well as to Grassmann et al. (2020) for more details on CTCR. For visual aid, Figure 2A to Figure 2H illustrate the effect of each dof and their combinations on a continuous slender structure.

To highlight the capabilities of the proposed design, we study its merit through two different prototypes throughout this paper. The first prototype combines type-III FSDs with 4dof to obtain a FAS prototype. The second prototype with fixed segment length has several type-II FSDs. Both prototypes are depicted in Figure 3.

The center of each disk i is denoted by O _i . A reference frame is attached to the base of each subsegment i at O _i−1 consisting of the local x _i−1, y _i−1, z _i−1 axes, shown in Figure 4A. The bending curvature components along the x-axis and y-axis are represented by κ _x,i and κ _y,i , while θ _i represents the geometric twist angle about the local z-axis, independent of the backbone torsion. The backbone is rotated by α in the actuation unit. This rotation results in each intermediate disks rotating about their local z-axis due to the tendon interactions. The rotation of each disk θ _i around the backbone is a passive dof and relative to the backbone rotation due to torsion. We assume that this rotation can be expressed as a function of the index of the disk and written as θ i = α f i . (1)

Since the base is fixed, f (0) = 0 and f(n) = 1 are assumed. Then the relative angle of rotation between the disks is given by θ i − θ i − 1 . Therefore, each subsegment experiences bending and twisting, represented by the configuration parameters X μ , i = [ κ x , i , κ y , i , θ ̂ i , l i ] , where θ ̂ i = θ i − θ i − 1 + ϵ i . The angle of twist ϵ _i is due to tendon tension imposed by the bending, i.e. κ _x,i and κ _y,i , whereas both geometric twist angle, i.e. θ _i and θ _i−1, are due to α. Therefore, θ ̂ i considers both contribution from the kinematics and statics.

Using ϕ _i = arctan2 (κ _y,i , κ _x,i ) and κ i = κ x , i 2 + κ y , i 2 for the sake of simplicity, the position of the end of a subsegment i written w.r.t. to the reference frame attached to disk i − 1 is given by p i i − 1 = cos ϕ i κ i 1 − cos κ i l i sin ϕ i κ i 1 − cos κ i l i 1 κ i sin κ i l i . (2)

The corresponding transformation matrix from the frame at disk i to disk i − 1 is given by T i i − 1 = R z ϕ i R y κ i l i R z − ϕ i + θ ̂ i P i i − 1 0 ⊤ 1 , (3) where R _y and R _z are two of the basic rotation matrices with index indicating the axis of rotation.

The transformation between any two reference frames is given by the product of transformation matrices between pairs of consecutive disks. The tendons are numbered in an anti-clockwise manner and are arranged uniformly at a distance of r _d with an angle of 2π/m between them, as shown in Figure 4B. The location of tendon k at disk i for a robot with m tendons is denoted by P _i,k and its coordinates w.r.t. the frame of reference at disk i − 1 is denoted by the v e c t o r i − 1 O i − 1 P i , k ⃗ . This vector is given by O i − 1 P i , k ⃗ i − 1 = T i i − 1 r d ⁡ cos 2 π m k − 1 r d ⁡ sin 2 π m k − 1 0 1 ⊤ . (4)

While the backbone representation maps the curvatures to the resulting pose in the task space, static modeling is required to obtain the curvatures for the corresponding input tensions. We do so by calculating the net moment ⁱ⁻¹ M _i acting on each subsegment i caused by the forces acting on a subsegment. These forces include gravity acting in the negative z-axis due to the weight of the disks, frictional forces written using the Coulumb friction model and the forces applied on the tendons, as described in Yuan et al. (2019). We then use the Hooke’s law to relate the net moment to the resulting curvature and torsion of each subsegment using the following equation (Rao et al., 2021), M i i − 1 = R z ϕ i R y κ i l i E I x x 0 0 0 E I y y 0 0 0 G J / l i 0 κ i ϵ i , (5) where E and G are the Young’s and shear modulus, I _xx and I _yy are the second area moments and J is the polar second moment of cross sectional area.

The equilibrium Eq. 5 constitute an implicit static model of FAS as a system of 3n non-linear equations of the form G X μ , p E , u = 0 , (6) where X _μ is the state space vector, such that X μ = X μ , 1 ⋯ X μ , n , p _E is the end-disk position and u is the vector of actuation inputs. This vector contains normally all the tendon tensions τ _i applied at the robot base. However, we can reduce in our case the dimension of the actuation space spanned by these tensions by looking at the FAS properties. Since the backbone is assumed to be inextensible and the torsion deformations due to tendon actuation can be reasonably neglected, as demonstrated later on, the m tendons can only control two bending degrees of freedom independently. As a result, the tendon actuation space can be represented by two variables. We chose here to express the tendon tensions with polar coordinates. We consider that pulling on the m tendons is equivalent to pulling on one single tendon, located at a distance r _d from the backbone, an angle γ around the backbone tangent from tendon 1, with a force T: τ i = T ⁡ cos 2 π m i − 1 − γ + T t m (7)

The pretension term T _t is a predefined value added to prevent slack in the tendons and to ensure that constraint ∑ i = 1 m τ i = T t holds. As a result, the tendon actuation can be parametrized using γ and T only. Considering now the actuated backbone rotation and translation, the actuation space of the FAS is four dimensional and spanned by u = T γ α β ⊤ .

The implicit static model (6) is used to solve numerically the forward and inverse static of the FAS. The forward statics are obtained by fixing u and solving the equations for X _μ , which allows in turn to obtain p _E . The inverse statics are obtained by fixing p _E , as well as three actuation inputs, and solve (6) for X _μ and the remaining actuation input. Both forward and inverse static models are solved using a Newton-Raphson algorithm on Matlab.

In this section, we verify our model. First, the influence of the backbone rotation on the disk rotation is determined. Afterwards, robot shapes are measured for different configurations. Some of them are used to calibrate the model. Finally, the model is evaluated.

Achieving FTL behaviour requires the continuum robot to keep a constant curvature at any arc-length along the path during the deployment (Garriga-Casanovas and Rodriguez y Baena, 2018a). This has been achieved so far for multi-segment TDCR by using specific optimization routines. The displacement to be applied on each tendon is calculated at each step of the deployment in order to minimize the error between the position of the robot backbone and the path to be followed (Palmer et al., 2014). As a result, approximate FTL behaviour is obtained, which accuracy depends on the accuracy of the model used to predict the robot shape and the range of extensibility achievable by each segment (Amanov et al., 2019). Although this optimization based approach can be generally applied to any continuum robot, it is complex and can be computationally heavy. We propose here a simpler deployment strategy for TDCR and, in particular, for the FAS, that takes advantage specifically of tendon actuation.

When a single segment TDCR composed of straight-routed and fully constrained tendons is considered, and the effects of gravity and friction are assumed to be negligible, it has been shown that the tension applied by the tendon can be represented by a constant moment applied (Camarillo et al., 2008). The constant moment results in constant curvature of the backbone. As a result, by applying a constant tendon force and by progressively increasing the length of this segment, the deployed robot bends in a plane with a constant curvature. It has the same curvature at any arc-length along the path. Consequently, a one segment TDCR for which the tendon force is controlled can be used to perform perfect FTL deployments along planar paths. Similarly for the FAS, we suspect that applying constant tendon tensions during the deployment will lead to a segment with constant bending and torsional curvature. Deploying the backbone, i.e. changing β, while keeping the backbone rotation, i.e. α constant, should thus lead to FTL deployments along spatial helical paths.

To assess the FTL deployment, we propose a measure which takes the covered projected area of the volume of the robot’s shape along a path during a deployment sequence into account. First, we define the overall covered area A given by A = ⋃ i = 1 n A i , (9) where A i and n are the area covered by the projected robot’s shape at the ith step and the number of steps to reach the end of the path, respectively. Hence, A is the projected area of the volume covered during the FTL deployment. Note that A i and A are defined as sets of points which constitute the respective area.

Second, we can define the area-based FTL error in terms of sets as L FTL = A n \ A ⋃ A \ A n A n , (10) where ⋅ is an operator acting on the set, which gives the area a numerical value depending on the sensors used to sense the shape. The numerator is the symmetric difference, also known as the disjunctive union, which represent the difference between the ideal area A n and actual covered area A . The denominator normalizes the difference w.r.t. A n . Note that (10) represents a simplification of a volumetric error measure, which considers spatial points and a continuous time during deployment. Further note that it is assumed that the robot is extending. To consider the contraction of the robot, A n is replaced by A 1 , whereas A reminds the same as changing the order of the operands in (9) does not change the result.

Finally, we define the area-based FTL error L FTL quantified by L FTL = A A n − 1 , (11) which is equivalent to (10) since the nominator of (10) in combination with the overall area (9) leads to A n \ A ⋃ A \ A n = A − A n . However, (11) gives a more practical implementation.

In this paper, we use a camera to sense the shape and images as sensor outputs. To give a numerical value to the area, we construct a binary image, where all pixels occupied by the robot are white and the rest are black. The operator ⋅ adds up then the white pixels in the binary image.

Using the FAS prototype described in Section 2.3.1, we load one tendon with an additional 5 N to the pretension. We indicate the termination point of the tendon as well as the end disk orientation using a red flag. Figures 7A,B shows a sequence of an in-plane deployment and an out-of-plane deployment, where α = 0 rad and α = −π rad, respectively. As can be seen from Figure 7, the orientation around the tangent vector of the backbone is constant during deployment, whereas the orientation changes continuously along the spatial path. The change in orientation indicates torsion along the backbone during the deployment.

For the evaluation of the FTL deployment, we utilize (11) and consider nine steps of each sequence, see Figure 7D. In order to consider all continuous intermediate steps between two discrete steps, we manually segment the prototype with a conceived sleeve including all disks, tendons, and the backbone. An example of a conceived sleeve is represented with a blue hull in Figure 7D. The sleeves of each sequence are overlaid and binary images are generated. The binary image indicate the occupied area representing the occupied volume during the deployment. Figure 7C illustrates the area indicated by white pixel in the binary image. We achieve an error of L FTL = 0.1537 and L FTL = 0.1399 for the in-plane and out-of-plane deployment, respectively.

Using the photographs taken from a motion sequence in (Amanov et al., 2016), the extensible segment TDCR achieves L FTL = 0.2280 , whereas a CTCR used in Girerd et al. (2020) achieves L FTL = 0.3285 in simulation and L FTL = 3.2728 with real hardware. Note that in (Amanov et al., 2016) the extensible segment TDCR has three segments and that a CTCR is typically incapable of pure FTL deployment as stated in (Gilbert and Webster, 2013). However, results give an intuition about which value of the error measure L FTL corresponds to an acceptable FTL deployment since it has been shown in (Neumann and Burgner-Kahrs, 2016) that an extensible segment TDCR (Nguyen and Burgner-Kahrs, 2015) is capable of FTL deployment. The achieved results L FTL = 0.1537 for the in-plane FTL deployment show a smaller error as the physical prototype has one segment. For the case α = 0°, the built prototype is essentially an extensible segment TDCR with one segment. Using three segments instead of one would probably lead to similar errors as the one reported by Amanov et al. (2016). Consequently, the value of L FTL seems reasonable. More important, the error for the out-of-plane FTL deployment being L FTL = 0.1399 is in the same range. This demonstrates that one FAS is capable of FTL deployment along spatial paths, which cannot be achieved with previously proposed TDCR segment designs.

The errors of FTL behaviour observed during the FAS deployment may be due to two reasons. First, as the torsion of the path to follow is constant, the geometric twist angle θ must vary linearly with the arc-length along the path. As for now, the geometric twist angle θ is determined by the backbone rotation α, which is kept constant during the deployment. Second, the curvature of the segment may be constant for one deployment step but might change slightly with the segment length as the tendons are not fully constrained. They are here locally constrained by the FSD. The goals of the analysis in simulation are two folds: to study the effect of these two sources of errors and to demonstrate the range of reference path that can be followed with the FAS.

In order to do so, we first update the FTL deployment strategy. In addition to varying the inserted length of the backbone and keeping the applied tensions constant, we use variations in α to follow the natural change in twist of the reference path. We consider the reference path as the shape of a segment with length l ^∗ actuated by pulling on the first tendon, with the central backbone rotated by α∗. The FTL deployment is then simulated by solving the forward static model while varying the insertion length l, where l < l ^∗. The applied rotation α for any intermediate configuration is scaled linearly w.r.t. the inserted length and is calculated as: α = α ∗ l l ∗ (12)

We then simulate FTL deployments with different number of disks along the backbone, to observe the effect of the tendon constraint, and for different reference paths. The obtained intermediate configurations in the FTL deployment have been shown in Figures 8A–D. We observe that for small numbers of disks, the obtained configurations do not perfectly align with the reference path. This deviation is due to the assumption of partially constrained tendon. As the tendons are not always normal to the end disk, the moments resulting from their forces, and as a consequence the backbone curvature and torsion, change with the deployed length. When the number of disks is increased such that the tendon path resembles a fully constrained design [as defined in (Rao et al., 2021)], exact spatial FTL is achieved.

The intermediate configurations have been super-imposed and indicated for different values of inserted lengths. Applying (11) to different projections of the covered area for nine equal-distance steps for a diameter of 5 mm, we get L FTL x y = 0.257 , L FTL x z = 0.012 , and L FTL y z = 0.091 for the projection onto the xy-plane, xz-plane, and yz-plane, respectively. Those values correspond to a model with n = 10 disks, where as L FTL x y = 0.0581 , L FTL x z = 0.0259 , and L FTL y z = 0.0303 correspond to n = 50 case. For comparison, for n = 50 and with the previous FTL strategy, where α = const. w.r.t. l during the deployment, we obtain L FTL x y = 0.474 , L FTL x z = 0.8128 , and L FTL y z = 1.538 . It is apparent that previous FTL strategy leads to higher FTL error and, therefore, varying the value of α linearly with length leads to better FTL performance. Further, the FTL error decreases with larger values of n. This reinforces the previous observation and lets us conclude that the higher the number of FSD, the better the accuracy for the FTL deployment. Finally, we can make an evident observation that the values of L FTL are view dependent due to the projection of the 3D volume of the robot’s shape onto a plane.

In Figure 8E, we show 20 spatial paths with α* ∈ [ − π rad, π rad] and randomized values of tensions applied to the first tendon for n = 50. We see that the robot observes FTL deployment for intermediate lengths along the various 3D spatial paths.

In order to obtain the robot configurations, standard approaches rely on a heavy discretization of the actuation space (Wu et al., 2017), spanned by u in our case, or of the task space (Li et al., 2017). In the first case, the static model is solved for each set of actuation inputs, leading to a dense set of tip positions. The robot configurations leading to tip positions in the vicinity of a desired position are gathered, leading to a set of tip orientations. In the second case, the tip orientation is parametrized by rotations around specific axes and discretized for each possible tip position in the workspace. The inverse static model is then solved for each tip pose, and converges eventually toward a robot configuration in case the pose is achievable. These two methods are both computationally expensive, and their result depends on the discretization strategy and density. The task space discretization approach requires to know the rotation axis of the robot tip, which cannot be completely controled with the four dof of the FAS. Moreover, the convergence of the inverse static model depends on the initial guess provided to the numerical solver.

To alleviate these problems, we compute the robot configurations by varying one actuation input and solving the inverse static model for a desired position p _E using a continuation method. The idea is to dedicate three actuation variables to maintaining the segment tip position at p _E , and to vary the fourth input to change the robot configuration and obtain different tip orientations. In particular, we vary the backbone rotation α as it has a major impact on the robot shape. The continuation method allows to vary α and to compute the corresponding set of robot configurations without relying on an a priori discretization and being sensitive to bad initial guesses. The set of robot configurations is considered as the smooth function B , called branch of solution of the inverse static model and defined as: B : R → R 4 n + 3 α ↦ X μ ⊤ T γ β ⊤ such that G X μ , p E , u = 0 (13)

It is computed using a prediction and correction process. Starting from a point of the branch, the next point is predicted by incrementing the actuation input by a given step size and by taking the corresponding point along the branch local tangent ∂ B / ∂ α . The inverse static model is then solved, starting from the predicted configuration, using a Newton-Raphson algorithm. The step size is automatically reduced in case the numerical solver has more difficulties to converge, i.e. the number of iterations required to converge increases. The process is repeated until a specified number of points along the branch have been computed. Once the robot configurations have been found, the resulting tip orientations are computed as rotation matrices R _E using Eq. 3. The prediction-correction method is implemented in the Matlab toolbox called MatCont (Dhooge et al. (2008)).

Finally, analysing the tip orientations requires to have an intuitive representation of them. Service spheres provide such representation as demonstrated in Li et al. (2017). A service sphere of a given radius r _S is defined, where its center is located at the robot tip position p _E . The intersection of the robot tip tangent with the service sphere creates then a point p _S , which represents the tip orientation. With forward and inverse kinematics based sampling method, the service sphere is usually discretized in a finite number of patches. The patches that contain a tip orientation point are then colored, which allows an easy visualization of the possible tip orientations and the evaluation of orientability metrics. We use the service sphere representation in this work to visualize and analyse the tip orientation of the FAS. The set of tip orientations obtained after varying α results in a set of consecutive points p _S on the service sphere, forming a one dimensional curve defined by: p S : R → R 3 α ↦ p E + r S R E e z (14) where r _S is the radius of the service sphere.

To illustrate the position redundancy of a segment with variable helical tendon routing, we conduct a reachability assessment. As before, we conduct a preliminary study with the physical FAS prototype described in Section 2.3.1. A more in-depth study in simulation follows afterwards.

We choose two target points in the robot workspace. Their position is indicated by a blue bead (diameter 2 mm). They lie about 58 and 51 mm in front of the prototype, and about 51 and 37 mm above the base of the prototype, respectively. Positions of the backbone origin, the hole for the first tendon in the base, and the projection of the point (blue bead) on the same xy-plane lies on a line, i.e. x-axis. The tendon force, the backbone rotation, and the backbone translation are then actuated manually in order to reach the target with different configurations. Table 3 summarizes all the configurations, configurations 1, two and three corresponding to the first target points, and the others to the second. Their realization using the prototype with one FAS is represented in Figure 9. Note that the tip position is the end disk position plus an offset, i.e. flagpoles with length 7 mm.

For the first target point, we start from configuration two and vary the backbone rotation and the tendon tension without changing the robot length. Configuration two is a pure in-plane bending in the xz-plane, where the first tendon is loaded with 3.21 N and α = 0 rad. We obtain the robot configurations 1 and 3, which lead to the same target tip position. While the robot shapes are close to each other and the tip orientation is rotated about the vertical axis of approximately − π/6 rad, the tip orientation varies significantly from − 2π/3 rad to 2π/3 rad. For the second target point, we start from configuration six which is again a pure in-plane bending in the xz-plane. The backbone is rotated with larger values, ranging from α = ±2π rad. As a result, the backbone must be translated to reach the target point, leading to configurations four to 8. Position redundancy is achieved while obtaining robot configurations with significant variations of tip orientation.

From the preliminary study with FAS prototype, we can conclude that the additional dof of the design greatly contribute to the position redundancy of the design. The dof for the extension of the segment has a negligible contribution for small values of α. However, leveraging β leads to a higher variation of α, and therefore to higher variations of robot shapes and tip orientations. Further, note that neither a non-extensible segment, which can only bend, nor an extensible segment, which can bend and elongate, exhibit position redundancy. In addition, note that the workspace of a non-extensible segment is a curved plane in the task space. Hence, a point on this curved plane can be reached with the FAS with different tendon tensions τ _i and a high variety of α, while β has a minor contribution.

We investigate numerically the FAS redundancy at three positions in its workspace. Making use of the symmetry in the robot workspace, we select three different planar configurations obtained by pulling on the first tendon, shown in Figure 10A. We consider that the frictional and gravitational forces are negligible, an assumption that we relax later on. The one-dimensional curve p _S (α) on the service sphere is computed with the continuation process for α ∈ [ − π, π]. For each configuration, the corresponding service spheres have been depicted in Figures 10B–D.

We obtain sets of helical robot configurations, as expected, that draw a circular curve on the service sphere. The center of this circle and its radius give indications about the main rotation axis of the segment at the target tip position and the angular displacement achieved around this axis respectively. We observe that the higher the initial curvature of the segment, the larger the disk radius and the angular displacement. We also observe that these large range of orientations are obtained with significant backbone rotations and small backbone translation. The evolution of the segment length with α during the continuation process is plotted in Figure 11 for the corresponding three configurations displayed in Figure 10B. We see that the variation in length is in the millimeter range, which aligns with the observations made in the preliminary study. It is interesting to note that these circular curves are not closed-curves. The same point on the service sphere, or a same tip orientation, can be approached using different set of actuation inputs and different robot configurations. Consequently, a specific property of the FAS is to be able to achieve the same tip position and tip tangent with different and discrete values of tip axial rotations, which coincides with the preliminary study.

Finally, we briefly investigate the influence of friction as they have been observed to have a significant effect on the segment shape. The changes of segment length during the backbone rotation and when considering friction are plotted with dashed lines in Figure 11. We observe that the changes in length is larger for configurations one and two when friction is considered to be non-zero compared to the frictionless case, while the opposite trend is observed for configuration 3. The different trends are observed due to the difference in backbone configurations, occurring due to presence of friction.

All dof can be actuated independently in an extrinsic manner. In addition, our proposed design is simpler than the prototypes presented in Starke et al. (2017) and Gerboni et al. (2015), which involve to combine helical and straight tendons and then may lead to complex tendon routing, especially if several segments are stacked. Special care needs to be taken for the friction between FSD and tendons. On top of that, the use of magnets in type-III FSD introduces new challenges regarding modelling the orientation and position of each FSD. Further, the repulsion forces decrease rapidly as the robot extends, limiting the robot’s maximum length. While the magnetic repulsion force can be improved by increasing the magnet’s thickness, it would in turn limit the robot miniaturization and increase its weight. Among other improvements, finding an alternative to magnets is a future direction to revise the current design.

We start with a fairly simple deployment strategy, i.e. constant tendon tension and backbone rotation while deploying the backbone. For the preliminary study, the resulting errors are in the range of inaccuracies of the used physical FAS prototype and comparable with the extensible segment TDCR (Amanov et al., 2019, 2016) provided that the deployment length and backbone rotation are small. Note that even in the case of not perfectly distributed FSD along the backbone as can be seen in Figure 7A the resulting errors are sufficiently small. Also note that these are valid postures as this assessment is for FTL use and independent of the model evaluation. In simulation, we show that the deployment strategy needs to be adapted to achieve perfect FTL deployment. The deployment is slightly modified, resulting in a constant tendon tensions with linear change in the backbone rotation and deployment. Indeed, as the backbone must have a constant torsional curvature during the deployment, the angle of the backbone base rotation α must be varied linearly w.r.t. the arc-length s. While on the one hand this can be seen as a limitation, on the other hand it can be used for FTL deployment strategies in future work.

Another key element to ensure perfect FTL deployment with a TDCR is to include more FSDs. The more FSDs, the closer the TDCR’s tendons to be fully constrained, see Rao et al. (2021) for a discussion on partially and fully constrained tendon paths. In simulation, we have found that perfect FTL deployment can be achieved with 70 FSDs. Also, slight discrepancies between the robot shape and the reference path are observed when 10 FSDs are used.

In the literature, FTL deployments are quantified by repeatedly measuring the robot shape during the deployment. In Amanov et al. (2019) and Garriga-Casanovas and Rodriguez y Baena (2018a), a 3D laser scanner is used to obtain a spatial point cloud of the robot. Afterwards, the centerline of the robot’s backbone is extracted relying on a thinning algorithm or on a model of a backbone. Downsides of this labor-intensive approach lie in the use of dedicated hardware and point cloud post-processing.

In contrast, the proposed area-based FTL error (11) is easy to evaluate, interpret, and visualize. The evaluation can be realized with binary images after segmentation and using a simple threshold after combining several binary images as well as summing up all occupied pixels in the respective images. The value can be interpreted as percentage of the normalized surplus of the occupied area during the FTL deployment. Lastly (11) can be visualized as shown in Figure 7. Nevertheless, sources for uncertainty are the manual image segmentation, the low discrete steps, and the dependency of the view. These issues could be alleviated by substituting the manual approach with a learning-based approach enabling an online multi-view evaluation of an FTL deployment. This will be the subject of future work.

Note that an area-based accuracy assessment is used in Chikhaoui et al. (2019) to compare two modeling approaches for a TDCR by projecting a measured point cloud onto several planes. In Girerd et al. (2020), superposition of all intermediate shapes of a CTCR is used to highlight the approximate FTL deployment. A similar approach is utilized in Amanov et al. (2019, 2016), where the overall occupied space over the course of deployment is used to visually show the feasibility. However, none of these qualitative approaches are used to quantify an FTL deployment.

Thanks to the four dof, one segment alone already shows interesting properties such as position redundancy as indicated in Figure 9 and Figure 10. Interestingly, the overlaid images shown in Figure 9 and Figure 11 expose that the change in orientation of the end disk is high in comparison to the shape deviation. Note that the segment length has a minor influence on the position redundancy, while α and θ, respectively, has a major impact as indicated in Figure 9 and Figure 11. The workspace of a TDCR with only two dof, i.e. κ _x and κ _y , is a two dimensional manifold. Therefore, a position on this manifold can be reached with different tip orientations, where α has a significant contribution. The position redundancy also known as orientability is higher for higher tendon tension as shown in Figure 10. Further, the combined applied tendon tensions are higher for the out-of-plane bending compared to the in-plane bending, see Table 3. Leveraging orientability capability for dynamic obstacle avoidance and for motion planning are promising future research directions.

During the preliminary study with the physical prototypes, we observed hysteresis behaviours, with velocity-dependent and bending-dependent configurations. We tried to mitigate those phenomena by using multiple measurements via a tactile device to induce noise and reduce measurement biases. The derived static model is calibrated to account for unmodeled effects. It achieves a reasonable accuracy, i.e. average tip error of 4.00 mm corresponding to a relative tip error of 4.39%, compared to 4.10 mm (1.7%) reported for a Cosserat rod-based approach in (Rucker and Webster, 2011). The assumptions of equidistant disk spacing and linear rotation distribution along the backbone are shown to be reasonable approximations. However, they are sources of error for the static model. Specifically modeling the inter-magnetic forces in magnetic spacer disks, and the influence of tendon actuation on the rotation distribution, should be investigated in future work. Additionally, more specific mechanics-based modelling approaches, e.g. (Rucker and Webster, 2011; Chikhaoui et al., 2019), can be adapted for the FAS design and then used to estimate the workspace including position redundancy measures based on (Burgner-Kahrs et al., 2014; Wu et al., 2017) or to extend the FTL deployment strategy proposed by Neumann and Burgner-Kahrs (2016); Amanov et al. (2019) to tendon tension.

In addition to extending the current static model and exploring a dynamic model, developing a kinematic model, which allows to investigate the redundancy, is desirable. This is motivated by an interesting fact stated by Garriga-Casanovas and Rodriguez y Baena (2018b): An extensible TDCR with six dof in the arc space is only capable of five dof at the end effector in the task space due to a loss in rank of the Jacobian matrix for the velocity mapping. Due to the high impact of a geometric twist angle on the orientability, using a FAS might provide a viable solution. Using two concatenated segments provides eight dof in the arc space and could be used to mitigate the above problem.