Design of a torsional stiffener for a cable-driven hyper-redundant robot composed of gear transmission joints

Abstract

This work describes the advancement in develo** a cable-driven gear transmission joint designed as a basic element for a long-reach hyper-redundant robot. Hyper-redundancy allows the robot to perform auxiliary tasks such as obstacle avoidance and joint limits satisfaction. This feature makes hyper-redundant robots particularly useful for performing tasks in confined and hazardous environments and areas that are not reachable by a human operator. The long-reach feature of the robot requires a detailed study of the overall structure and its components. The joint must be capable of transmitting forces and movements over a long distance without losing the precision and accuracy of the end-effector, so it is designed to optimise the robot’s performance in terms of stiffness, structural resistance, and functional characteristics. In light of the above considerations, the main focus of this work is to improve the structural performance of the entire robotic system. Consequently, since the most critical component of the robot in terms of torsional deformation is the gear transmission joint, this paper aims to design a torsional stiffener element to reduce its deformation and, thus, an increase of torsional stiffness of the overall robotic system. Tube-shaped and rectangular-shaped stiffener elements, which can fit the joint design satisfying its geometrical constraints, are proposed. A computer-aided engineering approach is implemented to improve the precision of positioning of the end-effector by adding stiffener elements in the joint. Two sensitivity analyses, varying the geometry of the proposed stiffener elements, are performed to evaluate their performance in terms of added mass and displacement reduction.

1 Introduction

In the last decades, with the development of service robotics, a whole new set of applications and tasks began to be faced by automated or teleoperated systems and robot [1,2,3]. One relevant field of application is industrial inspection [4]. When a dangerous or difficult-to-reach area needs to be inspected or maintained, using a robotic system can be a very convenient option [5, 6]. Inspection and maintenance represent a critical economic activity in several sectors, such as energy [7], water supply [7], oil and gas [8], civil engineering and infrastructure [9]. Robotic applications have massive potential to increase productivity and improve safety [10].

Although the need to operate in an unstructured and complex environment, recent advances in hardware design, computation capabilities, and control algorithms paved the way for the design, testing, and deployment of new robots, ranging from legged quadrupeds [11] to drones [12], elephant trunk [13, 14], crawler [15], octopus [16], and snake-like robots [17]. In particular, thanks to their mobility within constrained and hazardous environments, hyper-redundant robots can be used for inspecting and maintaining complex industrial plants. Long-reach cable-driven hyper-redundant manipulators have been widely employed in the field of industrial inspection [18], as well as in the nuclear sector [19, 20]. The cable design keeps the actuators safe from the environmental conditions of the workspace and reduces the weight of the manipulator structure. The kinematic redundancy allows the robot to perform multiple tasks simultaneously, like moving into constrained environments on a specific trajectory and avoiding possible obstacles. In addition, due to the March 2011 accident at Fukushima Daiichi nuclear power plants in Japan, the need to develop a specific long-reach robotic arm for decommissioning nuclear reactors has risen. Some examples are the Fukushima Repair Manipulator [21] and the robotic arm developed by Veolia Nuclear Solutions and Mitsubishi Heavy Industries [22].

All the above-mentioned long-reach robots are based on different designs. However, for all of them, the most crucial needs are to maintain a lightweight, to reduce the actuators’ size, increase their payload, and at the same time maintain sufficient stiffness to guarantee an accurate end-effector positioning. Therefore, since all the elements in a robotic system do not equally contribute to the overall stiffening of the structure, a key aspect is to identify which components or interactions between them need to be improved.

This work aims to improve the accuracy in the end-effector positioning of the fully actuated cable-driven hyper-redundant robot presented in [23]. Given that this robot moves at a relatively low speed and acceleration, the inertia effects can be neglected. Then, since it is a long-reach manipulator, the main sources of deformation are the static forces induced by the mass of each module. Gear transmission joints, particularly their intermediate links, are the most affected elements by the deformation due to torsional loads. The most important factors to take into account when studying the stiffness of a mechanical system are [24, 25]: (i) the structural deformations due to load-transmitting components (e.g. links and shafts); (ii) the deformations due to power-transmitting devices (e.g. actuators); and (iii) the deformations due to contacts between components (e.g. surface roughness due to imperfect machining). This work focuses specifically on the torsional deformation of the gear transmission joint. Torsional resistance is a common matter in the automotive industry. Indeed different works studied solutions for the torsional stiffening of the vehicles’ frames [26, 27]. When facing such design problems the state of the art is to take advantage of additive manufacturing combined with generative design, to obtain an effective geometry with few construction constraints [28] and relying on rapid prototy** for functional tests [29]. This paper represents a preliminary study of a torsional stiffening solution in a robotic system by develo** basic designs as a first step towards a more detailed investigation of the phenomenon and better-refined solutions. To increase the torsional stiffness of the gear transmission joint, two stiffening elements designs, meant to be manufactured using machining, are compared in terms of stiffness increase, added mass, and the displacement of the position of the end-effector.

Section 2 summarises the baseline on which this work is founded. Section 3 describes the critical aspects of the joint and the geometrical constraints that the stiffener elements have to satisfy. Section 4 presents the two proposed stiffener elements and compares their performance. Section 5 draws the conclusion and future works.

2 Baseline: robot concept and joint description

The concept of the fully actuated cable-driven hyper-redundant robot and the design of the gear transmission joint, described in [23], are revisited as a baseline on which this work is founded.

2.1 Robot concept

The robot comprises a chassis in which are mounted all the actuators and a robotic arm composed of five modules. Each module, composed of a gear transmission joint and a link, has one rotational Degree of Freedom (DOF) and is driven by a pair of cables. For the task of positioning the end-effector in the Cartesian space, this is a hyper-redundant robot. The gear transmission joint, described in Sect. 2.2, is a mechanism composed of two revolute joints, of which one is constrained to revolute on the other. Table 1 reports the modified DH parameters of the robot, according to [30], where $i_c$, with ${i=1,\dots ,5}$, represents the constrained revolute joint of the $i^{th}$ module, while EE indicates the end-effector.

Table 1 DH parameters of the robot

Given this assembly configuration, the robot can achieve 3D motions. A virtual prototype of the robotic system, shown in Fig. 1, has been developed with the parameters reported in Table 2, where m and CoM indicate the mass and the centre of mass position of each joint, respectively.

Fig. 1

Virtual prototype model of the fully actuated cable-driven hyper-redundant robot

Table 2 Parameters of the robot

2.2 Joint description

The robotic system presented in Sect. 2.1 comprises several modules connected in series. Each module, driven by a pair of antagonist cables and moved by a single motor, is composed of a gear transmission joint and a rigid link. The gear transmission joint, whose schematic is shown in Fig. 2, consists of two flanges, input and output ones, connected by an intermediate link of length $l_{int}$ through two revolute joints fixed in $O_1$ and $O_2$, respectively.

Two gears, $G_1$ and $G_2$, are mounted to the input and output flanges, respectively. This mechanism can be seen as a planetary gear, in which the intermediate link is the carrier and $G_2$ is the planet that moves around the sun $G_1$. The reference frame $R_0$, whose origin and axes are $O_0$ and ${x_0\text {-}y_0\text {-}z_0}$ respectively, is integral with the input flange. The reference frame $R_1$, whose origin and axes are $O_1$ and ${x_1\text {-}y_1\text {-}z_1}$ respectively, is integral with the intermediate link. The reference frame $R_2$, whose origin and axes are $O_2$ and ${x_2\text {-}y_2\text {-}z_2}$ respectively, is integral with the output flange. The origins, $O_i$, and the axes, ${x_i\text {-}y_i\text {-}z_i}$, with $i=0,\dots ,2$, are shown in Fig. 2. The two antagonist cables, with lengths $l_1$ and $l_2$, are anchored at the points A and B and pass through the points $A'$ and $B'$, respectively. The intermediate link freely rotates around $z_1$ by an angle $\theta$, while the output flange, integral with $G_2$, rotates around $z_2$ by the same angle $\theta$ with respect to the intermediate link itself. Consequently, the joint variables $q_{1,j}$ and $q_{2,j}$ are related as follows:

$$\begin{aligned} q_{1,j} = q_{2,j} = \theta \end{aligned}$$

(1)

and the total angle produced by the joint is $2\theta$. Indeed, assuming a gear ratio equal to 1, considering the input flange and $G_1$ both attached to the reference frame $R_0$, and exploiting the Willis formulation for planetary gears [31], the overall transmission ratio of the gear transmission joint can be described as follows:

$$ \tau = \frac{\omega _{f_{out}}}{\omega _{l_{int}}} = \frac{l_{int}}{r_{G_2}} = 2, $$

where $r_{G_2}$ is the radius of $G_2$, while $\omega _{f_{out}}$ and $\omega _{l_{int}}$ are the absolute angular velocity of the output flange and the intermediate link, respectively. To respect the design constraints, the joint limits are the following:

$$\begin{aligned} \begin{aligned}&-q_{m} \le q_{1,j} \le q_{m},\\&-q_{m} \le q_{2,j} \le q_{m}, \end{aligned} \end{aligned}$$

where $q_{m}= \pi / 4$. Defining the actuation coordinate lengths, $l_1$ and $l_2$, as follows:

$$\begin{aligned} \begin{aligned} l_1&= \left\| A'-A \right\| _2, \\ l_2&= \left\| B'-B \right\| _2, \end{aligned} \end{aligned}$$

(2)

it is possible to demonstrate that

$$\begin{aligned} l_1 + l_2 = 2l_{int} \quad \quad \forall \theta \in [0,2\pi ). \end{aligned}$$

(3)

Fig. 2

Gear transmission joint schematic

For a given joint angle $\theta$, from Eq. 3 follows that the shortening of $l_1$ and the lengthening of $l_2$ are equal to:

$$\begin{aligned} \Delta L = \frac{l_2-l_1}{2}. \end{aligned}$$

(4)

Using Eqs. 1 and 4, the direct kinematics, which relates the angle $\theta$ and the actuation coordinates in terms of $\Delta L$, is

$$\begin{aligned} \theta = {{\,\textrm{atan2}\,}}\left( \Delta L,\sqrt{d^2-\Delta L^2}\right) , \end{aligned}$$

(5)

where

$$\begin{aligned} d = \left\| A-B \right\| _2 = \left\| A'-B' \right\| _2. \end{aligned}$$

The corresponding inverse kinematics can be computed as follows:

$$\begin{aligned} \Delta L = d\sin \theta . \end{aligned}$$

(6)

Using Eqs. 4 and 6, the actuation coordinates can be derived as follows:

$$\begin{aligned} l_1&= l_{int} - d\sin \theta ,\end{aligned}$$

(7)

$$\begin{aligned} l_2&= l_{int} + d\sin \theta . \end{aligned}$$

(8)

3 Problem description

A statistical analysis has been performed to evaluate the maximum torsional load, $M_t$, applied on the origin $O_2$ along the $x_2$-axis of the first joint to obtain a more realistic scenario. First, to make the torsional phenomenon predominant, the joint variables of the first joint have been set to zero, $q_{1,1}=q_{1,2}=0$. Then, the joint configuration space of the remaining joints was explored using the Latin Hypercube Sampling method since it uniformly divides a given space into a fixed number of hypercubes of equal size, picking a random point in each hypercube. To obtain a statistically representative sample of the joint configuration space, 160, 000 samples have been computed. Next, for each configuration of the robot, the reaction forces and torques for each joint are evaluated, following the static model described in [23], using the parameters listed in Table 2. The distribution of torsional load applied on the origin $O_2$ along the $x_2$-axis of the first joint is reported in Fig. 3.

Fig. 3

Torsional load $M_t$ for the first joint as the result of the static numerical analysis

According to this analysis, $M_t$ is obtained when the values of $q_{i,j}$ are zeros for all j except for $j=2$:

$$\begin{aligned} q_{1,2}&= -\frac{\pi }{4}\text { rad},\\ q_{2,2}&= -\frac{\pi }{4}\text { rad}. \end{aligned}$$

Such configuration is schematised in Fig. 4. The corresponding load value is $M_t=100$ Nm.

Fig. 4

Robot configuration corresponding to maximum torsional load on the first joint, $M_t$

The resulting torsional deformation of the first joint, which can be evaluated through finite element analysis, induces an error on the position of the end-effector of the robot. The structural static analyses performed in this work are executed using PTC Creo^® 9.0 and are based on the following assumptions and parameters:

• The joint geometry is simplified considering the bodies most sensitive to $M_t$, i.e. the intermediate links connected by two shafts;

• All the contact interfaces are assumed to be bonded;

• P-elements polynomial order from minimum 3 to maximum 9;

• Mesh element type tetrahedron;

• Mesh element edge size 5 mm;

• Multi-Pass Adaptive method with percent convergence $10~\%$.

Table 3 Physical and mechanical properties of A7075 T6 (Ergal)

All the mechanical components considered in this study are made of aluminium alloy A7075 T6, also known as Ergal. This material has been chosen because performs well in terms of mechanical stiffness-to-weight ratio:

$$\begin{aligned} \frac{E}{\rho }=26\,\frac{\text {~MPa m}^{3}}{\text {kg}}, \end{aligned}$$

where E is the Young modulus and $\rho$ is the density. Such value is comparable to that of titanium. The physical and mechanical properties are reported in Table 3. The 3D Computer-Aided Design (CAD) model of the simplified joint, whose mass is $m_0=0.368$ kg, is reported in Fig. 5.

A first structural static analysis has been performed on the simplified joint where the load $M_t$ is applied on the contact surfaces between the outer shaft and the intermediate links, having constrained the inner shaft. Figure 6 reports the results of the analysis.

Fig. 5

3D CAD model of the simplified joint highlighting the point P

Fig. 6

Deformed model of the simplified joint showing the constraint on the inner shaft and the torsional load on the outer shaft. A scale factor of 10 is applied to better visualise the deformation

The displacement of the point P, which is one of the end points of the axis of the outer shaft, is

$$\begin{aligned} u_{P_0}=0.48 \text {~mm.} \end{aligned}$$

The corresponding torsional deformation results in a rotation of

$$ \theta _{P_0} = {{\,\textrm{asin}\,}}{\left( \frac{2 u_{P_0}}{l_s}\right) }=10.7 \times 10^{-3}\text {~rad} \,, $$

where $l_s=90$ mm is the length of the outer shaft. Considering the joints configuration of the robotic system for which the torsional load on the first joint is $M_t$, the maximum displacement of the position of the end-effector induced by $\theta _{P_0}$ is

$$ u_{{EE}_0}=20.9 \text {~mm.} $$

To increase the torsional stiffness of the joint without changing the original geometry of the components, it is necessary to add a stiffener element. Such an element must satisfy the constraints of the geometry of the whole joint, including the gears, the flanges, and the cables. Figures 7b and 8b report the geometrical parameters required to describe such constraints. In the following, two different designs that satisfy such constraints are investigated.

Fig. 7

Geometrical constraints of solution 1

3.1 Tube-shaped geometry

Assuming a ring section, with width and thickness w and t, respectively, drawn in the space between the two gears of the joint as shown in Fig. 7a, the relations represented in Fig. 7b are defined:

$$\begin{aligned} \begin{aligned} \varphi&= {{\,\textrm{acos}\,}}{\left( \frac{2 r_p - w}{2 r_t}\right) },\\ r_i&= \sqrt{\left( \frac{s}{2}\right) ^2+\left[ r_t \sin {\left( \varphi \right) } \right] ^2},\\ t&= r_e - r_i = \left[ \frac{d\cos {\left( q_m\right) } - d_c}{2} \right] +\\&- \sqrt{\left( \frac{s}{2}\right) ^2+r_t^2 \left[ 1-\left( \frac{2 r_p - w}{2 r_t}\right) ^2 \right] },\\ \end{aligned} \end{aligned}$$

(9)

where $r_t$ and $r_p$ are the tip and the pitch radii of the gear, respectively, s is the gear depth, $d_c$ is the cable diameter, and $r_e$ is the semi-minor axis of the ellipse passing through the positions of the cables when $q_1=q_m$.

Table 4 Geometric parameters describing the tube-shaped geometry constraints

Considering a minimum thickness $t_{\min }$ and a minimum width $w_{\min }$, w lies in the range ${\left[ w_{\min }, w_{\max }\right] }$ where

$$\begin{aligned} {w_{\max }=w\left( t_{\min }\right) }. \end{aligned}$$

The values of the geometric parameters required to describe the constraints that the tube-shaped geometry has to satisfy are listed in Table 4.

Furthermore, assuming a support similar in shape to a cylindrical tube with the ring cross-section described above, its faces must be cut to avoid contact between the support and the joint flanges. As shown in Fig. 7b, the cut angle $\theta _c$ is subject to the following constraint:

$$\begin{aligned} \theta _c = {{\,\textrm{atan2}\,}}\left( r_e , r_p - \frac{w}{2}\right) < q_m. \end{aligned}$$

Given the limits of w, it is trivial to prove that such a condition is always respected.

3.2 Rectangular-shaped geometry

Assuming two equal rectangular sections, with base and height b and h, respectively, drawn in the inner spaces of both the gears of the joint as shown in Fig. 8a, the relations represented in Fig. 8b are defined:

$$\begin{aligned} \begin{aligned} h&= 2 \sqrt{r_2^2-\left( r_1+b\right) ^2}, \end{aligned} \end{aligned}$$

(10)

where $r_2$ is the inner radius of the gear and $r_1$ is the radius of the circle tangent to the mounting support placed inside the gear.

Fig. 8

Geometrical constraints of solution 2

Table 5 Geometric parameters describing the rectangular-shaped geometry constraints

Assuming a minimum height $h_{\min }$, h lies in the range ${\left[ h_{\min }, h_{\max }\right] }$, with

$$ h_{\max }=2\sqrt{r_e^2-\left( \frac{l_s - 2 d_i}{2}\right) ^2}, $$

where $d_i$ is the depth of the intermediate link. The base b belongs in ${\left[ b_{\min }, b_{\max }\right] }$, with

$$\begin{aligned} b_{\min }&=b(h_{\max }),\\ b_{\max }&=b(h_{\min }). \end{aligned}$$

The values of the geometric parameters required to describe the constraints that the rectangular-shaped geometry has to satisfy are listed in Table 5.

Fig. 9

Sensitivity analysis for solution 1 in terms of $\varepsilon$. The orange circle highlights the values associated with maximum efficacy

Fig. 10

Sensitivity analysis for solution 1 in terms of $\eta$. The orange diamond highlights the values associated with maximum efficiency

4 Proposed solutions

The displacement $u_{P_0}$ can be reduced by increasing the torsional stiffness of the joint adding a stiffener that connects the two intermediate links. To compare different stiffener designs, the following performance indicators are evaluated for a given $i^{th}$ solution:

• the efficacy $\varepsilon _i$ is calculated as follows

  $$\begin{aligned} \varepsilon _i=100\frac{\left( u_{P_0}-u_{P_i}\right) }{\left( {u_{P_0}}\right) }, \end{aligned}$$

  where $u_{P_i}$ is the displacement of the point P with the stiffener, defined as the percentage reduction of the displacement with respect to the joint without the stiffener;

• the efficiency $\eta _i$ is calculated as follows

  $$\begin{aligned} \eta _i=\frac{\left( u_{P_0}-u_{P_i}\right) }{\left( m_i-m_0\right) }, \end{aligned}$$

  where $m_i$ is the mass of the joint with the stiffener.

Fig. 11

Design for solution 1 that maximises $\varepsilon _1$

Fig. 12

Design for solution 1 that maximises $\eta _1$

According to the geometrical constraints, assumptions, and parameters introduced in Section 3, two different solutions have been designed and analysed.

4.1 Solution 1: tube-shaped stiffener

The first proposed solution is a tube-shaped stiffener surrounding the gears. This element is obtained from a cylindrical tube with length $2 r_p$, external radius $r_e$, and thickness t, whose faces are cut each by two planes tilted of the angle $\theta _c$. In addition, two portions of material with a rectangular cross-section in the x-z plane are removed to avoid interference between the stiffener element and the gears, maintaining a width of w according to Eq. 9. A sensitivity analysis has been performed varying the parameter w within the range $\left[ w_{\min },w_{\max }\right]$ with 30 steps. The corresponding results in terms of performance indicators are reported in Figs. 9 and 10.

Fig. 13

Deformed model of the simplified joint with the stiffener solution 1 that maximises $\varepsilon$. A scale factor of 10 is applied to better visualise the deformation

Fig. 14

Deformed model of the simplified joint with the stiffener solution 1 that maximises $\eta$. A scale factor of 10 is applied to better visualise the deformation

Two structural static analyses have been performed with the designs corresponding to the parameters that maximise $\varepsilon _1$ and $\eta _1$. The two designs are depicted in Figs. 11 and 12.

The corresponding structural static analyses are shown in Figs. 13 and 14.

4.2 Solution 2: rectangular-shaped stiffener

The second proposed solution comprises a pair of rectangular-shaped stiffeners passing through the gears. Both elements have a rectangular cross-section with base b and height h, which are related by Eq. 10. A sensitivity analysis has been performed varying the parameter b within the range $\left[ b_{\min },b_{\max }\right]$ with 30 steps. The corresponding results in terms of performance indicators are reported in Figs. 15 and 16.

Fig. 15

Sensitivity analysis for solution 2 in terms of $\varepsilon$. The orange circle highlights the values associated with maximum efficacy

Fig. 16

Sensitivity analysis for solution 2 in terms of $\eta$. The orange diamond highlights the values associated with maximum efficiency

Fig. 17

Design for solution 2 that maximises $\varepsilon _2$

Fig. 18

Design for solution 2 that maximises $\eta _2$

Fig. 19

Deformed model of the simplified joint with the stiffener solution 2 that maximises $\varepsilon$. A scale factor of 10 is applied to better visualise the deformation

Fig. 20

Deformed model of the simplified joint with the stiffener solution 2 that maximises $\eta$. A scale factor of 10 is applied to better visualise the deformation

Table 6 Summary of the results of the sensitivity analyses for solution 1 and 2

Table 7 End-effector displacement corresponding to the first joint without stiffener and the first joint with solution 1 and 2 for $\varvec{\varepsilon }_{\text {max}}$ and $\varvec{\eta }_{\text {max}}$

Two structural static analyses have been performed with the designs corresponding to the parameters that maximise $\varepsilon _2$ and $\eta _2$. The two designs are depicted in Figs. 17 and 18.

The corresponding structural static analyses are shown in Figs. 19 and 20.

4.3 Results and discussion

Figures 9, 10, 15 and 16 report the sensitivity analyses for solution 1 and 2, respectively, in terms of $\varepsilon$ and $\eta$. Table 6 summarises the corresponding results.

Solution 1 performs better in both performance indicators. Furthermore, in the range of w between the points of maximum efficacy (orange circle) and maximum efficiency (orange diamond), $\varepsilon _1$ and $\eta _1$ are monotonically decreasing and increasing, respectively. Since, in this range, the absolute local slope of $\varepsilon _1$ is much lower than that of $\eta _1$, it is possible to have a slight loss in efficacy (i.e. less displacement reduction) while retaining high performance in efficiency (i.e. less mass added). Consequently, different stiffener designs can suit the different joints of the robot depending on the magnitude of the torsional load that affects the joint itself. In addition, the displacement on the end-effector has been evaluated considering the robotic system in the configuration depicted in Fig. 4 and mounting the two proposed stiffener on the first joint. Table 7 reports the comparison of the two displacements on the end-effector for both solutions with respect to the joint without stiffener.

Solution1 performs better also in terms of positioning accuracy of the end-effector. Furthermore, the proposed design for solution 1 can be easily manufactured starting from a cylindrical tube. Finally, this element can be mounted on the gear transmission joint without changing the assembly procedure so far defined. Figures 21 and 22 show the assembled simplified model of the joint for solutions 1 and 2, respectively.

Fig. 21

3D CAD model of the joint with the solution 1

Fig. 22

3D CAD model of the joint with the solution 2

5 Conclusion

This paper presents two designs for increasing the torsional stiffness of a gear transmission joint of a fully actuated cable-driven hyper-redundant robot. First, a summary of the fully actuated cable-driven hyper-redundant robot and a description of the joint are reported. Then, the critical aspects of the mechanical system are highlighted and the geometrical constraints the possible stiffener solutions have to satisfy are defined. Subsequently, a tube-shaped and a rectangular-shaped stiffeners are introduced. Exploiting the geometrical constraints, a single independent variable defines the whole geometry of the stiffeners. Performing a sensitivity finite elements analysis by varying such independent variables, the performance of the two solutions is compared in terms of added mass and displacement reduction. The tube-shaped stiffener provides a more significant displacement reduction while adding less mass. In addition, in a wide range of the independent variable, the displacement reduction and the added mass are anti-correlated. Consequently, different tube-shaped designs can suit the different joints of the robot depending on the magnitude of the torsional load that affects the joint itself. Moreover, the effect that the tube-shaped design has on the end-effector positioning accuracy is greater than that of the rectangular-shaped one. Finally, the proposed tube-shaped design can be easily manufactured and mounted.

Future works will include the stress analysis to perform a multi-parameters optimisation of the tube-shaped stiffener to obtain a uniform design removing the material that do not transfer loads at the supports further reducing the mass and displacement. Next, a physical prototype of the fully actuated cable-driven hyper-redundant robot implementing the proposed stiffener will be developed and tested to verify its performance. Furthermore, a significant improvement could be obtained by employing the generative design combined with the additive manufacturing technology to develop a miniature polymeric version of the prototype.