2. Robot design

The four robots presented in this article have been designed with the main goal of having a single print to highly simplify the manufacturing process and the assembly, as it allows us to simply add the tendons through their structure. The four designs are modular, made of a specific backbone unit that is then duplicated along the backbone up to the desired backbone length. For ease of comparison, this paper refers to prototypes with shared length (80.00 mm), external diameter (17.50 mm), and internal diameter (3.50 mm), set to match the design previously presented in ref. [Reference Perugini and Russo34]. However, these dimensions are purely representative, and the system can be scaled according to the capabilities and workspace of the available 3D printers, as well as printing-induced vibrations (which depend on printing speed and on the pressure exerted by the nozzle on the robot backbone while printing), the choice of material and further printing parameters such as infill percentage and pattern, printing temperature, plate temperature, and rafts/brims. Generally, designs from 4 mm up to 200 mm in external diameter can be found in literature, with the minimum size being limited by the minimum wall thickness that the 3D printer can manufacture and the maximum size bound by the 3D printer’s workspace.

The four designs have been obtained by using common geometries for compliant hinges, allowing for robot bending in one or two directions by reducing local stiffness; at the same time, regions with higher stiffness have been preserved to replicate the tendon-routing functionality of continuum robot vertebrae/disks. The four designs presented all achieve local stiffness variation by reducing cross-section area, according to profiles that allow for smoother or harsher changes: a sawtooth profile (V-notch, linear, Fig. 1), a square wave profile (T-notch, high/low, Fig. 2), a circle arc profile (hourglass, sine, Fig. 3), and a helical profile (helix, Fig. 4).

Figure 1.

CAD model of the V-notch design: (a) isometric view; (b) front view; (c) side view.

Figure 2.

CAD model of the T-notch design: (a) isometric view; (b) front view; (c) side view.

Figure 3.

CAD model of the hourglass design: (a) isometric view; (b) front view.

Figure 4.

CAD model of the double-helix design: (a) isometric view; (b) front view; (c) side view.

The first design, presented in Fig. 1, is a circular design with a V-shaped notch pattern on the sides. Consequent notches are designed to allow bending on two perpendicular planes, in order to obtain 3D motion. Fillets have been added at the inner extremity of each notch, allowing for more space for the tendon channels. The notches are set at a 45° angle to allow printing without additional support. Similarly to the other designs, the backbone length (80.00 mm) includes cylindrical segments (height of 2.00 mm each) at each extremity of the main body. An additional base (height of 4.00 mm) is used to attach the design to the actuation system, as well as adding stability and guaranteeing base plate adhesion during the print.

The second design, in Fig. 2, presents a T-shaped notch pattern. Similarly to the previous design, consequent notches are rotated by 90° around the backbone axis to allow bending on two perpendicular planes. The notches include fillets on their inner sides to facilitate the printing as well as making the prototype more resistant to fatigue induced by the bending of the robot. Additionally, it allows us to obtain a more precisely comparable result during the displacement finite element analysis (FEA), avoiding stress concentration at the corners of the notches.

The third design created in this article was inspired by the design of a traditional hourglass, as illustrated in Fig. 3. It presents a circular cross-section with varying outer diameter along the backbone, whereas the inner channel diameter remains constant. The thicker sections of this design are used for tendon routing, while the thinner sections facilitate the bending of the robot. Given its axial symmetry, this design can intrinsically achieve 3D motion without requiring alternating designs like the previous two models.

The fourth design, inspired by our DNA’s geometry, is based on a double helix. The two helical bodies present the same pitch and number of revolutions but differ in helix direction, as shown in Fig. 4. The resulting design, while less “discrete” in its development when compared to the V-notch and T-notch design, still results in alternating planes of bending generated by the helices’ intersections alternating along the two main axes of the cross-section.

As mentioned earlier, all four designs present the same main geometric features, with a cylindrical envelope with the same maximum outer diameter and inner channel diameter, as well as tendon-routing channel diameters and locations. These parameters are highlighted in the top view in Fig. 5, shared by all the designs (including [Reference Perugini and Russo34]). Throughout their conception, the secondary parameters of these designs have been tuned through FEA to obtain an equivalent behavior under load. As the aim of this paper is to compare the kinematic behavior of these four designs, the objective of this analysis is only to ensure that the selected geometries do not differ (i.e., same displacement of the control point at the distal end of the backbone), rather than fully simulating motion. For the actual robot characterization, experimental data are favored over FEA here due to the challenge of modeling highly nonlinear behavior such as tendon friction and superelasticity [Reference Raimondi, Russo, Dong and Axinte35]. As such, the design tuning has been set up by matching the displacement of the distal point of the backbone of each design under a set of known loads (up to 2 N) and with fixed base, through FEA, which was run in SolidWorks 2023, using the PBT material profile and with matching mesh parameters, solver, and boundary conditions (fixture on the lower surface of the base plate). As shown by the results in Fig. 6, the mechanical performance of these designs can thus be compared under equivalent conditions.

Figure 5.

Top view with characteristic dimensions for the proposed designs.

Figure 6.

Results of the finite element analysis of the four new designs as well as the design in ref. [Reference Perugini and Russo34] (on the left), showing an equivalent displacement under load for all the systems.

3. Actuation unit design

The selected actuation unit has been described in ref. [Reference Perugini and Russo34] and is composed of six layers, made out of six disks with a circular shape and a diameter of 75 mm. This allows for affordable and fast manufacturing through 3D printing or laser cutting, with an estimated cost of less than €1.00 per disk and a manufacturing time of less than 10 min (3D printing) or 30 s (laser-cutting). The disks are then separated by commercial electronic board spacers, making the design adaptable in case of changes in the choice of actuators, power supply, or controller (Fig. 7). The first disk at the top of the assembly is used to fix the base of the robot as well as route the antagonistic tendons into the robot segment. Underneath, the second disk routes the tendons out of the linear motors and avoids tangling and similar routing issues. The third and fourth disks are designed to fix the four linear servomotors that actuate the system (Actuonix PQ12). These motors have a 20 mm stroke, with a maximum output of 45 N; alternative linear motors, or rotational motors with pulleys, can also be used with minimal changes to the design of the actuation unit. As a control board, a Polulu Micro Maestro 6-Channel USB Servo Controller has been selected for this work; the microcontroller is fixed to the fifth disk. Alternative solutions include Arduino or Raspberry Pi boards, potentially with a PWM driver to control up to 15 servomotors if the system’s DOFs need to be scaled up. In the reported prototype, the Micro Maestro controller is linked to a computer via a USB cable to transfer the signal from the Maestro Control Center software to the motors. The fifth disk also allows the passage of wires through a 25 mm diameter hole. Finally, the six disk lodges the power supply for the controller and the linear servomotors, which comes in the shape of a 6V battery pack with four AA batteries but can be easily replaced with a rechargeable Li-Io battery or equivalent alternatives with no impact to performance.

Figure 7.

CAD model of the actuation unit.

The cost of these designs is estimated at 265€, including the material used for the manipulators (∼15 g) with the largest expense being the motors at 58.14€ each and the Polulu Micro Maestro controller at 27.68€, which, compared to a cost of 1682 US$ (equivalent to 1504,49€) in ref. [Reference Grassmann, Shentu, Hamoda, Dewi and Burgner-Kahrs32], denotes a significant cost reduction.

4. Operation and control

The robot has been modeled with piecewise constant curvature kinematics (PCCK), a common tool used in continuum and soft robotics [Reference Russo, Sadati, Dong, Mohammad, Walker, Bergeles, Xu and Axinte10]. Despite its limitations due to the lack of dynamic and material information on the robot, the model has been selected as the preferred tool in practical continuum robot applications because of its computational efficiency and scalability to multiple sections. PCCK describes the behavior of continuum robots by approximating the shape of their backbone to a circle arc. This assumption allows the definition of the backbone through two parameters only, namely, its angle of curvature $\theta$ and the direction of bending φ (see Fig. 8 for a graphical representation and Table I for the complete nomenclature). PCCK formulations can also be used to link tendon lengths, which are the actuation variables $l_{1}$ , $l_{2}$ , $l_{3}$ , and $l_{4}$ , to the motion at the robot’s tip, defined by the frame $\{S\}$ with its center in $(x, y,z)$ in the base frame $ \{S_{0}\}$ . The homogeneous transformation $\boldsymbol{T}\in \textrm{SE}(3)$ from $\{S_{0}\}$ to $\{S\}$ can be expressed by

Figure 8.

Piecewise constant curvature kinematic model for a tendon-driven continuum segment.

(1) \begin{equation}\textbf{T}\textbf{=}\left[\begin{array}{cc} \textbf{R} &\quad \textbf{t}\\ \textbf{0}_{\textbf{1}\times \textbf{3}} &\quad \textbf{1}_{\textbf{1}\times \textbf{1}} \end{array}\right],\end{equation}

where $\boldsymbol{R}\in \textrm{SO}(3)$ is the rotation and $\boldsymbol{t}\in \mathbb{R}^{3}$ is the translation along the segment arc. Parametrizing the motion by the bending angle $\theta$ and the direction of bending $\varphi$ , as illustrated in Fig. 8, the rotation from $ \{S_{0}\}$ to $\{S\}$ is

(2) \begin{equation}\boldsymbol{R}=\textrm{rot}_{z}\left(\varphi \right)\cdot \textrm{rot}_{y}\left(\theta \right)\cdot \textrm{rot}_{z}\left(-\varphi \right),\end{equation}

and the corresponding translation is

(3) \begin{equation}\boldsymbol{t}=\frac{l}{\theta } \left[\begin{array}{c} \cos \varphi \left(1-\cos \theta \right)\\ \sin \varphi \left(1-\cos \theta \right)\\ \sin \theta \end{array}\right],\end{equation}

where $l$ is the length of the backbone from {S} to {S0}. The transformation from bending angle and direction to tendon length can then be expressed as

(4) \begin{equation}\begin{cases} l_{1}=l-\theta r_{0}\sin \varphi \\ l_{2}=l+\theta r_{0}\cos \varphi \\ l_{3}=l+\theta r_{0}\sin \varphi \\ l_{4}=l-\theta r_{0}\cos \varphi \end{cases}\end{equation}

Table I.

Nomenclature.

This kinematic model can be used to control the motion of the prototype.