1 Introduction

Plants spend their whole life in the same place where their seed germinates; thus, to survive, they must adapt to different or even hostile environments [1,2,3]. Their adaptability is a consequence of their particular interaction with the surrounding environment, which involves distributed sensing, structural materials with adaptable stiffness, energy-saving mechanisms, movements without muscles, and indeterminate growth [2, 3]. Growth is the mechanism plants use to explore the surrounding environment in search of water and nutrients, and it is the most common and most important movement of plants [2], which occurs both above and under the ground. Regarding above-soil growth, the most interesting category from an engineering point of view is represented by climbing plants that can develop tendrils and vines searching for support and anchoring to it [2, 4]. On the other side, underground growth is carried out by plants’ roots which are very efficient diggers able to penetrate the soil and move around obstacles with very low effort and with unparalleled performances among living beings [1, 5].

The main advantage of this locomotion strategy of plants is that they grow from the tip, while the already grown part does not move, always maintaining the same shape without being affected by the tip motion [2, 3]. Growth is achieved by adding new cells at the root tip, and the growth direction is controlled by increasing the material on the opposite side with respect to the turning direction. This is carried out either by increasing the number of cells or with a cellular elongation mechanism, as shown in Fig. 1. Since only the tip moves relatively to the ground, the force required for locomotion is considerably reduced because the inertia force and the energy dissipation due to friction are minimized [1, 6,7,8]. This benefit is particularly evident for underground locomotion where friction with the soil is the predominant force [5]

Fig. 1
figure 1

Turning mechanism in plants’ roots

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Therefore, plant growth is a locomotion strategy suitable for every kind of terrain, allowing movements in narrow environments and through the soil, so it represents an interesting source of inspiration for robots designed for operations in unstructured or unknown environments, which find application in search & rescue, environmental monitoring, medicine, and exploration of places with limited accessibility or harmful to humans, e.g., archaeological sites or extraterrestrial planets [9,10,11].

Several growing robots have been developed, inspired by plant roots or tendrils which can elongate their bodies even tens of times the initial length [7, 10] and which differ from each other by the adopted actuation mechanism.

A robotic mechanism for soil penetration inspired by plant roots consists of a rigid hollow shaft covered by a thin skin that covers the internal surface of the cylinder and everts around its tip covering part of the external surface like a sleeve [5], and another strategy to mimic the roots’ growth is the deposition of new material by 3D printing adopted by the Plantoid robot [12,13,14]. A similar approach consists of using photopolymerization, a fluid is pushed through a cylinder to lengthen the robot and the already-grown part is solidified with UV light [15, 16]. A growing robot can also be built by using a special sprocket chain, which can be switched between flexible and rigid [17].

A different kind of growing robot, instead, is made of soft materials and is actuated by a fluid. Soft robots can move through obstacles adapting their shape to the surrounding elements because their body, made of deformable materials, is characterized by very high flexibility, and the materials constituting them are generally lightweight and can absorb most of the energy when they are involved in collisions [18]. Therefore, these robots can interact safely with obstacles, and collision can be useful to redirect the robot’s growth direction in an unknown environment [19]. Moreover, a fluidic actuation allows a faster deployment than material deposition, and the robot’s grown body can be retracted at the end of its use.

Many growing soft robots are actuated pneumatically, they grow by inflating air inside cylindrical chambers, and they can be composed of two balloons that can be inflated independently, giving directionality to growth [20, 21] or by a structure folded in an accordion-like fashion which elongates as air is blown inside [22]. A different principle of actuation for these robots is tip eversion: initially, the material of the body is folded and stored inside a dedicated case; then, as the robot is actuated, the material unfolds, elongating the robot [8, 23].

The Grow-Hose-I consists of an outer cylinder made of slippery cloth and two flat inner tubes bent at 180°. Air is inflated inside the inner tubes inducing the robot elongation, and turns are achieved by blowing air at a different pressure in the two tubes [24]. Another remarkable example of a pneumatically-actuated growing robot has a very deformable structure composed of a thin vessel folded inside itself. When air is blown inside the vessel, the reversed part unfolds, flowing out of the tip, elongating the robot’s body. The new material is added only at the tip, like during the roots’ growth. The bending system is constituted of some control chambers placed laterally with respect to the principal vessel, and they actuate a series of latches [7].

A robot exploiting a similar principle consists of a deformable vessel folded inside itself, with some Pneumatic Artificial Muscles (PAM), which constitute the bending system distributed along the robot’s surface [6]. A similar tip-everting robot is used for burrowing in the sand, and it steers using tendons placed on the outer diameter. Inside the inner diameter of the robot, a tube is positioned, from which pressurized air flows to the robot’s tip, where it fluidizes the surrounding sand, considerably reducing the drag force exerted by the material and decreasing the force required to dig [25, 26]. Steering can also be achieved thanks to a series of pouch motors that shorten when pressurized and are distributed along the robot’s external surface [10]. The same working principle is adopted by RoBoa, which elongates and deflects by actuating segments of pneumatic actuators and valve terminals [9]. Soft-growing robots also find application in medicine, like MAMMOBOT, a miniature growing robot used to inspect mammary ducts. In the middle of the robot, there is a more rigid catheter which is folded thanks to tendons actuated from servomotors placed on the robot’s base and give directionality to the robot’s growth [27]. Another principle to obtain robot elongation is storing the ungrown part of the body in a structure placed at the robot’s tip. The robot pushes forward this device which also hosts the bending system, a welding machine that welds the robot’s body reducing the inflatable length of one side and creating an asymmetric robot growth [28].

A different growth strategy consists of independently controlling the length of four tubings, allowing the robot to steer and move in 3D space [29]. A similar actuation principle is used by a soft crawling robot inspired by the leech, whose body is composed of three parallel inflatable chambers independently actuated, giving the robot the ability to stretch and bend in any direction, thus allowing 3D locomotion [30].

The robot presented in this article consists of a soft plastic cylinder that everts unfolding from the tip, the robot is pneumatically actuated, and its elongation is controlled from a fixed base. Part of the external surface of the cylinder is folded between Velcro straps, which can be opened selectively by pneumatic actuators at the robot’s tip, allowing asymmetric growth and achieving turns. This steering mechanism is the main innovation of the robot since it relies on a simple functioning principle that is easy to realize but very effective and allows controlling not only the turning direction but also the turning angle.

The remaining part of the article is organized as follows. In Sect. 2, the design of the robot is presented; in Sect. 3, the kinematic model of the robot is described; in Sect. 4, the assessment of Velcro straps is presented; in Sect. 5 the control system of the robot is shown, in Sect. 6 the results of the experimental tests are presented, and Sect. 8 is dedicated to the conclusion.

2 Design of the Robot

The soft-growing robot object of this research is designed to move in different environments, exploiting its peculiar feature of tip growing, with the already-grown part maintaining the same shape.

This prototype of the robot moves only in a plane, as its bending system does not allow bending in different directions; nevertheless, with simple enhancements to its bending system, rotations in three-dimensional space can be achieved.

The robot is mainly constituted of three parts:

  • A static base where the folded part of the robot’s body and the electronic components are placed;

  • The robot’s body that grows during functioning;

  • A structure at the tip that hosts the bending mechanism.

2.1 Robot Body

The robot’s body is the part that mimics the roots’ growth as it can elongate, and it is constituted by a deformable thin-walled polyethylene cylinder. In the initial state, one extremity of this cylinder is reversed, and from the inside, it is pulled toward the other extremity. The two extremities are sealed so that an air chamber is created. When a sufficient amount of air is blown inside this chamber, the plastic cylinder begins unrolling from the tip and increasing the length of the body.

This elongation strategy has the main advantage of roots’ growth: the lack of dynamic friction between the surrounding environment because the grown part is always still. This locomotion strategy is entirely different from traditional ways of locomotion, and the pneumatic actuation of the robot allows for maintaining a simple structure with the ability to exert high forces. The operating principle of the robot is shown in Fig. 2.

Fig. 2
figure 2

Functioning principle of robot elongation

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The cylinder is characterized by a thickness \(t = 0.1\) mm, and its diameter is \(D = 79.8\) mm. The maximum pressure it can resist is \(p_{max} = 140\) kPa, much higher than the pressure necessary to unroll. The mechanism used to give directionality to the robot’s elongation is also inspired by the plants’ roots. Roots elongate the cells on one side of their tip to obtain turns, creating an asymmetry in their growth. The bending system of the robot is positioned directly on the plastic membrane of the cylinder, and it consists of a pattern obtained by the repetition of a single module. In each module, part of the material of the cylinder is kept folded and periodically stored, forming equally spaced rings. When the robot unfolds, if the ring of this stored material at the tip is partially unfolded, the cylinder stretches asymmetrically, so the robot curves. The length of each module is inversely proportional to the number of modules per unit of length, so it is inversely proportional to the number of degrees of freedom of the robot.

In these rings, the material is kept folded by re-closable Velcro straps, distributed axially and tangentially, as shown in Fig. 3. The distance between two consecutive rings of fasteners is 50 mm, and the axial distance between two fasteners of the same pair is 3 mm.

Fig. 3
figure 3

Highlight to the Velcro straps that keep part of the robot’s body folded

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The working conditions of the bending system depend on the dimensions of the Velcro pairs. The load they are subject to can be classified as cleavage because the stresses are concentrated on the external edge of the Velcro strap, so their radial length does not affect their resistance, which only depends on their circumferential length. Conversely, the radial length of the straps determines the quantity of material stored in each fold. For a small curvature radius, storing a significant amount of material in the folds is necessary, so long Velcro fasteners are desirable. Nevertheless, having long Velcro straps increases the rigidity of the robot’s body considerably, leading to tip-clogging problems. Again, the fasteners’ width is a compromise between avoiding tip-clogging problems and having rings that resist the pressure inside the robot. The chosen dimensions of the Velcro straps are listed in Table 1

Table 1 Dimensions of Velcro straps
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2.2 Bending System

The Velcro straps at the robot’s tip can be selectively opened by pneumatic actuators, which are positioned in the structure shown in Fig. 4.

Fig. 4
figure 4

Structure at the tip hosting the balloons that act as pneumatic actuators

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This structure is a hollow cylinder positioned between the plastic vessel’s internal and external diameters. The friction on the inner side of the cylinder pulls the structure forward, and it guarantees that it is always placed at the robot’s tip. The top of this structure is characterized by two wedges that facilitate the flow of the unfolding plastic membrane and prevent the structure from getting stuck behind the fastener rings.

The pneumatic actuators must comply with strict constraints since they should provide enough force to open the Velcro strap, and at the same time, they must be small enough to fit inside the structure. The pneumatic actuators are realized by attaching a balloon to an elastic tube, which develops along the robot’s body, and its extremity is connected to the same source of pressurized air used to inflate the robot’s body.

2.3 Base of the Robot

The base contains all the components needed to let the air inflate the robot, maintain its internal pressure at the desired level, control the robot’s elongation, and activate the pneumatic actuators of the bending system. The pressurized air source is a compressor connected to the base. The principal element of the base is a rigid pressure vessel connected to one extremity of the robot. The folded part of the robot is folded inside this vessel, wound on a winch attached to a servomotor that controls the robot’s body elongation. The internal pressure of the robot and the pneumatic actuators is controlled with solenoid valves and measured with the pressure sensor BMP280. The electronic board used to control the valves’ opening and the servomotor rotation is an Arduino Uno.

3 Kinematic Model

Because of the continuous nature of the robot, the development of a control algorithm can be challenging; however, the robot object of this work is modular as it is constituted by the repetition of a single module. Therefore, this allows adopting a more straightforward approach to the solution of its kinematics.

First, the final position that every single module reaches is calculated for all the possible bending shapes; then, the obtained coordinates are moved from a local reference frame to the global reference frame. A single module’s position depends on the turning angle achieved by opening the Velcro fasteners. Therefore, it is necessary to develop a model that relates the number of opened fasteners to the relative angle between each module and the previous, which is the turning angle \(\theta\).

The first step involves finding the relation between the number of opened pairs and the opening angle \(\alpha\), which is half of the angle subtended to the arc of circumference free to elongate, as shown in Fig. 5.

Fig. 5
figure 5

Section of the cylinder with white rectangles representing open fasteners and black rectangles representing close fasteners

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Since the radial length of the Velcro pairs is much smaller than the cylinder diameter, the rectangular shape of the fasteners can be approximated as a circular sector without significantly affecting the result. Thus, the function relating the number of opened Velcro pairs and the angle \(\alpha\) can be obtained with simple geometric considerations. Since the robot moves on a horizontal plane, the symmetry about the horizontal axis should be maintained, and the number of opened Velcro pairs can be either odd or even according to the orientation of the ring with respect to the ground. In Fig. 5, an odd configuration with 8 pairs is represented with 3 opened pairs and 5 remaining closed. In an odd configuration, the horizontal and the vertical axes cut in half the Velcro straps, whereas in an even configuration, the axes cut in half the space between Velcros.

The opening angle \(\alpha\) is calculated as follows:

$$\begin{aligned} \alpha = {\left\{ \begin{array}{ll} 0, &{} \text {for } n = 0 \\ \dfrac{nw+(n+1)b}{D} = \dfrac{(n+1)\pi }{N}-\dfrac{w}{D}, &{} \text {for } 0< n < N \\ \pi , &{} \text {for } n = N \end{array}\right. } \end{aligned}$$
(1)

with b circumferential distance between the pairs, w the pairs width, N the total number of pairs, n the number of opened pairs, and D external diameter.

This model is based on the assumption that the elastic deformation of the plastic membrane and the Velcro fasteners is negligible since the internal pressure is much smaller than the yield stress of the material.

After opening some fasteners, the axial stress caused by the internal pressure makes the cylinder bend, achieving a turn. During bending, the module rotates about point B, as shown in Fig. 6, which is located between the open and the closed sides of the ring and acts as a pin.

Fig. 6
figure 6

Geometry of a module during a turn

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The segments a, b, and c of Fig. 6 are on the external surface of the robot, and they always remain parallel to its axis. When the robot grows straight, they are equal and have the same length as the module. Conversely, when the robot turns, the length of b remains equal to the module length, but the segment c slightly shortens. This occurs because the rotation about point B slightly compresses the closed side; however, the module cannot rotate freely because of the internal pressure of the robot, which opposes the closed-side shortening, as shown in Fig. 7a, where \(\Delta c\) represents the shortening on the closed side. Since a force opposes the rotation of the module, the open side does not elongate completely, as shown in Fig. 7b.

Fig. 7
figure 7

Shortening of the module during turns

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The shortenings depend on the distance from point B along the horizontal coordinate, with a zero value in point B and two maxima in points A and C, called \(\Delta o\) and \(\Delta c\), respectively. They can be expressed as functions of the bending angle \(\theta\) as follows:

$$\begin{aligned} \begin{aligned} \Delta _c&= (D-g) 2 \sin {\frac{\theta }{2}}\\ \Delta _o&= 2l-g\theta \end{aligned} \end{aligned}$$
(2)

with l equal to the axial length of the material stored between the Velcro straps and g projection on the horizontal axis of the open arc of the ring:

$$\begin{aligned} g = \frac{D}{2} (1-\cos {\alpha }) \end{aligned}$$
(3)

From Eq. 2, it can be noted that a sharper turn would increase the closed side shortening \(\Delta c\) and decrease the open side shortening \(\Delta o\), and vice-versa. The turning angle \(\theta\) is the angle at which the bending moments generated by the axial forces on open and closed sides reach an equilibrium.

The axial stress is considered constant throughout the section for the bending moment calculation, and the shortening ratios are assumed to be proportional to the bending moments. Since the axial stress is constant, the ratio between the shortenings \(\Delta c\) and \(\Delta o\) depends only on the leverage of the axial stress \(\sigma\) along the arc of circumference. Thus, the generated bending moment can be calculated as the integral along the circumference of the stress times the distance between the element of the circumference and the segment b projected on the horizontal plane:

$$\begin{aligned} M = \iint _A \sigma \left( x-x_b \right) \textrm{d}A \end{aligned}$$
(4)

where x is the horizontal component of the segment \(\overline{BA_2}\), \(x_b\) is the horizontal coordinate of the point B, and A is the area of the cross-section of the external side of the cylinder, as shown in Fig. 8. Thus, the generated moments on the closed side \(M_c\) and on the open side \(M_o\) are:

$$\begin{aligned} M_c&= \int _\alpha ^{2\pi -\alpha } \int _{R_i}^{R_e} \sigma \left( r \cos {\phi } -x_b \right) r \, \textrm{d}r \, \textrm{d}\phi \end{aligned}$$
(5)
$$\begin{aligned} M_o&= \int _{2\pi -\alpha }^{\alpha } \int _{R_i}^{R_e} \sigma \left( r \cos {\phi } -x_b \right) r \, \textrm{d}r \, \textrm{d}\phi \end{aligned}$$
(6)

where r is the radial coordinate and \(\phi\) is the angle coordinate.

Fig. 8
figure 8

Meaningful points of each module

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The integral of Eq. 4 can be solved using polar coordinates, giving the following result:

$$\begin{aligned}{} & {} \left\{ \begin{aligned} M_c&= \sigma (\pi -\alpha ) \left( R_e^2-R_i^2\right) x_b + \frac{2}{3} \sigma \left( R_e^3-R_i^3\right) \sin {\alpha }&\quad \text {for } 0 \le \alpha< \pi /2 \\ M_c&= \sigma (\alpha -\pi ) \left( R_e^2-R_i^2\right) x_b + \frac{2}{3} \sigma \left( R_e^3-R_i^3\right) \sin {\alpha }&\quad \text {for } \pi /2 \le \alpha < \pi \end{aligned} \right. \end{aligned}$$
(7)
$$\begin{aligned}{} & {} \left\{ \begin{aligned} M_o&= -\sigma \alpha \left( R_e^2-R_i^2\right) x_b + \frac{2}{3} \sigma \left( R_e^3-R_i^3\right) \sin {\alpha }&\quad \text {for } 0 \le \alpha< \pi /2 \\ M_o&= +\sigma \alpha \left( R_e^2-R_i^2\right) x_b + \frac{2}{3} \sigma \left( R_e^3-R_i^3\right) \sin {\alpha }&\quad \text {for } \pi /2 \le \alpha < \pi \end{aligned} \right. \end{aligned}$$
(8)

where \(M_c\) is the moment on the closed side, \(M_o\) is the moment on the open side, \(R_e\) is the external diameter of the cylinder, and \(R_i\) is the internal diameter of the external side of the cylinder, i.e., \(R_i = R_e -t\), with t equal to the thickness of the plastic membrane.

Having assumed that the shortenings are proportional to the bending moments and that the axial stress \(\sigma\) is constant and equal for both sides, it can be written that:

$$\begin{aligned} \frac{\Delta _c (\alpha , \theta )}{\Delta _o (\alpha , \theta )} = \frac{M_c (\alpha )}{M_o (\alpha )} \end{aligned}$$
(9)

Equation 9 can be combined with Eqs. 2 and 3, obtaining an equation that relates the bending angle \(\theta\) to the opening angle \(\alpha\):

$$\begin{aligned} M_c(\alpha ) \left( D - g(\alpha ) \right) 2 \sin {\left( \frac{\theta }{2}\right) } = M_o(\alpha ) \left( 2l - g(\alpha ) \theta \right) \end{aligned}$$
(10)

The equation is solved numerically because it is implicit, and the result is shown in Fig. 9, and the opening angles that can be obtained with 8 Velcro straps are listed in Table 2.

Fig. 9
figure 9

Relation between the opening angle and the bending angle

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Table 2 Bending angles for different opening angles
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Opening just one pair does not influence the robot’s orientation, and opening more than 5 pairs is detrimental to the structure’s resistance and achievable only with large pneumatic actuators. Moreover, the maximum bending angle is obtained with 4 opened pairs, and increasing the opening angle \(\alpha\) is useless. Therefore, an even configuration is adopted for the robot, considering opening only two or four pairs.

Due to the shortening of the cylinder occurring on both sides of the robot, opening Velcros not only generates a turn but also slightly affects the elongation of the module of the robot because of the dependence on \(\theta\) of the position of point F. The position of F can be calculated considering that the length of segment b remains unchanged during turns, so the length of the segments \(\overline{BM_1}\) and \(\overline{BM_2}\) is not affected by the module rotation and it is equal to the length of the module, which corresponds to \(d+q\) where d is the axial distance between two consecutive Velcro strap rings and q is the thickness of a Velcro pair.

Having calculated the relationships between the number of opened Velcro pairs and the bending angle and between the number of opened Velcro pairs and the position of F, it is possible to compute the position of every point of each module with respect to a local reference frame only with simple geometric considerations, as shown in Fig. 8 and listed in Table 3. Point \(M_1\) is the projection of point B on the horizontal axis, point \(M_2\) is the ending point of segment b, and point H is the intermediate point of the mid-cross-section of each module.

Table 3 Geometry of the module
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The final step to solving the kinematics of the robot is to assemble the structure in a global reference frame to obtain the final configuration. The origin of the global reference frame is positioned in point O of the first module. The points O, the origins of each local reference frame, are positioned in point F of the previous module. The angle of each module with respect to the horizontal axis \(\phi _i\) is the sum of all the bending angles of the previous modules.

$$\begin{aligned} \left\{ \begin{aligned} x^G_{O,i}&= x^G_{F, i-1} \\ y^G_{O,i}&= y^G_{F, i-1} \\ \phi _i&= \phi _{i-1} + \theta _i \end{aligned} \right. \end{aligned}$$
(11)

Since the global reference frame coincides with the first module’s local reference frame, it is possible to recursively transform the coordinates from a module’s local reference frame to the previous module’s local reference frame using a roto-translation matrix until the first module.

$$\begin{aligned} \begin{Bmatrix} x^{L,i-1}_{F,i} \\ y^{L,i-1}_{F,i} \\ 1\\ \end{Bmatrix} = \begin{bmatrix} \cos {\theta _i} &{} -\sin {\theta _i} &{} x^{L,i}_{F,i-1}\\ \sin {\theta _i} &{} \cos {\theta _i} &{} y^{L,i}_{F,i-1}\\ 0 &{} 0 &{} 1\\ \end{bmatrix} \begin{Bmatrix} x^{L,i}_{F,i} \\ y^{L,i}_{F,i} \\ 1\\ \end{Bmatrix} \end{aligned}$$
(12)

Therefore, the position of the robot’s tip, coincident with the point F of the last module for a robot with m modules, is obtained as shown in Eq. 13.

$$\begin{aligned} \begin{Bmatrix} x^{G}_{F,i} \\ y^{G}_{F,i} \\ 1\\ \end{Bmatrix} = \left( \overset{\curvearrowleft }{\prod _{i=1}^m} \begin{bmatrix} \cos {\theta _i} &{} -\sin {\theta _i} &{} x^{L,i}_{F,i-1}\\ \sin {\theta _i} &{} \cos {\theta _i} &{} y^{L,i}_{F,i-1}\\ 0 &{} 0 &{} 1\\ \end{bmatrix} \right) \begin{Bmatrix} 0 \\ 0\\ 1\\ \end{Bmatrix} \end{aligned}$$
(13)

4 Stress Assessment

A stress distribution model along the robot is developed to compute the force acting on Velcro straps to calculate their resistance because it is not desirable that a Velcro pair on the already grown part of the robot opens. The force generated by the internal pressure transmits axially along the structure, and the fasteners are pulled from one extremity, partially or totally opening after their limit is reached. The Velcro straps are DualLock made by 3 M, they can be considered rigid, and the load acting on them is cleavage, allowing high stresses that, for this model, are equal to 2.98 N/mm. This stress was assessed by attaching a mass to the extremity of the Velcro strap by means of a wire and measuring the load that caused the Velcro opening. The resistance of the fasteners to this type of action is not dependent on the length but only on their width.

Fig. 10
figure 10

Equilibrium of forces on a module

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The maximum pressure inside the robot \(p_w\) generates axial stress propagating along the cylinder’s wall to the base. Therefore, the force can be considered concentrated on the tip, where the only component is axial. To simplify the model, the tip surface is approximated as a perfect circle with the same diameter as the cylinder. Exploiting the modularity of the robot, the loads acting on each module can be computed as a function of the loads acting on the following module. The forces acting on each module can be considered concentrated on point F, and the reaction forces at point O can be computed with a static equilibrium, as shown in Fig. 10 and Eq. 14, where \((d+q)\) is the folded module length.

$$\begin{aligned} \begin{Bmatrix} N_O \\ T_O \\ M_O\\ \end{Bmatrix} = \begin{bmatrix} \cos {\theta } &{} \sin {\theta } &{} 0\\ -\sin {\theta } &{} \cos {\theta } &{} 0\\ -(d+q)\sin {\theta } &{} (d+q) \left( 1+\cos {\theta } \right) &{} 1\\ \end{bmatrix} \begin{Bmatrix} N_F \\ T_F\\ M_F\\ \end{Bmatrix} \end{aligned}$$
(14)

The forces acting on the last module m are:

$$\begin{aligned} \left\{ \begin{aligned} N_F^{(m)}&= p_w \pi \frac{D^2}{4} \\ T_F^{(m)}&= 0 \\ M_F^{(m)}&= 0 \end{aligned} \right. \end{aligned}$$
(15)

Hence, it is possible to compute the forces acting on each module i exploiting the modularity of the robot in the same way as already done for the kinematics, as shown in Eq. 16.

$$\begin{aligned} \begin{Bmatrix} N_O^{i} \\ T_O^{i} \\ M_O^{i}\\ \end{Bmatrix} = \left( \overset{\curvearrowright }{\prod _{j=m}^i} \begin{bmatrix} \cos {\theta _j} &{} \sin {\theta _j} &{} 0\\ -\sin {\theta _j} &{} \cos {\theta _j} &{} 0\\ -(d+q)\sin {\theta _j} &{} (d+q) \left( 1+\cos {\theta _j} \right) &{} 1\\ \end{bmatrix} \right) \begin{Bmatrix} p_w \pi \frac{D^2}{4} \\ 0\\ 0\\ \end{Bmatrix} \end{aligned}$$
(16)

The axial stress is constant along the section, whereas the stress induced by the moment varies radially. However, the thickness of the cylinder is so small that it can be considered constant along the radius and equal to the maximum value calculated on the outer side, and it is approximated as constant tangentially for the whole fasteners’ width. The stress assessment assumes that the fasteners pair is located at the most external points with respect to the vertical plane, bearing the maximum stress induced by the momentum. This is the case for an odd orientation; nevertheless, it can also be used for an even orientation since it ensures more safety. The maximum axial stress is compared with a cleavage resistance, which is multiplied by the cylinder thickness to allow a valid comparison.

Considering a flowing pressure \(p_w\) equal to 11 kPa, and knowing the kinematics of the robot for a given trajectory, it is possible to evaluate whether the Velcro straps resist or not.

In Fig. 11, two different configurations are shown. In Fig. 11a, the fasteners resist the stress, whereas, for the configuration in Fig. 11b, the high leverage creates a high-bending moment that opens one Velcro pair.

Fig. 11
figure 11

Robot shape in different configurations

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5 Control System

The control system regulates the growth of the robot through continuous servomotor rotation, maintains the internal pressure at the desired level, and inflates the pneumatic actuators at the right time with the right amount of air to obtain a direction change at the tip. The servomotor rotation is continuous, and its velocity is controlled in feedforward.

The robot’s elongation is controlled by the servomotor rotation; thus, the internal pressure is not set to a particular value, but it is sufficient that the control system maintains it in an interval. The lower limit is imposed by the flowing pressure, i.e., the minimum value of pressure that allows the robot to advance; on the other side, the upper limit is defined by the Velcro fasteners’ resistance, which cannot be opened by just the internal pressure of the robot. An external compressor provides the pressurized air, and a two-ways solenoid valve is used to control the flow inside the robot. On one side, they are connected to the pressurized air source; on the other side, they are connected to the robot. They can be either open and let air flow from the source to the robot or closed and prevent air from flowing. Since the valve can assume only two states, open or close, the control is straightforward: the valve is opened when the pressure drops below the lower limit because of the volume increase caused by the robot elongation, and it is closed when the pressure rises over the upper limit to prevent fasteners from opening, as shown in Fig. 12.

Fig. 12
figure 12

Pressure variation during robot functioning

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The pressure inside the robot is measured with a pressure sensor BMP280 that communicates through the \(I^2C\) protocol with the electronic board controlling the valves’ opening and closing. The board, the motor, and the valves are all powered by the same battery at 12 V; a MOSFET is present between each valve and the battery, acting as a switch, and a diode is connected in parallel to prevent possible returning current due to the high inductance of the solenoid valves. Two similar valves control the air flowing into the pneumatic actuators that open the fasteners, and they are controlled by an input of a joystick. The force generated by the pneumatic actuators depends on their internal pressure, related to how much air is infilled. Since they cannot be provided with internal pressure sensors due to their small dimensions, their control is feedforward. The parameter the controller acts on is the time the solenoid valves let air flow. The relation between the duration of valve opening and the number of Velcro straps unfastened has been found experimentally.

In Fig. 13, a block diagram of the control algorithm is presented. The black arrows represent signals, whereas the thick light-blue arrows represent airflow.

Fig. 13
figure 13

Block diagram of the control algorithm

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6 Experimental Tests

The designed robot has been assembled, and it is shown in Fig. 14, where the robot body is deflated. The structure at the tip holding the pneumatic actuators has been 3D-printed using ABS, and the Velcro fasteners have been glued to the external surface.

Fig. 14
figure 14

Robot base and body

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In Fig. 15, the base is presented, highlighting all its components.

Fig. 15
figure 15

Base with all its components

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To assess the kinematic model, the turning angle obtained for every possible number of Velcro straps opened was measured. The robot body is superimposed to a graduated angular chart, as shown in Fig. 16, and to improve accuracy, a long cylinder was used with only one Velcro strap ring attached to its external surface.

Fig. 16
figure 16

Turning angle measurement method

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The turning angles obtained experimentally for each opening angle are listed in Table 4. The reported values are the averages obtained after six measurements for each number of pairs opened.

Table 4 Bending angles for different opening angles obtained experimentally
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Finally, the pneumatic actuators that pen the Velcro straps have been tested, measuring the time required to inflate the actuators until they open the Velcro pairs. This required time is 4.8 s to open 4 pairs and 2.6 s to open 2 pairs. In Fig. 17, a sequence of images of the pneumatic actuator opening the straps is shown.

Fig. 17
figure 17

Sequence showing the fasteners’ opening. The actuator is inflated (a) until it reaches and opens the fasteners (b); then, it is deflated (c), and the robot advances, making a turn (d)

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7 Discussion

The turning angle measurement has revealed that the kinematic model is accurate when many straps are opened, whereas it is slightly underestimated when only a few straps are opened, as shown in Fig. 18. This mismatch is due to the simplifications made in the model, such as the approximation of the geometry of the Velcro ring to calculate the opening angle \(\alpha\). Another variable that was neglected in the model is the difference in axial stiffness between polyethylene and Velcro because of the presence of a stiffer material like Velcro on the external surface of the cylinder could affect the shortening on the open side \(\Delta o\). Moreover, another effect not considered in the model is the elastic deformation of polyethylene which alters the forces acting on the material and the geometry of the system.

Nevertheless, this discrepancy does not affect the performance of the control of the robot because, for trajectory planning, the real bending angles can be taken into account in place of the theoretical angles.

Fig. 18
figure 18

Comparison between predicted and measured bending angles

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Finally, the overall functioning of the robot is tested to assess the validity of the functioning principle. In Fig. 19, a sequence of images showing the robot growing and advancing is presented. For this test, the curves are always made by opening four Velcro pairs to maximize the curvature angle.

Fig. 19
figure 19

Sequence showing the robot growing. In the beginning, the robot is folded inside the base (a); then it starts unfolding and turns left (b); it goes straight, and then it turns right (c); finally, it goes straight and turns left (d)

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The final configuration reached by the robot is compared with the one predicted by the kinematic model in Fig. 20. It can be observed that the kinematic model is accurate as it correctly predicts the position and orientation of every module, with an error on the final position of the tip of 55 mm with a robot composed of 17 modules.

Fig. 20
figure 20

Comparison between the predicted and the experimental configuration of the robot

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This test was repeated three times, and the position and orientation of each module were measured and compared with the ones predicted by the kinematic model. In Table 5, the average distance from the base and absolute orientation of each module are reported.

Table 5 Average position and orientation for each module
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8 Conclusion

In this work, a growing biomimetic robot has been designed, taking inspiration from plant roots, which move by growing, and their locomotion strategy is characterized by excellent energy efficiency because the energy dissipations due to friction are minimized. The presented robot grows by tip eversion, and its body consists of a thin cylinder of polyethylene. The selected material is soft, so the robot has the ability to adapt its shape to the obstacles present in the environment and can move and explore narrow and constrained places.

The main innovation of the robot is its turning mechanism based on the selective opening of Velcro fasteners driven by the inflation of pneumatic actuators. A kinematic model of the robot motion in the plane was developed to find the relation between the number of opened Velcro straps and the turning angle and characterize the turning mechanism, and the validity of the model was confirmed by the experimental results. The implemented control algorithm synchronizes a servomotor and the opening of a valve to control the robot elongation and controls the valves for the airflow inside the actuators.

The robot was realized and tested, confirming the validity of the functioning principle and the reliability of the kinematic model.