An efficient soil penetration strategy for explorative robots inspired by plant root circumnutation movements

This paper presents a comparative analysis in terms of energy required by an artificial probe to penetrate soil implementing two different strategies: a straight penetration movement and a circumnutation, which is an oscillatory movement performed by plant roots. The role of circumnutations in plant roots is still debated. We hypothesized that circumnutation movements can help roots in penetrating soil, and validated our assumption by testing the probe at three distinct soil densities and using various combinations of circumnutation amplitudes and periods for each soil. The comparison was based on the total work done by the system while circumnutating at its tip level with respect to that shown by the same system in straight penetration. The total energy evaluation confirmed an improvement obtained by circumnutations up to 33%. We also proposed a fitting model for our experimental data that was used to estimate energy needed by the probe to penetrate soil at different dimensions and circumnutation amplitudes. Results showed the existence of a trade-off among penetration velocity, circumnutation period, and amplitude toward an energy consumption optimization, expressed by the lead angle of the helical path that should stay in the range between 46° and 65°. Moreover, circumnutations with appropriate amplitude (~10°) and period (~80 s) values were more efficient than straight penetration also at different probe tip dimensions, up to a threshold diameter (from 2 mm to 55 mm). Based on the obtained results, we speculated that circumnutations can represent a strategy used by plant roots to reduce the pressure and energy needed to penetrate soil. The translation of this biological feature in robotic systems will allow improving their energetic efficiency in digging tasks, and thus open new scenarios for use in search and rescue, environmental monitoring, and soil exploration.


Introduction
Engineering has looked at plants for making innovations for a long time (e.g. Velcro ® , Lotus Effect ® ). Recently, plants and their roots have been used as a model to design and build robotic technologies, including mechanisms and control solutions (Kim et al 2010, Ulrich et al 2010, Mazzolai et al 2011, Sadeghi et al 2014, Mazzolai 2017. Plants, being sessile organisms, have evolved different strategies of movement, based on an indeterminate growth, and capabilities to adapt their structures to the surrounding environments for anchoring and exploration, providing an important source of inspiration. Many biological observations, with the appropriate interpretation, can be relevant to engineering, including robotics. For instance, robots inspired by plant roots could be useful for subsoil exploration, penetration, and monitoring tasks (Sadeghi et al 2013). An improvement in performing these tasks can be achieved through the implementation of peculiar movements (i.e. circumnutations) in plant root-like robots. Circumnutations, also known by the general term nutations, are elliptical, circular, or pendulumlike movements performed by the plant organs active in growth (i.e. apical parts of shoot and apical part of roots) (figure 1). This phenomenon is induced by a differential elongation rate at the opposite flanks of the organ (Migliaccio et al 2013). Circumnutations have been known since Darwin's studies, but their role in roots is still not completely understood. Among several speculations, it also seems that they have an important role in facilitating soil penetration An efficient soil penetration strategy for explorative robots inspired by plant root circumnutation movements (Fisher 1964). In Oryza sativa L. a better establishment of seeds in paddy fields was observed when root tips rotated at high frequencies performing a spiral growth (Inoue et al 1999). Other studies, maybe less obviously connected to circumnutation roles, reported experiments with (i) mechanically stimulated roots, and (ii) unimpeded roots exposed to externally applied ethylene. These studies (Moss et al 1988, Sarquis et al 1991 revealed similar reactions on root growth: inhibition of longitudinal cell expansion, and promotion of lateral expansion. Ethylene is a phytohormone typically produced by plant roots in specific stress conditions, including mechanical stress (Veen 1982). These observations confirm that plant roots, when mechanically stimulated, produce ethylene that affects cells growth. Later studies demonstrated that ethylene increases root wave amplitude in Arabidopsis thaliana (Buer et al 2003). Root waving is a phenomenon present in very fine roots-e.g. Arabidopsis thaliana-and is ascribed to the interaction among gravity, tactile stimulation, and circumnutations (Mullen et al 1998). All the above observations support a correlation between soil mechanical properties and plant root circumnutations: high soil impedance induces stress in plant roots; in stress conditions a root produces ethylene; the increase of ethylene affects its growth and increases wave amplitudes; an alteration of waving can be linked to an alteration of circumnutation movement (Mullen et al 1998, Oliva andDunand 2007).
The role of circumnutations in plant roots can be empirically demonstrated by measuring resistances perceived by roots with and without circumnutations. Various studies have been conducted in the past to estimate the root resistance in different soils by means of penetrometers (for a review on this topic refer to Bengough and Mullins (1990) and Bengough et al (2011)). Despite the fact that penetrometers cannot provide the actual resistance experienced by plant roots, and usually give an overestimation (Bengough and Mullins 1990), these techniques still represent a valid method for soil penetration and comparative analysis with innovative solutions inspired by plant roots. For instance, Bengough et al (1991) proposed a comparison between a straight fixed tip and a straight rotating tip showing a lowering of the total forces perceived by the tester machine when penetrating sandy soil. Similarly, Jung et al (2017) analyzed the drag force needed by a conical probe to penetrate an artificial granular soil and glass beads in straight fixed penetration, by rotating the cone while preserving the vertical position. They also obtained a reduction of the forces when tip rotation was applied. However, studies on the effectiveness and efficiency of root circumnutations during soil penetration are still limited. Circumnutation is in fact a more complex movement with respect to a simple rotation in the vertical position of a probe tip, since it involves a misalignment of the tip with respect to the normal axis that is characterized by an amplitude, distance from the nutation axis, and movement frequency. In a previous study, we compared the axial forces necessary for a robotic root pushed from its top to penetrate an artificial soil using straight movements or circumnutations (Del Dottore et al 2016). In those experiments we measured up to 80 times less force when circumnutations were used. This demonstrates that such movement is convenient in terms of forces, inducing less stress on the penetration device and reaching higher depth compared to a straight probe movement with the same amount of external applied force. However, in this previous work we neglected the power needed to generate the probe tip's rotary motion.
The present work aims at estimating the efficiency of a root performing circumnutations during soil penetration in terms of energy consumption. We estimated the improvement by means of comparative experiments using a setup purposively developed. With respect to our previous work, we tested the mechanism in a real soil and in a larger (24 cm of diameter) and deeper (30 cm of penetration) environment, scaling down probe and tip dimensions (from 4 to 2 cm) to reduce chamber size effects (Salgado et al 1997, Bolton et al 1999. We isolated the axial forces applied at the probe tip and torque to evaluate the total energy consumed by the system. We present and discuss the experimental results and the model developed to fit data and predict root behavior at different tip sizes and nutation amplitudes. This work intends to contribute to the scientific discussion on the role of root circumnutations, and lays foundations for designing optimized plantlike robots for drilling and soil exploration.

Experimental setup
We estimated the energy that a sensorized artificial root requires to penetrate soil with and without circumnutations by performing two series of experiments. The energy of a straight penetration is given by the perceived axial force, times displacement. In the case of circumnutation, we obtained the total work combining two movements: straight penetration and a rotation. The relative energy was then obtained by simultaneously evaluating the axial force and the torque with the corresponding displacements, through a customized setup (figure 2). The setup included a rigid rotary shaft embedded inside a tubular metallic skin (mimicking the root lateral body) to protect the internal shaft from the lateral friction induced by interaction with the soil along the probe; a parabolic tip (connected to the rotary shaft) to mimic the root tip, with a diameter of 2 cm; an axial force sensor (LSB200-QSH01809-50 kg S-Beam Load Cell from FUTEK Advanced Sensor Technology, Inc.) embedded close to the tip to acquire only the force at tip level; a gear motor used to generate the rotary motion of the shaft; and a torque sensor (TFF350-FSH00645-100 in-lb reaction torque sensor from FUTEK Advanced Sensor Technology, Inc.) directly mounted on top of the rotary shaft to measure the torque applied at the tip level. The setup measured the torque necessary for rotating the tip, and eliminated the effect of any internal disturbing friction or external soil lateral friction from this measure. To accomplish this, the internal frictions on the rotary shaft were minimized by means of rotary bearings between the shaft and tubular skin, while at the same time the tubular skin protected the rotation of shaft from direct contact with soil, and thus eliminated the friction between soil and the rotating shaft. A slider mechanism was used to provide the penetration with constant velocity (a tensile force tester-Zwick/Roell Z005-was used in this case while the slider moved at a controlled and constant speed). Figure 2 depicts an overview of the whole setup and its architecture for data acquisition. The axial load cell, torque sensor (mounted on the bottom and top of the shaft), and an electronic board rotated with the shaft to avoid wire twisting during penetration. This board was equipped with two operational amplifiers (LMC6482 from Texas Instruments) to amplify the sensor signals; a microcontroller with a 16 bit analog to digital converter (CY8C3866 from Cypress); a wireless module to transmit data (RFD21733 from RF Digital Corp USA); and a battery for self-powering to avoid any connection to external power supplies. Data sent were collected by a twin wireless module in a second electronic board connected via USB to a PC for data storing. This board was equipped with a 32 bit microcontroller (PIC32MX460 from Microchip) and a motor driver (LV8548MC from ON Semiconductor) to command the shaft rotation by a metal gear motor (50:1 Metal Gearmotor 37D × 70L mm with a 64 CPR Encoder, from Pololu). Motor speed was controlled by implementing a PID control on the microcontroller. The system was managed by a custom user interface developed in VB.NET that permitted us to set the circumnutation period (T) and save the acquired data.

Experimental protocol and parameters
All penetration experiments were performed in topsoil, which was dried at room temperature for 48 h and then filtered with a 4 × 4 mm net to eliminate big particles and filaments. The experiments were Figure 2. Architecture of electronic and mechanical components of the experimental setup composed by an ad hoc penetrometer (on the left) assembled at the tensile force tester. The penetrometer is mainly composed by a rotary shaft, a parabolic tip, torque and axial sensors, and a tubular metallic skin that acts as an external protective skin to avoid direct contact of the soil with the internal rotary shaft, thus avoiding lateral friction during probe movements. conducted over 6 weeks in a closed and conditioned environment at 26-28 °C and 63-68% humidity. The soil was inserted into a cylindrical container with a 24 cm diameter and a total height of 52 cm. For each penetration experiment, soil was prepared at three different compactness levels with 0.38 g cm −3 , 0.40 g cm −3 , and 0.42 g cm −3 densities (ρ). We performed the penetration tests with different circumnutation parameters by placing the tip at (α) 10° and 20° with respect to the vertical axis, and varying a complete rotation period (T ) among 30 s, 60 s, 120 s, and 240 s. From our previous experiments (Del Dottore et al 2016), we observed that the penetration speed slightly affected the measured forces on the tip. Based on this consideration, we set the probe's velocity (v) at 40 mm min −1 , and its penetration at a maximum 30 cm; thus, each experiment lasted 7.5 min. Tests were repeated at least nine times for each experiment (table A1).

Data acquisition and analysis
Data were acquired every 100 ms (∆t) and saved in csv format with each row containing axial force (F Vi ) and torque (τ i ). Axial work (W V ) and rotary work (W R ) were then obtained with the following formulas: Where i identifies the ith row acquired from 0 to 30 cm of penetration, v is the penetration velocity, and ω = 2π/T is the angular velocity. The instantaneous work can be defined as the work for each time step as follows: The total work is simply obtained by Normality of data (W Tot ) was tested with the Shapiro-Wilk test (appendix), considering separately each set of samples grouped on the basis of combinations of period, amplitude, and soil density (table A1). Since the groups were not normally distributed, Levene's test (appendix) was then used to test the equality of variances (tables A2-A4). Considering the presence of non-normal distributions and unequal variances, Friedman's test (appendix) was used on the averaged energy for each group to compare the effects of period, amplitude, and density factors (tables A5 and A6). Huang et al (2004) presented a model for a conical penetrometer that showed how the resistance at the cone in homogeneous soil stabilized after 0.1-0.3 m of penetration. The point of stabilization changed according to different pressure and soil internal friction factors. The same behavior has been observed in penetrometer tests conducted over five different laboratories (Bolton et al 1999). This steady state can also be identified with the pressures obtained in our straight penetration experiments (figure 3(a)), where, after 0.15-0.25 m traveled, the force needed by the tip for pushing remains approximately constant; consequently, the instantaneous energy, obtained as in (3), reaches a maximum value (table 1). Figure 3(b) shows the almost linear increase of tip pressures obtained from our experiments as a function of soil density, at steady state. Similarly, Huang et al (2004) obtained a linear increasing cone resistance at steady state as a function of soil pressure. Table 2 shows an example of exclusively axial forces (coaxial with the shaft) perceived by the tip in straight penetration and with circumnutations, both taken as averaged forces at steady state (after half of the total experiment time-from 0.15 m); the percentage of improvement is from 66 to 78% in the three soils.

Experimental results
However, in the presence of circumnutations, forces involved in the rotary movement should be considered, and consequently the energy is affected. Figure 4 shows an example of instantaneous energy consumed by the system along the entire penetration with circumnutations, together with straight penetra- tion energy, in the three soils at different densities. With circumnutations, we can observe that after reaching a depth of ~0.15 m the energy stops increasing and tends to oscillate around a stable value. This kind of behavior confirms the existence of a limit depth below which soil resistance converges to an asymptote (Tardos et al 1998, Guillard et al 2013Guillard et al , 2015. The total energy obtained for each experimental group for the whole displacement of 30 cm is presented in figure 5(a). Analyzing the work done by the system, the energy consumed increases with soil density. Although we varied the tip angle and nutation period, we obtained an improvement for most of the combinations (figure 5(b)), up to 33.12% at 10° on the lighter soil with respect to the straight penetration (table 3).
Energy is then presented as a function of the nutation period (figure 6). In figure 6 the energy of straight penetration is used for comparison (black, solid and dashed lines). The same conclusions can been reached from the curves and statistical analysis with a 95% confidence interval: all period variation affect the energy among different amplitudes in the same way (P = 0.1447 in ρ = 0.38 g cm −3 and ρ = 0.40 g cm −3 , P = 0.1870 in ρ = 0.42 g cm −3 ). Meanwhile, amplitudes provide (P = 0.0455) slightly different effects among periods on ρ = 0.38 g cm −3 and ρ = 0.42 g cm −3 , but a greater effect (P = 0.3173) on ρ = 0.40 g cm −3 . Furthermore, the effect of different soil density is extremely significant (P = 0.0001). On each curve it is possible to observe a minimum located below each black line in a range from 60 s to 120 s, where lower energy is obtained.

Fitting model
To better describe the behavior emerging from our results, we analyzed the system's motion and the forces involved at steady state, where those forces became stable and reached the maximum value. The tip, while moving in soil with fine granularity, performs a circular movement with angular velocity ω and a linear movement coaxial to the shaft with velocity v. The combination of these two movements provides a helical path (figure 7(a)). For the formed helix, it is possible to define pitch P 0 = v · T and lead angle ε = a tan (P 0 /πd). The probe tip has approximately a parabolic shape with diameter d n and length l.
The problem of defining the total pushing force during the movement is equal to defining the resistance force acting only on the probe tip. Different from in viscous fluids, in granular soil there are complex combinations of force chains among particles that rearrange and propagate anisotropically and inhomogeneously when an object moves in it (Albert et al 1999, Takada andHayakawa 2016). This condition makes the problem of modelling an object moving into real soil still an open question. However, some work on this has been done; for instance, the independence between penetration velocity and the force needed by an object to move into a granular medium has been demonstrated when velocity is low (Albert et al 1999, 2001, Guillard et al 2015, as well as the independence from depth when a limit depth is reached (Tardos et al 1998, Guillard et al 2013, 2015, thus allowing approximation of the pressure in the soil below such limit depth with ~ρga, where ρ is the bulk density, g is the gravity acceleration, and a is a coefficient that should take into account soil parameters, e.g. internal friction angle, compressibility and water content. As a matter   (3)) experienced in straight penetration (light colors) versus circumnutations (dark colors). The example of 10° amplitude at 60 s of period is here presented.
of fact, bulk density alone cannot express soil compactness and pressure among different types of soil (clay, sand, loam, etc). Having a relatively low penetration velocity in our experimental conditions, we can assume independ ence from v, which is an assumption particularly true for plants; since we are considering only the part of the experiment in steady state condition, we can also assume independence from depth. With these assumptions and previous considerations, during lateral movement the tip needs to exert a force proportional to the pres sure exercised by the soil on the lateral area of the tip: Here, µ r is a dimensionless coefficient, and C r (m 3 ) is defined as follows: where f (x, y) is a logical function that defines the inclusion of a point in the area of the parabolic tip exposed to lateral soil interaction (f (x, y) = 1 if the point (x, y) is into the grey area in figure 7(b); f (x, y) = 0 otherwise), and x is the radius for each column of the tip. The integral    (4)) for the displacement in the steady state condition (from 0.15 to 0.3 m). The minimum point is indicated with a dark circle on each curve, along with the corresponding energy and period. Points were obtained by imposing a lower bound equal to 0 on µ v and µ r.
is the cross sectional area in a plane of the tip exposed to lateral soil interaction. The rotary work can be obtained by the following: where ∆s R is the distance traveled by the tip along the helical path where T exp is the time spent in the vertical distance traveled (∆s v ). For the axial movement, the force needed for pushing the tip (F v ) can be approximated with a function proportional to the area (A v ) directly affected by the axial pressure (P v ) (Whiteley et al 1981): In (10), P v is the pressure obtained from the energy of the straight penetration experiments. Nevertheless, taking into account that nutations affect soil compaction and particle rearrangement, we corrected this pressure with a factor depending on nutation parameters, that are expressed in equation (10) by ε, and are weighted with a dimensionless coefficient (µ v ): the high frequency of nutation movement can lighten the pressure under the penetrometer (ε → π/2 and cos ε → 1), while with a low frequency, soil pressure tends to reflect the pressure of straight penetration (ε → 0 and cos ε → 0).

The area (A v ) of the bottom-down projection of the tip (figure 7(c)) is obtained by
where d n is the tip diameter, and x * is obtained by subtracting the shaft radius from the maximal x point of the parabolic shape. The axial work can then be obtained by The vertical and rotational work found experimentally are fitted separately and then summed together. The total work is indeed given by the sum of (8) and (12).
Unknown parameters of the model are µ v and µ r , which are obtained through data fitting. The total work fitted at steady state is presented in figure 8. Each curve has a minimum under which the energy remains lower with respect to the energy found in straight penetration. On average, the point of minimum energy is located at ~77 s with α = 10 • , and ~122 s with α = 20 • . For the corresponding amplitude these periods bring an energetic advantage with respect to the straight penetration in every soil. An asymptotic behavior is present at the extremities: where Y is a finite value that represents the energy associated with a straight penetration performed with the tip bent. Considering the parameters obtained by the fitting, we can observe fluctuations of µ v around a relatively stable value: 1 with α = 10 • , and ~0.82 with α = 20 • ; µ r has instead an almost linear increase over soil density (with linear equation µ r = 1.1 · 10 3 ρ − 3.9 · 10 5 for α = 10 • , and µ r = 2 · 10 2 ρ − 6.9 · 10 4 for α = 20 • ). Both decrease with increasing amplitudes in each soil, suggesting that not only the soil compactness changes with different densities, but also that circumnutation amplitudes can affect soil compactness: a bigger amplitude means a greater area exposed to soil interaction, thus resulting in dislocation of a greater amount of soil, and therefore less pressure (lower µ v and µ r ) compared to a smaller amplitude (higher µ v and µ r ).

Setup and model considerations
This paper presented an experimental method and a data fitting model aimed at improving the scientific knowledge on the role of plant root circumnutations during soil penetration. Limitations of the experimental approach are the number of repetitions that can be performed, and the repeatability of the experimental conditions over a long period of time, since environmental conditions, such as temperature and humidity, can influence soil characteristics. We have confined this risk by performing experiments in an air conditioned room and within a relatively short period of time (6 weeks). Another limitation is probably induced by the protocol used for soil compaction, i.e. rotating the container, tapping on the side, and pressing from the top the soil. This procedure shows at the highest density (ρ = 0.42) some pressure inhomogeneity from one layer to another that is caused by an apparent decrease of soil pressure in straight penetration (figure 3, from 0.15 to 0.25 m). However, reaching a steady state in soil pressure has been demonstrated with lower densities, and the literature confirms the presence of a steady state penetrating granular soil (Tardos et al 1998, Guillard et al 2013, 2015, thus suggesting that going deeper or with a more uniform density, it could be possible to reach pressure stability also with ρ = 0.42. Our experiments aimed at comparing penetration tests performed with and without circumnutations, and varying nutation amplitude and period. Moreover, we proposed a fitting model based on the analysis of the forces required at steady state by the probe in axial configuration and in rotary movement. Since in our experiments only the soil density was known, all the other soil-related characteristics (i.e. soil internal friction and cohesion, particles interaction, and soilsystem interaction properties) were embedded in two unknown coefficients, µ r and µ v , which were then found by our fitting function. Despite its simplicity, the model demonstrates its efficacy by fitting the energy values found at different experimental conditions (i.e. soil density, and nutation amplitude and period).Variation of µ v and µ r parameters also let us appreciate the variability of soil compactness that increases with soil density (increasing µ v and µ r ) and decreases with circumnutation amplitude (decreasing µ v and µ r ), suggesting the reduction in soil pressure induced by circumnutations. We estimated the range of angular velocities to which the system optimizes energy to move in three different soil densities at a fixed penetration velocity. As demonstrated by our results, with a circumnutation period less than 30 s the system significantly increases the rotary work with consequent high energy consumption; on the other hand, for a circumnutation period greater than 120 s, the system does not obtain any particular advantage in using circumnutations during penetration. This observation suggests that a balance between penetration and angular velocities is required for energy optimization. Since the relation between penetration and angular velocities is included in the lead angle (ε) value, we can consider energy as a function of ε (figure 9) and find minimum energy values with ε between 46° and 65°. With ε being a function of nutation period T and amplitude α, if we fix a period of 77 s, we can find that circumnutation amplitudes between 6° and 13° can provide an optimal energy value.
Although we developed the setup to purposively reduce errors in the experimental phase, penetration tests in confined environments are influenced by side effects on soil resistance values. A complete absence of side effects is obtained with a minimum ratio between the container and tip diameter of 40 (Bolton and Gui 1993). In our experiments, we obtained a ratio of 12 for a probe tip with a 2 cm diameter. This ratio was mainly imposed by the dimension of a commercially available load cell that fits in our design of the container, which has to be kept maneuverable. Despite these constraints, the obtained results demonstrate the absence of side effects because of the presence of the steady state. Besides, the comparative analysis between the two different penetration strategies with the same probe further reduces the importance of these side effects, since we are not interested in defining an absolute cone resistance value.
Additionally, we disregarded experiments using tips with different diameters in the same container deliberately to avoid the occurrence of side effects. Such effects can only be nullified by preserving a minimum ratio between container and tip diameter as a function of soil relative density (Salgado et al 1997, Bolton et al 1999. We estimated the influence of diameter variation on root behavior using the developed model function. Figure 10 shows an example (in soil ρ = 0.40) of predictions obtained with different nutation amplitudes assuming the behavior of µ v and µ r to be linear. Function evaluations are performed by scaling up or down the entire tip size (d n and l), fixing period T to 77 s, utilizing a constant ratio between pitch and tip diameter (v · T/d n = 2.6), having a constant number of circles performed (T exp /T = 2.9), and by obtaining µ v and µ r through linear fitting of the corre sponding values obtained at 10° and 20° for the three soil densities. As previously stated for the straight penetration, based on Whiteley et al (1981), we assumed that the force exercised on the tip of different dimensions is linearly dependent on the tip diameter (equation (10) with ε = 0). Results show that energies increase with tip diameter. All selected amplitudes are convenient for diameters comparable to those of plant roots (figure 10(a)) but the positive improvement of circumnutations is preserved only until a threshold diameter that varies with the amplitude is reached. The existence of an optimal ampl itude that minimizes energy consumption is evident in table 3, which shows the greatest improvement for each soil with an amplitude of 10° or 20°. The curves in figure 10(c) also confirm this trend where the lowest energy for a tip diameter greater than 3 cm was obtained with the amplitude of 11°, which brings an advantage for diameters less than 6 cm.

Circumnutations in plant roots
The obtained results lead us to assume that circumnutations may represent a key mechanism to generate cavity expansion and crack propagation in the soil plastic zone, i.e. the cylindrical zone surrounding the penetration cavity where stresses are sufficient to cause failure (Salgado et al 1997).
Study of the circumnutation role in plant roots required simplification of several conditions with respect to natural behavior. In primis, we used a faster penetration velocity compared to plant growth; besides, our system is forced to penetrate coaxially with the shaft while plant movements are influenced by environmental stimuli such as nutrients, water, or gravity, and also takes advantage of soil cracks. Moreover, root interaction with soil is completely different, in terms of friction, particle interactions, and the mechanism of penetration, compared with the effects generated by a mechanical device that moves in the same medium. Plant roots move in soil through cell division and elongation phenomena at their tip, and thus, they reduce stress and friction at this level. Through these simplifications, growing and elongation effects were reproduced by positioning the load cell close to the tip and by employing a protective skin, thus allowing the artificial system to only perceive forces and torque acting at the tip level. By doing so, a comparison between the natural and artificial mechanisms-in terms of efficiency of straight versus circumnutation-based soil penetration-is still possible and even sustainable.
From our previous biological experiments (Del Dottore et al 2016) we observed circumnutations in Zea mays primary roots with a period between 60-80 min and amplitudes with a radius of 0.2 mm on average. Assuming that circumnutations occur in the central elongation zone (Okada and Shimura 1990), which is the root region with the highest cell elongation rate, and placed in maize at about 4 mm from the tip (Ishikawa and Evans 1993), a radius of 0.2 mm corresponds to ~3° of nutation amplitude. Considering a growing rate for Zea mays of 1.59 mm h −1 (Del Dottore et al 2016), ε is ~52°, a value perfectly in the range of optimal ε found with our previously presented analysis.

Circumnutations for robotic roots
Soil is still an environment little explored in robotics, especially for objective difficulties in penetration. However, autonomous devices able to move in subsoil are of interest in either terrestrial and space applications, for exploration (mapping of an area) or source localization (water or other relevant substances), in environmental monitoring, rescue, and other tasks. The presented results confirm the idea that circumnutation is an advantageous mechanism in terms of energy needed to penetrate; indeed, our tests demonstrated a reduction up to 33% of work done by the probe while using circumnutations with respect to straight penetration. Even just considering the axial forces at the tip level, we obtained significant results: 78% less force with circumnutations in the case of a 10° amplitude and 60 s period. We also observed that the energetic advantage can be preserved through a variety of different tip sizes by tuning circumnutation parameters (figure 10). Clearly, an actual evaluation of energetic performance will depend on the efficiency of the actuators in use. An accurate energetic analysis should be done to evaluate, per each actuation system, the feasibility of the nutation mechanism and the effective power saving. All presented results and considerations suggest that robots, as well as autonomous penetrometer devices, can increase their performance by stress reduction and save energy by implementing this natural movement. Taking inspiration from plant roots we can effectively extract new technologies and strategies for subsoil exploration. Future works will be focused on integrating growing abilities (Laschi et al 2016, Sadeghi et al 2017 together with circumnutations in root-inspired robots, exploiting and maximizing the advantages of both strategies.

Appendix. Statistical analysis
The total energies (W Tot ) obtained for the whole displacement of 30 cm are tested to verify their significance. As first test, we should evaluate the normality of data. Since not all groups have the same cardinality, the Shapiro-Wilk test (Royston 1982(Royston , 1992(Royston , 1993a(Royston , 1993b(Royston , 1995 is used. Results are reported in table A1. Next, we should evaluate the heteroscedasticity (equality of variances) among multiple combinations of the groups (tables A2-A4). Since not all the groups Figure 10. Energy behavior as a function of tip dimension obtained from the fitting function, with different circumnutation amplitudes and a fixed period of 77 s. The example of ρ = 0.40 is reported in the figure. The same general behavior is obtained with ρ = 0.38 and ρ = 0.42. α = 0 • (black dashed line) corresponds to straight penetration. Diameter and tip length are scaled proportionally, and penetration velocity is adapted to maintain a constant ratio between pitch and tip diameter. Column (a) shows closed views of energies with a tip diameter of 2 mm (representative of Zea mays root diameter); column (b) shows a general view of energies from 0 to 20 cm; and column (c) shows a closed view of energies with a tip diameter of 2 cm (our experimental probe tip) up to 5 cm.
were normally distributed, Levene's test is used for heteroscedasticity (Leven 1960). The test is performed with the function vartestn in MATLAB with the Levene option test type. Equality of variances is accepted if the P value is greater than 0.05.
Due to the presence of non-normal and non-heteroscedastic groups, Friedman's test is used to test the effects of period, amplitude, and density factors (tables A5 and A6). The test is performed in MATLAB with the designated function friedman. Table A2. Variances of straight penetration (G 1 , G 10 , and G 19 ) are compared with the variances of nutating groups for each soil separately.