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Edited by: Thomas Speck, University of Freiburg, Germany

Reviewed by: Manfred Bischoff, University of Stuttgart, Germany; Karl J. Niklas, Cornell University, United States; Wilfried Konrad, University of Tubingen, Germany

This article was submitted to Soft Robotics, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In recent years, there has been a rise in interest in the development of self-growing robotics inspired by the moving-by-growing paradigm of plants. In particular, climbing plants capitalize on their slender structures to successfully negotiate unstructured environments while employing a combination of two classes of growth-driven movements: tropic responses, growing toward or away from an external stimulus, and inherent nastic movements, such as periodic circumnutations, which promote exploration. In order to emulate these complex growth dynamics in a 3D environment, a general and rigorous mathematical framework is required. Here, we develop a general 3D model for rod-like organs adopting the Frenet-Serret frame, providing a useful framework from the standpoint of robotics control. Differential growth drives the dynamics of the organ, governed by both internal and external cues while neglecting elastic responses. We describe the numerical method required to implement this model and perform numerical simulations of a number of key scenarios, showcasing the applicability of our model. In the case of responses to external stimuli, we consider a distant stimulus (such as sunlight and gravity), a point stimulus (a point light source), and a line stimulus that emulates twining of a climbing plant around a support. We also simulate circumnutations, the response to an internal oscillatory cue, associated with search processes. Lastly, we also demonstrate the superposition of the response to an external stimulus and circumnutations. In addition, we consider a simple example illustrating the possible use of an optimal control approach in order to recover tropic dynamics in a way that may be relevant for robotics use. In all, the model presented here is general and robust, paving the way for a deeper understanding of plant response dynamics and also for novel control systems for newly developed self-growing robots.

Though the field of robotics has long been inspired from the capabilities of biological organisms, it is only recently that the plant world has become a source of inspiration, particularly due to the ability of plants to continuously change their morphology and functionality by growing, thus adapting to a changing environment (Del Dottore et al.,

Plants, on the other hand, excel at these types of tasks. Though plants exhibit a variety of types of movements as part of their interaction with their environment (Darwin,

In order to emulate these complex growth dynamics in a 3D environment in a way that is meaningful from the robotics standpoint, a general mathematical framework is required. Recently developed models of growth-driven plant dynamics are limited to specific aspects of tropisms or circumnutations. Bastien et al. have developed models for tropism in 2D, such as the AC (Bastien et al.,

Here, we present a general and rigorous mathematical framework of a rod-like growing organ whose dynamics are driven by both internal and external cues. Though this model is inspired by plant responses, it is not based on biological details and is therefore relevant to any rod-like organisms that respond to signals via growth, such as neurons and fungi. The model does not include elastic responses, but the mathematical framework we adopt here allows a natural integration of elasticity, which we plan to do in future work. The paper is organized as follows: section 2 describes the dynamical equations of our model based on a 3D description of an organ in the Frenet-Serret formalism, implementing differential growth as the driver of movement, and relating external and internal signals. In section 3, we present the numerical method required to implement this model, and in section 4, we perform numerical simulations of a number of key case examples, including responses to external stimuli, such as a distant stimulus, a point stimulus, and a line stimulus, as well as circumnutations (the response to an internal oscillatory cue). We also present an example where we superimpose two different types of cues, namely the response to an external stimulus and circumnutations. Lastly, in section 5, we consider a simple example illustrating the possible use of an optimal control approach in order to recover tropic dynamics in a way which may be amenable to robotics use.

In this section, we develop the dynamical equations that form the basis of our model. We first introduce a 3D description of an organ in the Frenet-Serret formalism and then detail the implementation of growth and differential growth as the driver of movement. Finally, we relate external and internal signals to differential growth, which drives the desired movement. We then show that our model is a generalization that consolidates different aspects of existing models, allowing the characteristic time and length scales of our model to be identified and discussed.

We model an elongated rod-like organ as a curved cylinder with radius

where

where κ is the local curvature of the curve, and

For the sake of legibility, we interchangeably omit writing the explicit dependence of variables on (

Geometrical definitions for a 3D cylindrical organ.

The Frenet-Serret framework describes the change in this local frame of reference as a function of the arc-length

so that the local coordinate system changes accordingly along the curve. Here, κ(

We now introduce growth, using similar definitions to those introduced in Silk (_{0} as the arc-length of the initial centerline of the organ, and the current arc length _{0}, _{0} in time, with initial conditions _{0}, _{0}. One can think of the arc-length _{0}, _{0} due to the growth of all previous parts of the organ (see _{0}, _{0} can be thought of as the Lagrangian, referential, or material coordinate and

where _{gz} as the length over which the growth rate _{gz} ≤ _{0}), the time derivative of Eulerian fields [functions of _{0},

Growth description. Illustration of a growing organ with a sub-apical growth zone, marked in green. The centerline (dashed line) can be parameterized by a material coordinate, _{0}, or by the arc-length, _{0}, _{1}, _{1}, _{3}, _{3}, _{2} and _{3} flow within the organ. _{2} flows out of the growth zone and will stay fixed.

As mentioned in the Introduction, plant tropisms are the growth-driven reorientation of plant organs due to a directional stimulus, such as light, gravity, or water gradient. In particular, the reorientation of the plant organ is due to differential growth, i.e., one side of the cylindrical organ grows at a higher rate than the other side, resulting in a curved organ. Following Bastien and Meroz (

Following this definition, for Δ(

In order to describe the active reorientation of an entire organ, we relate the shape of the organ and its growth dynamics, expressed by the dynamics of its local curvature,

where the equations have been linearized by assuming that the radius of curvature 1/κ is always larger than the radius of the organ (κ

Differential growth. Differences in growth rates across a cylinder lead to a change in curvature. At time _{0}, marked in green. The differential growth vector _{0}(1 +

In the last section, we represented the anisotropic growth pattern by the local differential growth vector

Environmental signals can be mathematically described as fields. For example, vector fields describe light and gravity, while a scalar field describes the concentration of water or nutrients, and the direction of increasing concentrations is again described by a vector field of the gradients. Lastly, tensor fields may describe stress and strain; however, we will not discuss these here since our model does not include elasticity. Here we focus on vector fields, where we can write the directional stimulus in the form

where we have defined _{⊥}(

Effective signal and response vector. An example of a signal that can be described by a constant vector field (such as sunlight and gravity) of the form _{0} is the magnitude of the stimulus and _{⊥} = _{0} sin (θ(_{⊥}(

Two central biophysical laws describe sensory responses to input signals, which we term here the sensitivity function λ(_{0}), referred to as the Weber-Fechner law (Norwich and Wong, ^{b}, known as Stevens' law (Stevens,

where the sensitivity function takes the effective stimulus sensed by the organ λ(_{⊥}(

However, it has been found that a so-called _{0} is the intensity of the bending, and ψ(

Together with Equation (9), Equation (12) completes our model for active growth-driven movements of rod-like organs in 3D taking into account external signals, internal cues (circumnutations), and posture control. For multiple stimuli, again assuming additivity, one can replace

Schematic of the governing equations. We present the main stages involved in the model.

Lastly, the distribution of sensory systems along the organ also requires attention. Sensory systems in plant organs are generally either distributed along the organ, providing local sensing (Sakamoto and Briggs,

Different models of growth-driven plant dynamics have been recently developed, encompassing different aspects of tropisms and circumnutations. Bastien et al. (

In order to compare with 2D models, we focus on the case where the dynamics of our model are restricted to a 2D plane, which occurs when the direction of the stimulus

identical to the ACE model developed in Bastien et al. (

We now consider Bressan et al. (

where λ > 0 is a constant measuring the strength of the response, similar to our tropic sensitivity, while ^{−η(t−σ)} is what they call a stiffness factor. The simplest way to compare with this model is by looking at its 2D projection. Taking

We note that this model considers accretive growth, where material is added at the tip, and elongation is disregarded. This means that growth is only taken into account implicitly as the driver of the tropic movement, and a material derivative is not required, which is a good approximation of the dynamics in certain cases (Bastien et al.,

Comparing Equations (15) and (16), we see that the equations are similar: the response, appearing on the l.h.s., is identical, and on the r.h.s., the tropic stimulus is represented by sinθ(^{−η(t−s)} represents a smooth growth zone with a characteristic size of 1/η: in the youngest parts (s=t at the tip) the stiffness factor is 1, while in older parts of the organ (as s goes to zero), the stiffness factor goes to 0. We also notice that Bressan et al. do not use a proprioceptive term, generally required for stable dynamics; however, they were able to circumvent this problem by using small growth zones.

In section 2.4, we show that in the case where the dynamics of our model are restricted to a 2D plane, our model recovers the ACE model developed by Bastien et al. (_{max}, and its inverse, the radius of curvature, corresponds to a characteristic length scale termed the _{c} = 1/κ_{max} = γ/λ, where γ and λ are the proprioceptive and tropic sensitivities, respectively. There are two time scales. One is associated with the time it takes for the organ to reach its steady state, termed the _{c} = _{v} = _{gz}λ. The ratio between the convergence length _{c} and the length of the growth zone _{gz}, as well as the ratio between the convergence time _{c} and arrival time _{v}, introduces a dimensionless number

Low values of _{c} > _{gz}, i.e., the growth zone is not big enough to contain the full arc-length associated with bending toward the stimulus with a given curvature, or alternatively that _{v} > _{c}, i.e., the organ dynamics converge before it is able to arrive to the desired orientation in the direction of the stimulus. High values of _{c} < _{gz}, i.e., the growth zone can contain the full bending, or alternatively that _{v} < _{c}, i.e., the organ arrives at the desired orientation before the dynamics converge, therefore also exhibiting damped oscillations. In other words, we see that the balance number

As stated in section 2, our model for active growth-driven dynamics, described by Equations (9) and (12) and schematically illustrated in

Here, κ_{1}(_{2}(

where ϕ is the angle between

In order to solve the dynamics, we integrate Equations (24) and (25).

The organ is divided into segments of length

We describe the location of the organ using the local coordinate system:

The dynamics of the organ is described through the evolution of the local coordinate system. We rewrite Equations (18)–(20) in matrix form, which describe the change in the local frame of reference as a function of

where D(

and U(

In order to integrate Equation (28) while keeping the orthonormality of the local frame, we take inspiration from Gazzola et al. (

Since U(

This can be interpreted as a rotation around the axis _{1} and κ_{2} to describe the organ in time. To integrate κ_{1} and κ_{2}, we discretize Equations (24) and (25), adopting the following numerical time and arc-length derivatives (where

leading to:

The growth speed appearing in the material derivative, _{gz} and uniform growth rate

in the case _{gz} ≥ (

finally resulting in κ_{1}(_{2}(

As discussed in section 2.2, growth is implemented via a material derivative with a local growth rate described in Equation (6), representing the elongation of cells in the growth zone, creating a one-dimensional growth flow within the organ. When cells reach a certain threshold size, they stop elongating, thus leaving the size of the growth zone _{gz} constant. Since the total length of the organ increases over time, in the numerical scheme, we add a new segment

where N(m) is the total number of segments in the organ at time step m, and therefore the total length is _{1}(_{2}(_{gz}_{1}(_{1}(_{2}(_{2}(

In the simulations presented in the next section, the initial conditions include a straight vertical organ κ(_{1}(_{2}(_{0} = 1.0 and a growth zone _{gz} = 1.0. Boundary conditions are defined with a clamped base κ(_{1}(_{2}(_{0} = 0.1 or λ_{1} = 0.05. The ratio of the proprioceptive and tropic sensitivity values substituted in Equation (17) correspond to balance numbers _{max} = λ_{0}/γ = 10, yielding κ_{max}_{gz} = 0.1. In the next section, we discuss simulations of specific cases. The code is freely available at

Here, we discuss various representative cases of internal and external cues. Since the differential growth term is the driver of the dynamics, it is the only term that needs to be defined accordingly. We present the specific form of the differential growth vector for each case, as well as a snapshot of a numerical simulation. Videos of the full simulation dynamics can be found in the

The simplest type of stimulus is a constant stimulus placed at infinity. In this case, the stimulus is a parallel vector field originating from direction

The sensitivity λ_{0} may depend on the intensity of the stimulus, for example, in the case of phototropism, following either the Weber Fechner or Stevens' Law, as discussed in section 2.3. This is not the case for gravitropism, since plants sense inclination rather than acceleration (Chauvet et al.,

Examples of numerical simulations for various scenarios. Here, we showcase snapshots of simulations for various cases. The subapical active growth zone is in green, while no growth occurs below that in gray. The arrows on the apex are the apical tangent direction

We consider the case of a stimulus whose source is a point located at

Here again, λ_{0} is constant in space; however, this can be generalized to depend on space, for example, in the case of a diffusive chemical where

We can generalize the point stimulus to any geometrical form. Here, we show an example of a stimulus in the form of an attracting straight line. Let us assume that the line is parallel to an arbitrary direction

The response vector will then be

where

Circumnutations are circular periodic movements of the tips of plant organs, generally associated with search processes, for example, climbing plants searching for a support or roots searching for nutrients. Unlike tropisms, these are inherent movements due to internal drivers, not external stimuli, and can be described as _{0} is the intensity of the bending, ψ(

In our simulations, we took ψ(

As already suggested in the example of a line stimulus, where a directional stimulus is added, we can consider multiple types of stimuli by assuming that they are additive. We present here another example based on plant behavior, where we consider an organ responding to a distant external signal while also exhibiting internally driven circumnutations. In this case, we simply add to Equation (45) the term for the distant stimulus in direction

A snapshot of the resulting dynamics is shown in

In this last section, we take a step back and consider a simple example illustrating the possible use of control theory to recover tropic dynamics—in a way that may be amenable for robotics use. In what follows, we no longer use the Frenet Serret formalism developed in this paper, relaxing the assumption of a constant arc-length parameterization. Instead, we consider the general case where the curve of the organ is parameterized using the Lagrangian coordinate _{0}, as described in section 2.2, without further reparameterizing the curve as it evolves over time. This general case may be pertinent to some robotics systems. We consider an organ with apical sensing, a fixed length L (neglecting an explicit account for growth, as discussed before), and dynamics restricted to 2D, similar to the case of apical sensing discussed in Bastien et al. (

We further limit the family of possible control strategies to those for which:

where

In this case, the cost function has a geometric meaning: when the dot product goes to zero, together with Equation (47), we have

where

for all

for all _{f}] and β ≥ 0, where

which is identical to the dynamics described in Bastien et al. (

In this work, we presented a general and rigorous mathematical framework of a rod-like growing organ whose dynamics are driven by a differential growth vector. We constructed the differential growth vector by taking into account both internal and external cues, as well as posture control, as schematically illustrated in

We ran numerical simulations of a number of key cases. In the case of the response to external stimuli, we considered a distant stimulus (such as sunlight and gravity), a point stimulus (such as a point light source), and a rod stimulus that emulates

While building a physical robotic representation that can behave as the model predicts is well out of the realm of current technology, the current model can be simplified so as to be relevant for current technologies, yielding limited behavior. As an example, current additive manufacturing technologies are generally limited to the addition of material at the tip, with no elongation. This accretive growth can be represented in our model by taking the growth zone to an infinitesimal size. In order to account for a robotic structure made of a number of rigid components with hinges, nodes, etc., the infinitesimal segments

Following this line of thought, we note that the framework presented here disregards parameters pertinent to robotic structures, such as energy, friction, weight, etc. In this paper, we present a simple example illustrating the possible use of optimal control theory in order to recover tropic dynamics in a way that may be relevant for robotics use. Optimal control theory optimizes processes where some cost function is minimized, and it is therefore useful in engineering problems. The example

This general framework allows a deeper understanding of plant dynamics in response to their environment. Indeed, while current investigations on tropisms are generally restricted to 2D, our model enables the quantitative study of tropisms in 3D, i.e., where single or multiple stimuli are placed outside of the organ plane. Furthermore, careful attention has been paid to relating environmental stimuli to differential growth, discussing stimuli with different physical characteristics categorized by their mathematical description, such as vector fields (light and gravity), and scalar fields (concentration of water or nutrients). Indeed, the latter finally allows a rigorous characterization of plant biosensors in tropisms that are less understood, such as hydrotropism and chemotropism, as well as a currently lacking quantitative analysis of their dynamics.

Understanding plant movements is essential for a rigorous understanding of plant behavior—a field that has only recently become the focus of research. Basic behavioral processes in animals are generally studied through their motor responses to controlled stimuli, and a solid understanding of plant movements (in response to both internal and external cues) paves the way to designing controlled behavioral experiments. For example, simulations incorporating both circumnutations and tropisms will allow quantitative investigation of the role of circumnutations in the successful search for nutrients or light.

Though the framework we develop here successfully describes various scenarios of growth-driven movements of plants, it of course differs from its botanical inspiration. One main difference is that here we do not consider branching. Furthermore, as noted throughout the text, this framework does not currently include mechanics or elasticity, disregarding any elastic responses of the organ to physical forces. However, this can be naturally implemented in the Frenet-Serret frame of reference (Chelakkot and Mahadevan,

The numerical code for simulations presented in this work is accessible at:

AP and YM developed the mathematical framework. AP carried out the numerical analysis and simulations and prepared all the figures and videos. FT, MP, and PM developed the optimal control calculation and the comparison with the model by Bressan et al. AP and YM drafted the manuscript. All authors contributed to the authoring of the final manuscript and contributed to the research design.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Renaud Bastien for helpful conversations.

The Supplementary Material for this article can be found online at:

The supplementary material includes one pdf file and 5 videos.

Includes three Appendices:

Derivation of growth dynamics both in the Frenet-Serret frame and the natural frame, relevant to sections 2.2 and 3.

Shows that our 3D simulations converge to the known analytical solution in 2D with an exponential growth profile.

Pontryagin's Maximum Principle, part of the calculation in the optimal control approach in section 5.

The videos show simulations for the different cases presented in section 4, and

Infinitely distant constant stimulus.

Point stimulus.

General stimulus geometry: twining around a line stimulus.

Internal processes: circumnutations.

Superposition of internal and external stimuli.