A general 3D model for growth dynamics of sensory-growth systems: from plants to robotics

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, which direct growth in the direction of 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. 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 which 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 both the response to an external stimulus together with circumnutations. Lastly 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 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, as well as novel control systems for newly developed self-growing robots.