# Swarming Behavior Emerging from the Uptake-Kinetics Feedback Control in a Plant-Root-Inspired Robot
## 1 Introduction  
A decentralized control system is a system in which the components act on the basis of local information when accomplishing global tasks. In such systems, the collaborative behavior emerges from independent local decisions without the need for centralized processing [1]. This definition is intrinsically linked to the idea of self-organization. Many natural systems have been studied in terms of their ability to perform complex tasks without a centralized control but due to simple rules followed by many distributed agents with communication capabilities [2]. Well-known examples include ant colonies that are able to accomplish foraging tasks by following pheromone trails [3], or honeybees that indicate the direction of the nectar source by dancing [4]. However, systems that also have no nervous system are considered, such as bacteria that organize themselves in order to maximize nutrient availability [5]. The analysis of these behaviors has inspired routing algorithms [6], load balance problem solutions [7], ant colony optimization [8] or particle swarm optimization [9] algorithms. This approach typically provides a scalable and robust solution to large-scale complex problems. It has also opened the door to a relatively new discipline, called swarm robotics, which applies swarm intelligenceprinciples to robotics [10]. Collaborative exploration specifically in unstructured environments such as disasters or dangerous areas is a particularly important application of this discipline. Like animals, plants also need to explore the environment for foraging purposes. They actively interact with the environment perceiving, for instance, the presence of obstacles and adjusting their growth when mechanically stimulated [11; 12]. Moreover, they need to optimize their energy due to the uncertainty of nutrient availability. Specifically, production, mobilization and allocation among the tissues of photosynthesis products (e.g., carbon and sugars), that regulate plant growth and development, are highly affected by sugar and hormone signals in response to environmental cues [13; 14; 15]; in addition, nutrient uptake and usage are regulated according to the availability of the nutrient [16]. However, unlike animals, plant locomotion is irreversible, since it takes place through organ growth, which suggests that plants should focus more on decision-making activities compared to animals. Plants have been already taken as source of inspiration in engineering [17; 18; 19], and they have been also explored for optimization algorithms [20]. Macro-rules for the design of metaheuristics have been extracted for instance from pollination processes [21], the colonization of invasive weeds [22] or strawberry plant propagation strategies [23]. Plant roots have also been considered, in particular, their distribution in searching for optimal soil, water, and fertilizer conditions. Specifically, [PERSON] et al. [24] proposed the Root Mass Optimization Algorithm (RMO) where the search for optimality is driven by operators inspired by the concepts of growth and branching, and the search evolves through generations of roots, as with a classical genetic algorithm. Roots, which represent different initialization points in the search domain, can grow in conditions of optimal soil impedance, water, and fertilization. This optimal condition is monitored by a fitness function. In addition, roots can decide to generate a branch with a random probability in a random position; each root is then evaluated and the best are selected for the next generation. Similarly, [PERSON] et al. [25] proposed the Root Growth Algorithm (RGA) based on root branching and root hair growth operators. In this case, the length and distance of hairs and roots are also important in obtaining wider spatial distribution and increasing the diversity of fitness values. However, to our knowledge, in the robotic community, plants have not yet been explored as swarm intelligent systems, while at the same time extensively analyzing plant root behavior.