## 3 The Robots and the Communication
### The Minimal Communication System  
The communication system proposed in this paper, inspired by how mobile robots communicate using IR technology, is minimal because of the following main reasons:  
* The communication is local, with a remarkably small communication range. This means that two robots can communicate if their distance is lower than a threshold. The communication range is fixed to 80 cm (see [40]).
* Only one message can be received at each time instant. This means that, regardless of the number of neighbors, the robot and, thus, the CTRNN is only aware of one neighbor at each simulation step.
* The possible directions of message reception or the number of IR sectors are restricted to 4 sectors, highly complexifying the tasks.
* The received context corresponds uniquely to the current received message.  
Robots' controllers are fed by the received message and its context and elaborate a new two-dimensional message to be isotropically broadcasted using the IR transmitter. Before transmitting the message, it is subject to a quantization mapping that converts the raw message content to one symbol in the set \(\mathcal{C}\) (see Equation (1)).  
\[\mathcal{C}=\left\{0,\;\frac{1}{K-1},\;\cdots,\;\frac{K-2}{K-1},\;1\right\}^ {M} \tag{1}\]  
where \(M=2\) is the dimension of the message and \(K=4\). This leads to 16 possible symbols in the communication. A symbol in \(\mathcal{C}\) is randomly sampled assigning higher probabilities to the nearest symbols and lower probabilities to the symbols that are more distant. As proved in Section 5, 16 symbols are enough for the addressed applications and using the algorithms and techniques of this paper. It is shown how, even though there are 16 available symbols, the emerged communication uses, at most, only two of the symbols.  
Alternatively, the IR receiver is sectorized, meaning that the coverage is split into equiareal sectors so that the agent can distinguish the orientation from where a message is sensed. We propose a minimal communication system and, consequently, the number of sectors is fixed to 4 in the tasks. This means that the reception orientation is discretized to \(\theta_{RX}\in\{\theta_{r},\,\theta_{r}+\pi/2,\,\theta_{r}+\pi,\,\theta_{r}+3\pi/2\}\) radians. Two agents can directly communicate if their Euclidean distance is lower that 80 cm and the misalignment is lower than \(\pi/4\). By misalignment, we refer to:  
\[\mathit{misalignment}=\pi-\min\{|\theta_{RX}-\theta_{TX}|,\,2\pi-|\theta_{ RX}-\theta_{TX}|\} \tag{2}\]  
where \(\theta_{RX}\) and \(\theta_{TX}\) are the orientations of the sectors from where the message was received and transmitted, respectively.  
Moreover, we propose the incorporation of a communication state (\(\mathit{MODE}\) when referred as a binary variable) as follows: an agent can be either in send mode or in relay mode. Send mode means that the robot controller creates a novel message, based on the incoming message from its neighborhood, and broadcasts it to its vicinity. On the contrary, relay mode refers to broadcasting the input message sensed by the receiver. The commutation between these states will be performed by the agent controller. Moreover, a maximum number of hops is fixed so that old messages eventually stop from being relayed. Messages have an attached identifier of the robot that initially generated the message content in send mode. Therefore, an agent avoids receiving echoes of its own messages by filtering out incoming packets with its own identifier. As robots can only perceive one message per time cycle, a random selection among all the messages currently captured from all sectors is implemented. Moreover, only the context information corresponding to the chosen message if provided to the robot's controller.