Concurrent design of control software and configuration of hardware for robot swarms under economic constraints

Anytime-Selection

The robot swarm must aggregate in one of the two circular black zones. The size of two black zones and their position in the arena are given in Fig. 2A. The performance of the swarm is measured by the following objective function: (2)${\displaystyle F_A=\sum _{t=1}^T\left|\right.\left(N_a\left(t\right)-N_b\left(t\right)\right)\left|\right.,}$ where N_a(t) and N_b(t) are the number of robots in zone a and b at any time t; T is the total duration of the experiment.

In Anytime-Selection, the performance is measured at every control step. Due to this reason, the robots are expected to promptly aggregate in one of the black zones and stay there until the end of the experiment.

End-time-Aggregation

The robots must aggregate in one of the two black zones. The dimensions of two black zones and their position in the arena are given in Fig. 2A. The performance of the robot swarm is measured with the following objective function: (3)${\displaystyle F_E=\left|\right.N_a\left(T\right)-N_b\left(T\right)\left|\right.,}$ where T is the duration of an experiment and N_a(T) and N_b(T) are the number of robots in zone a and zone b at time T.

Unlike Anytime-Selection, the performance in End-time-Aggregation is computed at the end of the experiment. Due to this reason, the robots can take some time to explore the arena and converge in a black zone: the experiment duration is not a significant constraint in this mission. However, the real challenge is to keep the robots assembled in a zone until the end of the experiment.

Foraging

The swarm must collect a maximum number of objects from two sources and drop them in the nest. We abstract the Foraging experiment by considering that an object is retrieved when an individual robot visits a source, and an object is dropped when a robot visits the nest. The two sources in the arena are represented as two black zones, while the nest is represented as a white zone. A light is also placed behind the nest as a cue for robots. The dimensions and position of the two source zones and nest are given in Fig. 2B. The performance of the robot swam, the number of objects (N_o) retrieved by the swarm, is expressed by the following objective function: (4)${\displaystyle F_F=N_o.}$

In Foraging, an individual robot can retrieve a single object at a time. Therefore, the performance of the swarm heavily rely on the number of robots and on the duration of the experiment.

No-constraint

This experiment (NC) is performed without any constraint: the monetary resources and battery capacity are unconstrained.

Monetary-constraint

In these experiments, the limited resource is the monetary budget, M_limit, available to purchase the robots and range-&-bearing modules. The design process only considers the combinations of hardware modules that keep the total cost of the swarm, P_swarm, equal or below the available monetary budget—i.e, P_swarm ≤ M_limit. The total swarm cost, P_swarm, is computed with the following equation: (5)${\displaystyle P_{\mathit{swarm}}=N\times \left(P_r+P_x+P_y\right),}$ here N is the total number of robots in swarm, that is, 15 to 30 robots; P_r is the price of extended version of e-puck without range-&-bearing modules, that is, 2,000€; P_x and P_y are the prices of a range-&-bearing receiver and a range-&-bearing transmitter respectively—see Table 3.

The minimum cost of a swarm is 43,500€, when the minimum number of 15 robots are equipped with the least-capable range-&-bearing receiver and transmitter modules. The maximum cost of a swarm is 108,000€, when the maximum number of 30 robots are equipped with the most-capable range-&-bearing receiver and transmitter modules.

For each mission, we perform two experiments, M₈₀ and M₆₀, where the monetary budget is 80,000€  and 60,000€  respectively—see Table 4.

Power-constraint

In these experiments, the limited resource is the battery capacity, P_bc. There is no constraint on the monetary resources: the design process can choose any combination of the range-&-bearing modules and the number of robots between 15 and 30—see Table 4. The operation time, T_r, of each robot in the swarm depends on its hardware configuration, available battery capacity, and the execution of the individual-level behaviors. The operation time of a robot can be computed by the following equation: (6)${\displaystyle T_r=\frac{\left(P_{bc}\times 3600\right)}{\left(I_{\mathit{cpu}}+I_{lm}+I_{rm}+I_{y\left(\mathcal{R}\right)}+I_x\right)},}$ where I_cpu is the current rating of robot’s cpu and other fixed sensors, that is, 100 mA. The CPU and other fixed hardware modules will always consume the same power. I_lm and I_rm are the current ratings of the left and the right motors of the robot, that is, 150 mA at maximum speed. I_x and $I_{y\left(\mathcal{R}\right)}$ are the current ratings of range-&-bearing receiver and transmitter modules respectively. $\mathcal{R}$ is the range of range-&-bearing transmitter—see Table 3. The experiment terminates, when every robot in the swarm consumes its battery power. The total experiment time, T, is expressed as: (7)${\displaystyle T=max\left(T_{r\in \left\{1,2,3,...,N\right\}}\right).}$ For each mission, we perform two experiments with different levels of battery capacities: P₂₀ and P₁₅—see Table 4.

Monetary-&-Power-Constraint

In these experiments, both monetary budget and battery capacity are limited. The design process is required to choose the hardware modules that are not only affordable but also allow robots to operate for a sufficient amount of time. For each mission, we perform four experiments with dual constraints: M₈₀P₂₀, M₈₀P₁₅, M₆₀P₂₀, and M₆₀P₁₅—see Table 4.

No-constraint

Waffle tends to configure robot swarms whose total cost is close to the maximum possible—see Fig. 3D. Indeed, the robot swarms comprise 25 to 30 robots—see Fig. 3G—equipped with high-tier range-&-bearing receivers and transmitters—see Fig. 4. At visual inspection, the robots first form clusters and then slowly converge on a black zone: when the robots find a black zone, they spin in place and block the way for the remaining robots, which are therefore unable to enter the zone. This behavior is obtained with Exploration, Stop, and Attraction—see Fig. 5A. As expected, the performance of the swarm is considerably better than the one achieved when constraints are imposed—see Fig. 3A.

Monetary-constraint

Under the constraints imposed by M₈₀ and M₆₀, Waffle tends to configure the robot swarm so that the total cost is close to the maximum available budget—see Fig. 3D. The number of robots in the swarm decreases proportionally to the monetary budget—see Fig. 3G. The robots are equipped with high-tier range-&-bearing receivers and long-range range-&-bearing transmitters. In M₆₀, however, Waffle also selects two low-tier range-&-bearing receivers—see Fig. 4. The robot swarms designed under NC and M₈₀ behave in a similar way. In M₆₀, however, the robots prefer to use the Attraction low-level behavior to remain in a black zone, but as the robots are equipped with low-tier range-&-bearing receivers, they often leave the black zone: due to the high loss-probability of low-tier range-&-bearing receivers, the robots often fail to perceive the presence of their peers in their neighborhood. The performance of the swarms designed under M₈₀ and M₆₀ is considerably lower than the one achieved under NC: in M₈₀ and M₆₀, the swarm comprises fewer robots as compared to NC—see Fig. 3A.

Power-constraint

In contrast to NC, the swarms configured under P₂₀ and P₁₅ have a total cost that is noticeably lower than the maximum possible—see Fig. 3D. This is because the robots are equipped with low-price range-&-bearing transmitters, which reduces the overall cost of the swarm—see Fig. 3D. Cheaper range-&-bearing transmitters have a shorter transmission range but have low power consumption, which allows a longer battery life. We observe a major shift in the dominant individual-level behaviors in the produced instances of control software. The robots stop in the first black zone they encounter and limit their movement to save energy—see Fig. 5A. Consequently, the swarm splits and becomes unable to gather on the same zone. As a result, the performance drops by approximately 50% as compared to the performance achieved under NC—see Fig. 3A.

Monetary-&-power-constraint

In all experiments, Waffle tends to use all the available monetary budget—see Fig. 3D. In M₈₀P₂₀ and M₈₀P₁₅, Waffle designs swarms that comprise 23 to 24 robots—see Fig. 3G—equipped with any of the range-&-bearing receivers and low-range transmitters. In M₆₀P₂₀ and M₆₀P₁₅; however, the number of robots decrease considerably, and the robots are equipped with low-tier range-&-bearing receivers and low-range transmitters—see Figs. 3G and 4. The control software generated under the Monetary-&-Power-Constraint behave similarly to those of the Power-Constraint experiments—see Fig. 5A. Due to the increased battery capacity, the swarms produced under M₈₀P₂₀ and M₆₀P₂₀ perform slightly better than the ones produced under P₁₅, M₈₀P₁₅, and M₆₀P₁₅—see Fig. 3A.

No-constraint

Waffle tends to configure robot swarms whose total cost is close to the maximum possible—see Fig. 3E. The hardware configuration is similar to the one generated under NC for Anytime-Selection. Indeed, the robot swarms comprise 28 to 30 robots—see Fig. 3H—equipped with high-tier range-&-bearing receivers and transmitters—see Fig. 4. At visual inspection, the robots first form clusters and then converge on a black zone: robots tend to remain there by spinning in place until the end of the experiment. This behavior is obtained with Exploration, Attraction, and Stop—see Fig. 5B. The performance of the swarm is considerably better than those achieved when constraints are imposed—see Fig. 3B.

Monetary-constraint

Under the constraints imposed by M₈₀ and M₆₀, Waffle tends to configure the robot swarm so that the total cost is close to the maximum available budget—see Fig. 3E. The number of robots in the swarm decreases proportionally to the monetary budget—see Fig. 3H. The robots are equipped with long-range range-&-bearing transmitters and high-tier receivers, except a small minority of configurations in which the robots are equipped with low-tier receivers—see Fig. 4. At visual inspection, in M₈₀ and M₆₀ the robots converge on a black zone and stay there until the end of the experiments. Contrary to NC, the robots stop on the black zone instead of spinning in place: dominant individual-level behaviors are Exploration, Attraction, and Stop—see Fig. 5B. The amount of available monetary budget has a direct impact on the performance of a swarm. Indeed, due to the limited monetary budget, the number of robots decreases in the swarms designed under M₈₀ and M₆₀, which results in a considerable performances drop as compared to the performance achieved under NC—see Fig. 3B.

Power-constraint

Similar to NC, under the constraints imposed by P₂₀ and P₁₅, Waffle tends to configure the robot swarm so that the total cost is close to the maximum possible—see Fig. 3E. Indeed, the robot swarms comprise 28 to 29 robots—see Fig. 3H—equipped with high-tier range-&-bearing receivers and long-range transmitters—see Fig. 4. However, this selection of hardware has a direct impact on the duration of the experiments due to its high current rating. As the maximum power is consumed by the motors, the designed control software skips the Exploration behavior to move robots in the arena. In some instances of control software, the robots use Phototaxis and Anti-Phototaxis individual-level behaviors to move straight and avoid obstacles. Moreover, the most dominant individual-level behavior is Attraction which is used to keep the robots assembled on one zone—see Fig. 5B. The performance achieved under P₂₀ is relatively higher than the one achieved under P₁₅. Due to the limited battery capacity, which affects the total experiment duration, the swarms designed under P₂₀ and P₁₅ have a lower performance than those designed under NC—see Fig. 3B.

Monetary-&-power-constraint

In all experiments, Waffle tends to use all the available monetary budget—see Fig. 3E. In M₈₀P₂₀ and M₈₀P₁₅, Waffle designs swarms that comprise 22 to 24 robots—see Fig. 3H—equipped with high-tier range-&-bearing receivers and long-range transmitters—see Fig. 4. In M₆₀P₂₀ and M₆₀P₁₅, the number of robots decreases considerably, and the robots are equipped with high-tier range-&-bearing receivers and long-range transmitters, except a small minority of configurations in which robots are equipped with low-tier receivers—see Fig. 4. The instances of control software produced are similar to those produced under Power-Constraint: the movement of robots in the arena is identical and the prominent individual-level behavior is Attraction—see Fig. 5B. The performance achieved under M₈₀P₂₀ and M₈₀P₁₅ is slightly better than the one achieved under M₆₀P₂₀ and M₆₀P₁₅: the level of monetary budget is the key factor that determines whether Waffle selects few or more robots—see Fig. 3B.

All categories of constraints

Under all the categories of constraints considered, Waffle produced robot swarms sharing the same hardware configuration. This because, in Foraging, the robots do not rely on local communication. As a result, the selected hardware configuration typically does not include range-&-bearing transmitter and receiver—see Fig. 4. The total cost of a swarm is between 80,000€  and 60,000€—see Fig. 3F. The swarm comprises the largest possible number of robots—see Fig. 3I.

All instances of control software that are produced in all experiments have an unexpected behavior. Although in all experiments of Foraging, the robots are not equipped with range-&-bearing modules, the most prominent individual-level behaviors are Attraction and Repulsion, which the robots use to explore the arena—see Fig. 5C. The swarm uses these behaviors in a way that is completely different from the one originally intended (Francesca et al., 2014). The reason behind this anomaly is that the individual-level behaviors in the design space are not strictly associated with the related hardware. In the absence of range-&-bearing receivers and transmitters, the Attraction and Repulsion behaviors are actuating robots to move straight using proximity sensors to avoid obstacles. In all Foraging experiments, Waffle selects the Phototaxis individual-level behavior to locate the nest in the arena, as shown in Fig. 5C.

There is no prominent performance difference between the experiments under the No-Constraint and Monetary-Constraint categories—see Fig. 3C. However, we observe a considerable performance drop by the swarms designed under categories of experiments that have limited battery capacity—that is, Power-Constraint and Monetary-&-Power-Constraint. Indeed, the performance achieved in experiments with 20 mAh battery capacity—i.e., P₂₀, M₈₀P₂₀, and M₆₀P₂₀—is considerably better than the performance achieved under experiments with 15 mAh battery capacity—i.e., P₁₅, M₈₀P₁₅, and M₆₀P₁₅—see Fig. 3C.