An adaptive weighting mechanism for Reynolds rules-based flocking control scheme

Behavioral rule prioritization

The five above behavioral rules are divided into three groups of behaviors according to their contextual significance.

• Both Separation and Avoidance should be given the highest priorities due to safety issues. In cases where they are both obstacles and neighbors violating the safety distance, the drone will attempt to avoid the nearest object.

• The Migration rule is the second most crucial behavior since its role is to navigate the group of drones to destinations as planned.

• The Alignment and Cohesion rules are used to make the drones travel in a swarm and maintain the compactness. They are the lowest priorities.

Adaptive weighting mechanism

The weights of the five behavioral rules, including w_se (Separation), w_av (Avoidance), w_mi (Migration), w_al (Alignment), and w_co (Cohesion) are determined by (2)–(6), respectively.

(2) $$w_{se}=T_s\underset{i\in I,i\ne k}{max}(c_s-{\displaystyle \frac{d_{i,k}^2}{c_s}})\mathrm{\forall }k\in m$$

(3) $$w_{av}=T_a\underset{i\in I,i\ne k}{max}(c_a-{\displaystyle \frac{d_{i,k}^2}{c_a}})\mathrm{\forall }k\in m$$

(4) $$w_{mi}=\{\begin{array}{l}M,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}if\hspace{0.17em}}m\text{\hspace{0.17em}=\hspace{0.17em}0}\\ M+\frac{M}{m},\text{if\hspace{0.17em}}m\ne 0\end{array}$$

(5) $$w_{al}=\{\begin{array}{c}0,\mathrm{i}\mathrm{f}m=0\\ {\displaystyle \frac{M}{m},\mathrm{i}\mathrm{f}m\ne 0}\end{array}$$

(6) $$w_{co}=\{\begin{array}{c}0,\mathrm{i}\mathrm{f}m=0\\ {\displaystyle \frac{M}{m},\mathrm{i}\mathrm{f}m\ne 0}\end{array}$$where

• m denotes the number of nearby objects at a certain moment. An object within a quadcopter’s perception will be considered as its nearby neighbor in Separation, Cohesion, Alignment, and Migration rules or obstacles in Avoidance rule.

• c_s and c_a are perception coefficients for the Separation and Avoidance rules, respectively.

• d_j,k is the distance between quadcopters ith and kth (in Separation); or between quadcopter ith and obstacle kth (in Avoidance), k ∈ {1,2,…,m}.

• M is the optimal number of neighbors surrounding each quadcopter. In an ideal case, each of the six quadcopters will try to maintain the same relative distance with the one at the center and with two adjacent ones, consequently forming a hexagon with quadcopters at its center and vertices, as illustrated in Fig. 1. Therefore, for a swarm with a large enough number of members as in our implementation, M will be six (i.e., M = 6).

• T_s and T_a are the transformation factors for Separation and Avoidance rules, respectively.

Transformation factors

It is seen in (2) and (3), we introduce the transformation factors that are equal to the ratio of the desired maximum weight W to the corresponding perception coefficient (i.e., c_s or c_a). In detail, the solution of (c_s − d²/c_s) in (2) is a set of [0,c_s] and that indicates the impact of distance based on the Separation rule perception, whose measurement unit may vary. For example, with c_s = 5,000 m and maximum weight W = 30, the solution set [0,5000] is completely different from the weight range [0,30]. Therefore, T_s is used to keep the impact of Separation rule unaffected by the measurement unit, which makes our scheme able to adapt to different settings.

To calculate T_s, it is required to determine the maximum weight W and the distance d_min when Separation will become largely effective for safety issues. Since there is probably the situation where Migration, Alignment, and Cohesion vectors simultaneously point to the same direction, the Separation vector needs to point to the opposite direction to prevent any possible collisions. Therefore, the w_se should satisfy (7).

(7) $$w_{se}>w_{mi}+w_{al}+w_{co}$$

As inferred from (4) to (6), the maximum values of w_mi, w_al, and w_co are 2M, M, and M, respectively. Consequently, w_se should be greater than 4M so as to take control of the drone. As a result, we have $(c_s-d_{min}^2/c_s)\in [0,c_s]$ and 4M ∈ [0,W]. Additionally, since w_se is proportional to (c_s − d²/c_s) according to (2), the maximum weight W can be computed by (8).

(8) $$W={\displaystyle \frac{4M}{1-{\displaystyle \frac{d_{min}^2}{c_s^2}}}}$$

Then, T_s is obtained by (9).

(9) $$T_s={\displaystyle \frac{W}{c_s}={\displaystyle \frac{4M}{c_s-{\displaystyle \frac{d_{min}^2}{c_s}}}}}$$

In short, T_s can be computed based on particular preference for the shortest range d_min at which we want the drones begin to exponentially react to Separation rule. The act of choosing d_min can be done after M and c_s are determined. For example, if d_min = 0.4c_s then T_s ≈ 4.76M/c_s. Similarly, the transformation factor T_a for Avoidance is determined by (10).

(10) $$T_a={\displaystyle \frac{4M}{c_a-{\displaystyle \frac{d_{min}^2}{c_a}}}}$$

Rule implementation

The explanations for rules’ implementation will be given in prioritized groups. In both Separation and Avoidance, the drone travels away from nearby objects to avoid crashes, and a repulsion vector for each of them is formed. As we only consider the direction at this stage, these vectors are then normalized. Moreover, an impact called dist_impact is also calculated by (11).

(11) $$dist\mathrm{\_}impact=c-{\displaystyle \frac{d^2}{c}}$$where c is the rule perception that can be c_s or c_a depending on either Separation or Avoidance, and d is the distance between two drones or between a drone and an obstacle. Subsequently, the normalized vectors are multiplied with dist_impact since we are expecting the smaller the distance is, the larger the magnitude of this repellent force is. Then, all resulting vectors are combined and the sum vector is scaled by (12).

(12) $$v_s=\{\begin{array}{cc}\hat{v}& \mathrm{i}\mathrm{f}||v||>0\\ (0,0)& \mathrm{i}\mathrm{f}||v||=0\end{array}$$

where $\hat{v}$ is the unit vector of v.

In addition, we need to find the nearest object among all detected ones with the most significant impact as max_impact inferred from (11) so that the drone will try to avoid the nearest object. Finally, the product of the scaled vector v_s, max_impact, and T produces the return vector. Algorithm 1 describes the implementation of Separation and Avoidance rules, which are different by the transformation factor T (i.e., T can be T_s or T_a).

The Migration rule’s effect will attract the drones to the predefined target by forming a vector from the drone to the target point. The vector is then scaled using (12) and assigned a weight affected by the current number of nearby neighbors. If there are any detected neighbors, Migration weight will be M + M/m to get individuals in the group to the desired location rather than being stuck by the effects of Alignment and Cohesion. Otherwise, if there is no detected neighbor, the weight will be M. Both cases guarantee that Migration weight will always be equal or greater than the sum of Alignment and Cohesion weights. This weighting scheme ensures that even in case both Alignment and Cohesion velocity vectors point to the same direction but are opposite the target point, the drone will still be able to head towards its target. The implementation of the Migration rule is shown in Algorithm 2.

Alignment and Cohesion rules share the same weighting mechanism due to partial similarities in their effect. The implementation of these two rules is represented by Algorithm 3. In detail, the drone will fly in the same direction as its neighbors due to Alignment. The vector for this behavior is computed by calculating the average velocity of each drone’s nearby neighbors. Regarding Cohesion, the drone tries to fly towards the average location of its neighboring drones. The resulting vector is scaled and then multiplied with its weight.

Evaluation metrics

Regarding the evaluation process, the proposed scheme is compared to the conventional one using the weighted sum of all rule vectors to assess their performance differences in terms of two metrics, including flock compactness and collision reduction represented by the number of crashed drones.

• Flock compactness shows the drones’ capability in maintaining the optimal distances among one another by calculating the average distance of each pair of drones in the flock. The smaller the average distance is, the less space the flock uses. This is particularly useful for flights in confined environments or using wireless communication. This metric is represented by the average distance computed by (13).

(13) $$\overline{d}={\displaystyle \frac{1}{N_s}\sum _{k=1}^{N_s}{\displaystyle \frac{2}{n(n-1)}\sum _{i=1}^{n-1}\sum _{j=i+1}^n{d}_{i,j}}}$$where N_s is the number of loops in the swarm algorithm, n is the number of drones, and d_i,j is the distance between drone i and j.

• The number of crashed drones is counted if there are any other objects (i.e., obstacles or neighboring drones) within the collision distance around a drone.

Besides, the magnitude of each rule’s velocity vector before being combined into one final vector will be investigated over how each swarm behavior influences the drones’ movement trajectory during flights.

Methodology

In the simulation program, flight tests are conducted within 1,200 x 700 on a 2D plane measure by unit. The unit can be pixels or scalable to different measures since the proposed transformation factors keep the rules weighting computation unaffected by the measurement unit, as presented above.

For both conventional and proposed schemes, all common parameters that represent each drone’s characteristics are summarized in Table 1. Drones are assumed to fit in a circle with a diameter of 8 units, and the center coordinates are used as their position. Besides, a collision is determined once any objects within each drone’s safety zone are defined by a collision distance of 8 units (same as drone’s size). In addition, Alignment and Cohesion behaviors start taking effect when there are neighbors closer than their perceptions (50 units). Meanwhile, Separation and Avoidance will only become operative with a smaller perception (40 units) to prevent the unnecessary impact that may cause drones to be directed away from their group early. The speed of each drone will be scaled to 1.2 units/step if it exceeds this value.

To evaluate our proposed scheme’s effectiveness compared with the conventional one adopting fixed weights, it is required to determine an optimal set of fixed rule weights that produce the conventional scheme’s best experimental results. Therefore, we conduct several simulations with Separation and Avoidance’s weight being gradually increased, whereas other rules’ weights are fixed. The reason is that with higher priorities, Separation and Avoidance rules have the most considerable influence on both the flock’s compactness and the number of crashed drones.

It is reasonable to start with the weight set of {w_se:w_av:w_mi:w_al:w_co} as {4:4:2:1:1} based on the proposed rule prioritization and the rule’s weights computed by (2), (3), and (7). For every weight set, 100 simulations are executed with initial randomized positions of drones and obstacles. As can be seen in Fig. 2, the drones being too close leads to the large number of crashed drones. Thus, Separation and Avoidance weights are increased by 0.4 in each next experiment. It is seen that when the Separation and Avoidance weights begin to reach 5.2 and higher, the number of crashed drones remains steady at nearly 10. However, the flock compactness tends to rise along with the increase in weights of Separation and Avoidance. Regarding safety concerns and the flock compactness, Separation and Avoidance weights will be set as 5.2 (i.e., w_se = w_av = 5.2) for further performance comparison with our proposed scheme.

The optimal weight set of {5.2:5.2:2:1:1}, along with our adaptive weighting mechanism, are ready for further comparisons. For each simulation, ten drones are initially created with random positions near the upper left corner of the window, and then they proceed to move towards the target point at the lower right corner. There are two scenarios, as follows:

• Flights without obstacles: this is the ideal environment for the deployment of any flocking control scheme. As there is no Avoidance behavior, the Separation velocity vectors keep being dominant during the entire flights, leading to no collision between drones. Therefore, flock compactness is the only metric for evaluation and comparison in this scenario.

• Flights with obstacles: obstacles are randomly generated in an area of 800 × 300 in the middle of the window for each simulation, as depicted in Fig. 3. For this scenario, all two metrics are used.

Simulation results

An ensemble of 200 simulations has been done for each scheme in all scenarios. Fig. 4 summarizes the simulation results regarding the flock compactness in the obstacle-free scenario. It can be noted that the drones have an average distance between each other of about 81 units with the conventional flocking control scheme. Meanwhile, our proposed approach produces approximately 73 units. This indicates that by leveraging our improved scheme, the flock is better at maintaining its compactness with roughly 10% smaller average distance.

Figure 5 shows the average distance of each scheme when it comes to flying areas with obstacles. It can be noticed that in terms of the flock compactness, the proposed scheme continues to outperform its conventional counterpart. In fact, every pair of swarm members in the proposed scheme manages to maintain an average distance of 85 units, which is 14.14% lower than the conventional scheme’s result of 99 units.

Besides maintaining a smaller relative distance with other neighbors, the drones maneuvered by the proposed scheme happen to encounter fewer crashes. In detail, the conflict of Separation and Avoidance rules in the conventional scheme leads to 15 crashed drones during 200 flight tests. Meanwhile, due to the flexibility in weight allocation, the proposed scheme keeps the flock free from collisions.

To explain the reason behind this better performance of the proposed scheme, we investigate every swarm rule’s impact on a drone’s behavior against the conventional control scheme. As shown in Fig. 6, it is observed that there are numerous times where the sudden detection of both nearby neighbors and obstacles leads to unexpected behaviors of the drone. For instance, at around simulation step 300, the sudden and considerably weighted appearance of Avoidance rule in purple leads the drone to be directed far from its neighbors. This is to the extent that Separation force is barely needed and Cohesion and Alignment forces are predominant to keep the drone closer. In contrast, regarding the proposed model, as investigated in Fig. 7, the adaptive weighting mechanism enables the drone to stay close in the swarm throughout the flight. This is indicated by all of the swarm behaviors’ presence during and after avoiding the obstacles (around steps 200, 300, and 410).