Fault-Tolerance Generalization

The generalization to assess 𝑘-fault-tolerance in the initial deployment of robots can be achieved by having the ranger to ear the beacon messages from the last 𝑘 unloaded robots, thus computing the minimum signal quality value between them, i.e., min(𝑞_𝑖, 𝑞_𝑖+1, … , 𝑞_{𝑖+𝑘−1}) = 𝑞_𝑚𝑖𝑛.

Proposition 5.1. In a 𝑘-connected MANET, with 𝑘 ≤ 𝑁_𝑆− 1, when the ranger deploys the first robot, instead of moving apart from it until the signal quality reaches the minimum desired value, in linear units (e.g., Watts), the ranger will unload the second robot when the signal quality reaches a level of:

𝑞₁ = 𝑘 ∙ 𝑞_𝑚𝑖𝑛, (5.6)

thus guaranteeing a quasi-balanced distribution of the scouts.

Proof: Considering a maximum communication distance 𝑑_𝑚𝑎𝑥 problem in a 𝑘-connected MANET, one needs to ensure that the 1^𝑠𝑡 robot can communicate with the 𝑘^𝑡ℎ robot, with a distance between the 2^𝑛𝑑 robot similar to the other inter-robot distances such as Figure 5.2 depicts.

Based on the law of cosines and considering equation (5.1), one can define the following relation between 𝑑 and 𝑑_𝑚𝑎𝑥.

𝑑 = ^{𝑑𝑚𝑎𝑥}

√4+𝑘−√8𝑘+16∙cos ∑ 𝑎𝑟𝑐𝑡𝑎𝑛(¹

√𝑗) 𝑘+1𝑗=2

,

(5.7)

such that 𝑑 = 𝑑_𝑚𝑎𝑥 when 𝑘 = 1.

Figure 5.2. EST deployment in a 𝑘-connected MANET.

d φ₁ d 1

d d

φ₂

Section 5.1. Initial Deployment 116

Note that 𝑑 does not depend on the direction of the spiral 𝜎_𝑠 since the cosine function is unaf-fected by the signal. Also, as the requirements of the MANET connectivity are softened (for smaller 𝑘), the denominator of equation (5.7) presents a more linear relation with 𝑘. This relation can be easily observed in Figure 5.3.

Figure 5.3. Relation between 𝑑 and 𝑑_𝑚𝑎𝑥 varying the MANET connectivity 𝑘.

Hence, one could consider the following approximation since the multi-connectivity level 𝑘 in MRS, contrarily to wireless sensor networks, is usually small (e.g., below 5):

𝑑 = ^{𝑑𝑚𝑎𝑥}

𝑘 , (5.8)

The same analysis can be conducted in a minimum signal quality 𝑞_𝑚𝑖𝑛 problem. However, note that this may only work when using signal quality units that present an approximately inversely pro-portional relation with the distance. In other words, most of wireless equipment returns the RSSI signal in Watts (W) or decibels (dB) – while in most situations the signal quality in Watts is almost inversely proportional to the distance between nodes, decibels present a logarithmic relation. Taking this into account, one can estimate a desired signal quality between robot 1 and 2 as equation (5.6) shows.

■ When facing obstacles, the ranger should follow an optimized decision about whether it should rotate left or right, thus choosing the smallest rotation it needs to perform based on the configuration of the sensed obstacles. When the ranger is unable to compute the best turning decision, it randomly

𝑘 𝑓(𝑘)

𝑓(𝑘) =^{𝑑𝑚𝑎𝑥}

𝑑 𝑓(𝑘) = 𝑘

117 Chapter 5. Deployment and Fault-Tolerance chooses between rotating left or right. In this way, the growth of the radius at a certain robot 𝑖 can be approximated using the following equation:

∆𝑅_𝑖 ≈ 𝑑_𝑚𝑎𝑥^𝑖,𝑖+1(√𝑖 + 1 − √𝑖), (5.9)

thus allowing to understand the distribution of the robots over a scenario by calculating the approxi-mated total radius of the EST strategy within subgroup 𝑠:

𝑅_𝑇 ≈ ∑^𝑁_𝑖=1^𝑠 ∆𝑟_𝑖+1^. (5.10)

It is noteworthy that equations (5.9) and (5.10) are only approximated measures since the dis-tance between pairs of robots may change²⁸ and the existence of obstacles may increase or decrease the growth of the EST radius.

Therefore, to further assert the distribution of the EST strategy, a set of deployment trials were numerically computed changing the number of robots within a subgroup in an environment with a large density of randomly deployed obstacles. A fixed maximum communication range was used, i.e., 𝑑_𝑚𝑎𝑥^𝑖,𝑖+1 = 𝑑_𝑚𝑎𝑥 ∀ 𝑖, since it is easier to implement in simulation. Also, fault-tolerance was not con-sidered, i.e., 𝑘 = 1. As the random initial deployment of robots is the most common strategy in the literature (cf., section 2.2.4), its distribution was compared with the EST deployment. In the random distribution, robots are successively randomly deployed within a circumference of radius 𝑑_𝑚𝑎𝑥 and centered in the previously deployed robot while avoiding the overlap with obstacles. For a more de-tailed description of this random initial deployment please refer to (Couceiro M. S., Figueiredo, Portugal, Rocha, & Ferreira, 2012).

The number of scouts to be deployed was set to 𝑁_𝑠 = {5,10,15,20,25} with 100 trials each for both strategies. Figure 5.4 presents a couple of simulated examples of a team of 10 scouts deployed using both strategies. Blue scouts were deployed using the ETS while red scouts were deployed using the random distribution. As it is possible to note, contrarily to the ETS in which scouts are scattered throughout the scenario, the random deployment turns out to reveal an unbalanced distribution of scouts. Nevertheless, to further measure the dispersion of both deployment strategies, a metric based on the average distance from each scout to the centroid 𝑥_𝑐[0] was used:

28 The signal depends not only on the distance, but also on the multiple paths from walls and other obstacles.

Section 5.1. Initial Deployment 118 𝜎_𝑠 = ¹

𝑁_𝑠∑^𝑁_𝑖=1^𝑠 ‖𝑥_𝑖[0] − 𝑥_𝑐[0]‖^. (5.11)

Figure 5.4. Computational outputs of the EST (left) and random (right) deployment of 10 scouts over a given scenario endowed with obstacles.

Figure 5.5 depicts the dispersion of the robotic team 𝜎_𝑠 for both strategies.

Figure 5.5. Relation between 𝑑 and 𝑑_𝑚𝑎𝑥 varying the MANET connectivity 𝑘.

As one can observe, the dispersion of the team of robots using the ETS deployment is signifi-cantly higher than using a random distribution. In fact, the random deployment does not present a substantial increasing dispersion as the number of robots increases.

Algorithm 5.1 generalizes the ranger EST behaviour to deploy a whole subgroup of scouts in an unknown environment while ensuring that the MANET is 𝑘-connected (Couceiro M. S., Figueiredo, Rocha, & Ferreira, 2013 (Under Review)).

𝑁𝑠

𝜎_𝑠[𝑚] 𝐸𝑆𝑇

𝑟𝑎𝑛𝑑𝑜𝑚

119 Chapter 5. Deployment and Fault-Tolerance to 𝑖 + 2 while avoiding obstacles

𝑠𝑒𝑛𝑑_𝑝𝑜𝑠𝑒(𝑖 + 2, 𝑥_𝑖+2[0], 𝜃_𝑖+2[0]) // informs scout 𝑖 + 2 of its initial pose 𝑠𝑒𝑛𝑑_𝑠𝑡𝑎𝑟𝑡() // broadcast information to start the mission

Algorithm 5.1. Initial deployment EST algorithm for ranger 𝑛.