APLICACIÓN DE NANOPARTÍCULAS EN RECUPERACIÓN

In addition to efficiently mapping the environment, the lead robot must also transmit the map data and the primary sensor’s data back to the access point with minimal delay. In order to reduce the amount of data that needs to be transmitted, only the changes in the best estimate of the map are transmitted. In contrast to transmitting the full map on every update, the differential approach requires data transmission only when new information is obtained. This results in periods of low rates when the map is unchanged and periods of high rates when new, previously unexplored, areas are encountered. Since the primary objective of the team is to provide real-time situational awareness, the primary sensor’s data flow is prioritized over the map updates. As such the update only utilize the residual bandwidth available after the primary sensor’s data is transmitted. There are periods where the residual bandwidth is insufficient for the map updates, thus they must be queued and transmitted as bandwidth becomes available. Queueing the data introduces a delay between the update of the map at the leader and the update of the map at the access point. A delay in the map updates at the access point can be detrimental to the support robots since they are required for successful localization and channel estimation. Additionally, if the delay experienced by the map updates is large enough, any benefit of the lead robot quickly mapping the environment may be lost due to the data taking longer to reach the access point, as a result of minimal residual bandwidth.

The flow of the primary sensor and the map updates are combined into a single flow since they are both destined for the access point. While the system is capable of handling the two flow separately there is no difference in the results, thus the two flows are combined for the sake of clarity. During operation the experienced end-to-end rate for the primary sensor’s flow is

ae

pri = min{ae, apri}, whereae is the currently available end-to-end rate andapri is the primary

sensor’s required rate. The remaining bandwidth,ae

map=ae−aepriis allocated to the map updates.

must be sufficient for the primary sensor’s flow,a1

l,m> as. Using this requirement will allow the

system to satisfy the primary sensor, but the delays to the map updates can be arbitrarily large. To avoid this, the value ofa1

l,mcan be increased so that the maximum expected delay of a map update

is below some threshold,dmap. This can be achieved by estimating the maximum amount of change

between sequential maps and determining the number of packets,nmapnecessary to transmit that

information. This leads to the requirement that the time needed to transmit nmap packets at a

rate of rmap be below the threshold, dmap ≥nmap/rmap. Therefore, the value of rmap must be

chosen such that this requirement is satisfied. Since the communication channels are capable of transmitting rppackets per seconds, the required residual end-to-end rate can be computed from

the end-to-end packet rates as amap =rmap/rp. The additional rate, amap, combined with the

primary sensor’s rate, apri, becomes the new value of a1l,m = min{1.0, amap+apri}. It can be

seen that as the maximum tolerable delay dmap is decreased, the value of a1l,m increases which

restricts the total area that can be explored by the leader. This is due to the maximum separating distance between two robots being inversely proportional toa1

l,m. This approach also suffers from

the drawback of applying conservative communication requirements even when there is minimal new information, such as revisiting a previously mapped location.

These limitations lead to the desire to dynamically adjust a1

l,m based on the current com-

munication conditions. This approachmore effectively balances the competing desire of mapping the largest area with minimizing the delay of the map updates at the access point. First, for the value of a1

l,m to dynamically adjust, the local controller must be adapted from its current

form to allow for changes in a1

l,m without compromising network integrity. Since a decrease in

a1

l,m can only increase the margin afforded to network integrity, due it relaxing the communi-

cation requirements, we only focus on times then the value is increased. The goal then is to guarantee that the value of a1

l,m will never increase enough to cause the current formation to

become infeasible. To provide this guarantee, we begin by defining a path between two robots as a series of links that connect robots i and j, without repeating a link. Written explicitly,

pi,j ={(ip, jp)}pP=1, where i1 = i, jP =j, ip =jp−1, (ip, jp)6= (iq, jq) for all q 6= p. Using the

communication rate for the value of the link,v(ip, jp) =R(xip(t), xjp(t)), we define the score of a

paths(pi,j) = minpv(ip, jp) = minpR(xip(t), xjp(t)). This score can be interpreted as the rate at

which data can be transmitted from robot ito robot j if only the pathpi,j is used. Since there

are multiple paths the data can travel, we define the set of all possible paths from robotito robot

j as_Pi,j. Continuing with the interpretation of the score of a single path, we define the score of

the set _Pi,j as the maximum score of its paths, mi,j = maxpi,j∈Pi,js(pi,j), which we refer to as

the max-min path score. Therefore, given the current formation, we can see that the maximum end-to-end rate between robotsi andj ismi,j. Thus, as long as a1l,m≤ml,N, the change in the

communication requirements will not make the current formation infeasible.

With access to only local information it is difficult for the leader to estimate its max-min path score. To overcome this without introducing global coordination, the local controllers maintain their current max-min path score with the access point,mi,N, in a distributed manner. To compute

mi,N we begin by constructing a subgraph of the current communication graph. The nodes in

this graph are robot i, the access point, and only the neighbors that robot i can send data to. To determine this set of neighbors we compare their dual variables with robot i, specifically only those in whichλk

i > λkj. It can be seen whenλki < λkj, the optimal solution to (4.1.13) will include

αk

ij = 0. Thus, robotiwill not transmit any received data to robotj and robotj can be ignored

in the subgraph. In the subgraph, the value of the links between robot iand its usable neighbors is the current estimate of the channel rateR(xi(t), xj(t)), and each neighbor has a link back to the

access point with a value equal to its current max-min path score,mj, N. Robotithen computes its max-min path score over the subgraph to compute its currentmi,N. These values are computed

and exchanged much like the dual variables, but at a much lower rate. This process results in an accurate estimate of the max-min path score for each robot, which can be used by the lead robot to limit the value of a1

l,m and guarantee that the formation will never become infeasible.

information to the leader. Thus, we introduce a tolerance >0, that provides a margin of error when comparing the value of ml,N with the requested value ofa1l,m. When the requested value

fora1

l,m is larger thanml−,ml−is used, otherwise the requesteda1l,m is used.

With the ability to dynamically adjust the value ofa1

l,m, we shift our focus to the development

of a mechanism that will compute the desired a1

l,m. For this computation we derive inspiration

from back pressure routing, originally developed in [82]. In this system, the size of the transmit queue is used to exert pressure on the communication requirements, namely the larger the queue size the more conservative the communication requirements must become. Thus, we return to the acceptable level of delay for the map updates, dmap. The value of dmap can also be interpreted

as the maximum time a packet should remain in the transmit queue. Thus, the value computed for amap should allow for the queue to be cleared in dmap seconds. Using this interpretation,

along with the current size of the queue, qmap and the nominal packet per second rate of the

communication channel, rp, we can compute the end-to-end rate required to clear the queue as

amap=qmap/(rpdmap). This formula foramap provides the desirable properties ofamap = 0 when

the queue is empty, linear growth with the queue size, and maximum communication requirement when the queue size is very large.

Since the map updates are discrete in time, the number of packets in the queue can increase dramatically when a new area is encountered. To mitigate these large jumps, the value requested for a1

l,m = min{1.0, apri+ ˆamap}, where ˆamap is the output of a low pass filter with inputamap.

The result of these modifications is the teams ability to dynamically adjust to the size of the transmit queue, while avoiding the imposition of requirements that are infeasible for the given formation.