La responsabilidad social y ética que tiene el revisor fiscal en las

In order to provide situational awareness, the robots must not only maneuver safely through the environment while completing the task, but must also maintain a minimum Quality of Service (QoS) level. The QoS we consider is end-to-end data rate, namely what is the rate at which a robot can transmit its sensor measurements to any other member of the team via the ad-hoc network. For this we defineK QoS requirements, one for each flow of sensor data. Note that we define the QoS as a flow of sensor data, this is done to allow for a single set of sensor measurements to be transmitted to multiple destinations. For thekth _{QoS requirement we define the minimum}

end-to-end data rate that robot i must maintain for sensor data flow as ak

i,m. Since each sensor

data flow has an origin and a destination, we define_Sk as the source andDk as the destination of

flowk. Since only the source of the data flow requires bandwidth,ak

i,mis non-zero fori∈ Sk, and

ak

i,m = 0 fori6∈ Sk. As an example, assume there is only one sensor data flow of interest,K= 1,

originating from robot i=sdestined for the access point, i=N. With only one flow,k= 1 we would seta1

s,mto the data rate necessary to support the sensor and all othera1i,m= 0. We would

also set _S1={s} to indicate robotsbeing the source, and D1={N} to indicate that the access

pointi=N is the destination.

begin with a normalized point-to-point rate function R(xi(t), xj(t)) :R4→[0,1]. This measures

the rate at which a robot i located at xi(t) can transmit data to robot j located at xj(t). By

computing the rate between every pair of locations in the formation, x(t), we can compute a rate matrix R(x(t))∈RN×N, where elementRij(x(t)) =R(xi(t), xj(t)). While it is common to

assume that R(x(t)) is symmetric, i.e. Rij(x(t)) =Rji(x(t)), due to channel reciprocity, we do

not require this assumption to allow for extension to heterogeneous transmitters. The data flow over the network is specified by a set of routing variablesαk

ij(t)∈[0,1] which

indicate the fraction of time that robotiis sending data to robotjfor sensor data flowk. Similar to the rate matrix we collect the routing variables into a routing matrix α(t) _∈RN×N×K with

entriesαk

ij(t). Sinceαkij(t) is a fraction of time robot iis transmitting to robotj for sensor data

flowk, the sum over allj andkmust not exceed 1,PN

j=1

PK

k=1αkij(t) =

P

j,kαkij(t)≤1, for alli.

Using this definition we can now compute the rate of data flow over the communication link from robot ito robotj for sensor data flowkas αk

ijR(xi(t), xj(t)). The amount of data flowing out of

robotifor sensor data flowkcan then be computed asPN

j=1α

k

ij(t)R(xi(t), xj(t)) and the amount

of data flowing into robotifor sensor data flowkcan be computed asPN

j=1α

k

ji(t)R(xj(t), xj(t)).

If we consider the difference between the outgoing and incoming sensor data flows, minus the data destined for robot i, we can compute a communication rate margin,

aki(α(t),x(t)) = N X j=1 αkij(t)R(xi(t), xj(t))− N X j=1, i6∈Dk αkji(t)R(xj(t), xi(t)). (2.1.7)

The exclusion ofi_{∈ D}k in the second summand of (2.1.7) captures the understanding that data

destined for robot i does not impact the communication rate margin. This is because upon reaching i, the destination, there is no more need to relay the data and it can be removed from consideration. To provide network stability and prevent unbounded accumulation of data at a single robot, we require that ak

i(α(t),x(t)) ≥ 0 for all i and k. This constraint allows us to

reinterpret ak

data flowkwithout compromising network stability. Combining (2.1.7) with the QoS requirements for a data flow, we define the concept of network integrity, which is achieved when,

aki(α(t),x(t))≥aki,m,

X

j,k

αkij(t)≤1, for alli, k. (2.1.8)

We see that when (2.1.8) is satisfied fort_∈[t0, tf], real-time situational awareness is achieved for

the duration of the deployment.

Incorporating the network integrity constraints with the motion planning problem in (2.1.6) creates the concurrent mobility and communication problem,

min ˙ x(t),α(t) Γ (x(tf)) (2.1.9) s.t. x(t) =x(t0) + Z t t0 ˙ x(s)ds, x(t)_{∈ F}, for allt_∈[t0, tf], aki(α(t),x(t))≥aki,m, X j,k αkij(t)≤1, for alli, k.

The problem in (2.1.9) seeks to find optimal motion control inputs, ˙x∗_{(t), and routing variables,} α∗(t), that not only allow the team to reachx(tf) that completes the task, Γ(x(tf)) = Γmin, but

also maintain network integrity for the duration of the deployment.