Appendix A. Proof of Proposition 1
Figure A1.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 6 cluster of robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses the strategy for the data collection. With respect to this initial position of the UAV at , the positions of the CH robots are units.
Figure A1.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 6 cluster of robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses the strategy for the data collection. With respect to this initial position of the UAV at , the positions of the CH robots are units.
Figure A2.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 6 cluster of robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses the strategy for the data collection. With respect to this initial position of the UAV at , the positions of the Ch robots are units.
Figure A2.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 6 cluster of robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses the strategy for the data collection. With respect to this initial position of the UAV at , the positions of the Ch robots are units.
In this example, we take the initial position of the UAV as . With respect to this initial position, the positions of the CH robots are .
With respect to this positions, we obtain the distance between the CH robots. , , , , , , , where is the initial point of route (trajectory) of UAV.
Assume that
and let’s investigate the cost of the following two strategies,
as shown in
Figure A1 and
as shown in
Figure A2.
The UAV consumes the following energies under these two strategies.
From Equations (
A3) and (
A4),
which equals to the battery capacity of the UAV,
B. In this case,
because it visits maximum number of CH robots, 4.
Let’s compare the cost of these two strategies and for Problem 1.
The total energy consumption of the CH robots under the strategy
The total energy consumption of the CH robots under the strategy
From Equations (A7) and (A10),
As it can be seen from
Figure A1,
for Problem 1. Remind that
is the optimal strategy for the TSP which aims to visit the maximum number of CH robots, 4 in this example. From Equation (
A11),
which yields
cannot achieve optimality in this example. Hence, it is proved.
Figure A3.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 7 cluster head (CH) robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses an optimal strategy such that it visits CH 1, CH 2, CH 3, CH 4, CH 5, CH 6, and CH 7 in order (The path has a length of 244 m). With respect to the initial position at , the positions of the CH robots are .
Figure A3.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 7 cluster head (CH) robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses an optimal strategy such that it visits CH 1, CH 2, CH 3, CH 4, CH 5, CH 6, and CH 7 in order (The path has a length of 244 m). With respect to the initial position at , the positions of the CH robots are .
Appendix B. Proof of Lemma 1
The proof will be done by contradiction. Consider a multi-robot system in
Figure A3 which consists of a fusion center (FC) where the UAV starts its route and 7 cluster head (CH) robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster (to focus on the CH robots, the other robots are not shown in
Figure A3,
Figure A4 and
Figure A5). It is assumed that the initial position of the UAV, the position of the fusion center is
. With respect to this initial position, the positions of the CH robots are
. The UAV uses an optimal strategy such that it visits CH 1, CH 2, CH 3, CH 4, CH 5, CH 6 and CH 7 in order. To follow that route, the UAV needs exactly the following amount of energy
From Equation (
A13), if
, then the UAV need to desist from visiting at least one CH robot. Assume that the UAV decided to desist from visiting CH 3 and then search for an optimal route to visit all CH robots except CH 3.
The UAV uses a strategy such that it visits CH 1, CH 2, CH 4, CH 5, CH 6, and CH 7 in order. In this strategy, instead of visiting CH 3, the UAV visits CH 4 just after visiting CH 2 (reconstructing the route with a direct line from CH 2 to CH 4) To follow that route, the UAV needs exactly the following amount of energy
Figure A4.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 7 cluster head (CH) robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses an optimal strategy such that it visits CH 1, CH 2, CH 4, CH 5, CH 6, and CH 7 in order (The path has a length of 220 m). With respect to the initial position at , the positions of the CH robots are .
Figure A4.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 7 cluster head (CH) robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses an optimal strategy such that it visits CH 1, CH 2, CH 4, CH 5, CH 6, and CH 7 in order (The path has a length of 220 m). With respect to the initial position at , the positions of the CH robots are .
Figure A5.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 7 cluster head (CH) robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses an optimal strategy such that it visits CH 1, CH 2, CH 5, CH 4, CH 6 and CH 7 in order (The path has a length of 210 m). With respect to the initial position at , the positions of the CH robots are .
Figure A5.
The whole multi-robot system consists of a fusion center (FC) where the UAV starts its route and 7 cluster head (CH) robots around FC. The red circles show the locations of the cluster head (CH) robots in their robot cluster. The UAV uses an optimal strategy such that it visits CH 1, CH 2, CH 5, CH 4, CH 6 and CH 7 in order (The path has a length of 210 m). With respect to the initial position at , the positions of the CH robots are .
The UAV uses a strategy such that it visits CH 1, CH 2, CH 5, CH 4, CH 6, and CH 7 in order. To follow that route, the UAV needs exactly the following amount of energy
From Equations (
A14) and (
A15), using a direct line from CH 2 to CH 4 does not provide the optimal route for visiting all CH robots except CH 3. By this contradiction, the lemma is proved.