This document proposes algorithms to optimize communication quality in robot networks by computing optimal formations of relay vehicles. For a single remote unit, it computes a relay chain using a modified breadth-first search on a layered graph. For multiple units, it computes a minimum-cost relay tree using Steiner tree algorithms. The algorithms run in polynomial time and were tested in simulations and hardware experiments.

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Relay Vehicle Formations for Optimizing
Communication Quality in Robot Networks
Md Mahbubur Rahman
Leonardo Bobadilla
Franklin Abodo
Brian Rapp
SCIS, Florida International University
September 27, 2017

2. Introduction
Introduction
Relay Chain
Computation
Relay Tree Computation
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3. Motivation
Introduction
Relay Chain
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Relay Tree Computation
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Communication relay has vital importance in military, mining,
surveillance and rescue missions, where robots are remotely
controlled by an operator (e.g, drone) who stays in a safe location.
Problem: Wireless signal over distance degrades. Obstacles, terrain,
weather condition further attenuates the signal.
Solution: Relay robots are placed in between the operator and
remotely placed robotic units.
Unmanned Ground Vehicle Intermediate relays

4. Two Research Problems
Introduction
Relay Chain
Computation
Relay Tree Computation
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1. Single Remote Unit: Chain of intermediate relays.
2. Multiple Remote Units: A spanning tree of relays and units.
** Where to place the given number of relays to achieve maximum signal
quality?
Relay Chain Relay Tree

5. Literature Review
Introduction
Relay Chain
Computation
Relay Tree Computation
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Burdakov’10: Discretizes the environment into a grid and uses k-Hop
Bellman Ford algorithm to find the shortest sequence of grid points.
No reusable data structure is developed like us.
Tekdas and Volkan ’10: A dynamic programming solution is proposed
and the running time of the algorithm which is O(nm), compared to
our polynomial time algorithm with a running time of O(mn2)
(n=number of points on a grid and m=number of available relays).
Dixon and Frew’12: Do not consider obstacles and use aircraft as
relays that are not static and orbit around a control point. Therefore,
the solution cannot guarantee uninterrupted service for large
geographical area.
Common Drawbacks:
***Additionally, no reusability of solution is possible for the above works.
***Existing literature do not consider multiple remote units.

6. Problem Formulation
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m relay vehicles, A1, A2, . . . , Am and p mobile units
B1, B2, . . . , Bp and one static operator K.
K, Ai and Bj all are represented with configurations (x, y) ∈ E.
Free space communication cost, fF between ρ1 and ρ2,
fF (ρ1, ρ2) =
γd2(ρ1, ρ2) if d(ρ1, ρ2) < dth
∞ otherwise
(1)
The signal loss in the presence of obstacles, fO(ρ1, ρ2, O), includes the
costs resulting from diffraction (fDF ), fading (fFA):
fO(ρ1, ρ2, O) =
0 if ρ1ρ2 has LoS
fDF (O) + fFA(O) otherwise
(2)
Finally, the total communication cost fC is defined as:
fC(ρ1, ρ2) = fF (ρ1, ρ2) + fO(ρ1, ρ2, O) (3)

7. Complexity and Solution to Relay Chain Problem
Introduction
Relay Chain
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Relay Tree Computation
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A bi-criteria shortest path decision problem was proven to be
NP-Complete (arkin’91) when we need to decide if a path with m + 1
links (for m relays) is the shortest.
The optimization version of calculating the shortest m + 1 link path is
NP-Hard.
A geometric Discretization process is applied to decompose the
plane.
A common method is to decompose the environment into a grid of n
points.
Uniform Grid Adaptive Grid Triangular Mesh

8. Relay Chain Computation
Introduction
Relay Chain
Computation
Relay Tree Computation
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9. Steps of Computing the Relay Chain: Communication Graph
Introduction
Relay Chain
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Relay Tree Computation
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A graph G is computed where all the decomposed cells are
considered as the vertices.
An edge between two vertices represents the communication and the
weight is measured by the communication cost fC:
fC(ρ1, ρ2) = fF (ρ1, ρ2) + fO(ρ1, ρ2, O)
0
2
6
1
3
4
9
7
5
8
Figure 1: (a) A uniform grid; (b) Connected communication graph G with the
weights in fC;

10. Relay Chain: Layered Graph
Introduction
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Relay Tree Computation
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Create a (m + 2)-layered graph if m relays are available.
Layer l0 contains a single node corresponding to the operator’s
position.
Each other layer li has n nodes corresponding to n grid points.
Edges between subsequent layers li and li+1 will follow G.E.
0
2
6
1
3
4
9
7
5
8
Figure 2: (b) Connected communication graph G with the weights in fC; (c)
Directed layered graph G generated from G.

11. Relay Chain: Modified BFS
Introduction
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Relay Tree Computation
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Apply a modified BFS Algorithm with a hashmap that assigns the parents.
The running time is O(V + E) as every node and edge is visited once.
There are total (m + 1)(n − 1) + 1 nodes for m + 2 layers and the total number of edges is
at most |E| = (number of edges in m + 1 layers) + (number of edges in layer
l0) = m(n − 1)(n − 2) + n − 1 = O(mn2).
Algorithm 1 multiRelaySingleUnit(G(V, E))
1: G(V, E) = calculateGraph(G)
2: vs.cost = 0, and v.parent = NULL; ∀v ∈ V
3: Enqueue(Q, vs)
4: h[v.id] = ∞; ∀v ∈ G.V
5: while Q = ∅ do
6: u = Dequeue(Q)
7: for v ∈ u.Neighbors do
8: if u.cost + fC (u, v) < h[v.id] then
9: v.parent = u
10: v.cost = u.cost + fC (u, v)
11: h[v.id] = v.cost
12: Enqueue(Q, v)
13: end if
14: end for
15: end while
Directed layered graph G
Communication map M0
c

12. Extraction of a Relay Chain
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Check unit position in layer lm+1: If found, backtrack all the way to
layer l0, otherwise check in a lower layer.
Resolution Completeness: Returns success if we find the unit node
in between layers l1 and lm+1. Otherwise return failure (no solution in
the current grid resolution).
At location 5 At location 8 At location 9

13. Relay Chain: Using Communication Map
Introduction
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Relay Increase: Two, three and four. Unit at cell 5 and operator at cell 0.
m=2 m=3 m=4

14. Relay Tree Computation
Introduction
Relay Chain
Computation
Relay Tree Computation
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15. Second Problem: Connecting Multiple Units
Introduction
Relay Chain
Computation
Relay Tree Computation
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Connect the p units through m available intermediate relays to a
static base operator.
Formation will be a tree instead of a chain. Has similarity to the
(p, m)-Steiner tree problems discussed in [Watel ’13].
This tree spans over p fixed terminal nodes and m variable nodes.
However, We must limit the unit nodes from branching.
Here is an instance of 2-units, 2-relays and 1-operator problem. The units
are at positions 4 and 9.

16. Relay Tree: Steps of Computing a Tree
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Compute a min-cost tree, with exactly m relays, p units, and one operator.
p + 1 fixed nodes be VT = {vs} ∪ VB, where vs is the operator node
and VB ⊂ V is the set of nodes corresponding to the remote units.
Select m relay locations from the remaining n − p − 1 nodes. This
means
n−p−1
m sets of nodes are selected.
Create
n−p−1
m possible directed graphs. Unit nodes in a graph are
not allowed to have outgoing edges.
Compute min-Arborescence tree for all the
n−p−1
m graphs.
Select the best one in terms of communication cost.
0
2
1
4
9
0 2 3
9
4
Communication Graph Two Candidate Graphs
VT = {0, 9, 4}

17. Relay Tree: Complete Algorithm and Example
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Algorithm 2 multiRelayMultiUnit(G(V, E))
1: VT = {vs} ∪ VB
2: ϑm = {ν ∈ P(V VT ) : |ϑ| = m}
3: for νi ∈ ϑm do
4: Vi = νi ∪ VT
5: Gi = computeDiGraph(Vi)
6: if Gi.connected() then
7: Ti = minArborescence(Gi)
8: T .add(Ti)
9: end if
10: end for
11: return failure if T = Null
12: return argmin
Ti∈T
[f
T
C (Ti)]
The loop of line 3 runs
n−p−1
m
times, which can
be simplified as O(nm
).
Tarjan’s algorithms runs in O(E + V log V ) =
O(m2
+ mp + p).
The worst case running time is O(nm
(m2
+
mp + p)).
0
2
1
4
9
0 1 2
4
9
(a) (b)
0 2 3
9
4
0 2
3
9
4
(c) (d)
Figure 3: (a) A sub-graph G1 con-
structed with ν1 = {v1, v2}; (b) Re-
sulting min-arborescence tree T1 of G1;
(c) Another candidate sub-graph G2 with
ν1 = {v2, v3}; and (d) Candidate tree
T2

18. Chain Formation on Line of Sight Based Systems
Introduction
Relay Chain
Computation
Relay Tree Computation
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(a) (b) (c)
(d) (e) (f)
Figure 4: Multi relay chain simulation: (a) Four relays forming a chain; (b) and (c) Number of relays
are reduced to three and two, respectively; (d) Shadow region Φ1 for one relay; (e) and (f) Adaptive
grid decomposition for three relays and one relay.

19. Relay Chain Formation (Radio Network)
Introduction
Relay Chain
Computation
Relay Tree Computation
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(a) (b) (c)
(d) (e) (f)
Figure 5: Communication can now be established through the obstacles: (a) Four relays, (b)
three relays and (c) one relay connecting the unit to the operator. (d), (e) and (f) are the adaptive grid
decomposition with four, three and two available relays, respectively.

20. Multi-Unit Multi-Relay Tree Formation
Introduction
Relay Chain
Computation
Relay Tree Computation
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(a) (b) (c)
(d) (e) (f)
Figure 6: Multi-Relay Multi-Unit simulations. (a) and (b) show min-arborescence tree for two relays
serving four and six units, respectively; (c) shows three relays serving three units and (d) is a case of
three relays connecting five units; (e) and (f) are min-arborescence tree for four relays connecting the
units to the operator.

21. Hardware Experiment on Miniature World
Introduction
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Computation
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(a) A* path planning (b) A chain formation
(c) Multi-Unit system (d) A tree formation
Video Link:https://youtu.be/r44K-HVONc4.

22. Experiment in ECS Corridor
Introduction
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(a) (b)
(c) (d) (e)
Figure 7: Large area deployment: (a) A corridor map; (b) Placements of relays as a chain formation;
(c) Maximum signal loss has been measured at 23 dB (decibel); (d) The maximum communication cost
was measured at 37 dB; (e) Three relays are deployed and the maximum link cost has been reduced
to 25 dB from 37 dB.

23. Comparison of Running Time with Best Known Solution
Introduction
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Our layered graph, G, computation takes slightly more time as it has redundant nodes.
Computation of reusable map Ms
c is faster for smaller environments as we use a modified BFS
algorithm on G, compared to a modified Bellman-Ford algorithm used by Brdakov’10.
We achieve significant improvements in the subsequent computations than Burdakov’10, as we
only need to extract a chain of relays from Ms
c instead of recomputing the entire data structure.
Table 1: Analysis of Running Time (in seconds)
Nodes Our Method Burdakov et el.
Building G+
Ms
c computation
Subsequent
Runs
Building G +
k-hop BF
Subsequent
Runs
361 6.70+0.438 0.0052 2.39+1.23 1.05
400 8.15+0.58 0.0067 2.66+1.62 1.50
625 23.82+2.41 0.0081 6.54+4.12 4.06
729 35.57+3.70 0.0079 11.39+7.78 7.23
900 49.86+8.01 0.0095 13.52+8.93 8.85
1089 85.01+14.07 0.012 23.03+14.51 14.87

24. Summary
Introduction
Relay Chain
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Relay Tree Computation
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Solution for a relay chain system using a m + 2-layered graph.
Developed a polynomial time algorithm through modifying a breadth
first search algorithm.
Estimate the optimal placements using min-Steiner tree algorithms
to serve multiple units.
Develop a hardware test-bed to perform real world experiments.

25. Future Research Direction
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3D Relay placement can be computed with minimal modification.
Use other vehicles in an outdoor setup to test the impact of different
motion dynamics on signal strength.

26. Thank you!
Introduction
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Computation
Relay Tree Computation
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Thank you very much!