In Chapter 3, we have seen that submodularity theory is a powerful tool in solving some multi-agent tasks. To date, optimal coverage problems, sensor deployment prob- lems (Krause et al., 2008b; Krause et al., 2008a), and task assignment problems (Qu et al., 2015) have been solved by greedy algorithms in a centralized fashion. Design of
distributed or asynchronous greedy algorithms for multi-agent systems with probably performance bound is an interesting direction. In addition, extending our optimal coverage solution to 3-dimensional mission space is also an interesting research direc- tion, especially when the sensor network operates in environments with mountains, forests, high-rise buildings, or an array of wind turbines.
In Chapter 4, the leader is assumed to be fixed and given a trajectory in the mission of optimal dynamic formation control. Another interesting direction is to relax this assumption to make the whole agent system to be fully autonomous. The leader could be time-varying, thus making the problem more challenge in communication preservation. If multiple leaders are allowed, it could be an optimal containment control problem where mobile leaders form a convex hull and followers are controlled to move within this convex hull while the whole team optimizing their joint coverage of the mission space.
In both optimal coverage problem and optimal dynamic formation control prob- lem, the obstacles are assumed to be time-invariant. Another interesting direction could be taking the moving obstacles into account.
Appendix A
Mathematical Proofs
A.1
Proof of Equivalent Definitions
Definition 1 ⇒ Definition 2
Suppose that S ⊆ T , y /∈ T , and f satisfies (3.7). Replacing S in (3.7) by S ∪ {y} gives
f (S ∪ {y} ∪ T ) + f (S ∪ {y} ∩ T ) ≤ f (S ∪ {y}) + f (T ). (A.1) Rearranging the terms in (A.1), we obtain (3.8).
Definition 2 ⇒ Definition 1
Suppose that f satisfies (3.8), and y ∈ S/(S ∩ T ). It is easy to verify that S ∩ T ⊆ T , and y /∈ T . Replacing S in (3.8) by S ∩ T , and using (3.8) repeatedly to all elements y ∈ S/(S ∩ T ), we obtain
f (S ∩ T ∪ (S/(S ∩ T ))) | {z } f (S) − f (S ∩ T ) ≥ f (T ∪ (S/(S ∩ T ))) | {z } f (T ∪S) − f (T ) . (A.2)
Rearranging the terms in (A.2) gives (3.7).
A.2
Proof of Theorem 5 in Chapter 4
Before we prove Theorem 5, we provide some notation and prove three lemmas. For simplicity, we omit t from si(t) in what follows. For any i ∈N , we define the maximal
detection quality
Mi = R
Z
Ωi
pi(x, si)dx (A.3)
Lemma 2 If pi(x, si) = p(x, si) for all i ∈N , then Mi = M for all i ∈N .
Proof: In the local polar coordinate system, Mi is given by
Mi = R Z ΩL i ripi(ri)dθidri = R 2π Z 0 C Z 0 ripi(ri)dθidri = 2πR C Z 0 ripi(ri)dri (A.4)
where pi(ri) = p(ri) if pi(x, si) = p(x, si), so the statement holds and Mi is spatially
invariant.
Lemma 3 Assume that (i) the feasible space is F = Ω = R2, C ≤ 2δ and R(x) = R
for all x ∈ F , and (ii) pi(x, si) = p(x, si) for all i ∈ N . Then, the global optimal
solution to problem (4.9) when N = 1 is any position vector (s0, s1) such that ks0−
s1k = C and the flow variable is ρ = (1, 0).
Proof: Recalling the sensing model (4.12) and the assumption pi(x, si) = p(x, si),
the objective function in (4.10) for N = 1 is H(s) = Z F R(x)P (x, s)dx = R Z Ω0 p(x, s0)dx + R Z Ω1 p(x, s1)dx − R Z Ω0∩Ω1 p(x, s0)p(x, s1)dx (A.5)
where the first two terms are the constants M0 = M1 = M by Lemma 1 and we define
a function M2(si, sj) to represent the third term:
M2(si, sj) = R
Z
Ωi(si)∩Ωj(sj)
Using M and M2(si, sj) in (A.5), we get
H(s) = 2M − M2(s0, s1) (A.7)
Let sC = (s0, sC) and sz = (s0, sz) be two feasible solutions where ks0− sCk = C and
ks0 − szk = z, 0 ≤ z < C. We will show that H(sC) is a global optimal solution,
i.e., H(sC) > H(sz) holds for any z. To facilitate this proof, we establish a Cartesian
coordinate system where s0 = (0, 0), sC = (C, 0) and sz = (z, 0), as shown in Fig.
A·1. Accordingly, we define the sensing range intersections Aa = Ω0(s0) ∩ Ω1(sC) and
Ab = Ω0(s0) ∩ Ω1(sz). Note that Aa⊂ Ab due to the fact that z < C. It follows from
(A.7) that H(sC) − H(sz) =R Z Ab p(x, s0)p(x, sz)dx − R Z Aa p(x, s0)p(x, sC)dx =R Z Aa p(x, s0)[p(x, sz) − p(x, sC)]dx + R Z Ab\Aa p(x, s0)p(x, sz)dx (A.8) Observing that if z > C, sz is an infeasible solution, we will prove next that H(sC) −
H(sz) > 0 for any 0 ≤ z < C. Define a function ˜p(x, sz, sC) = p(x, sz) − p(x, sC)
and observe that it has the following properties which are direct consequences of the monotonicity of the function p(·) in kx − sik, i = C, z:
P 1 :˜p(x, sz, sC) > 0 if xx < (z + C)/2 = 0 if xx = (z + C)/2 < 0 if xx > (z + C)/2 P 2 :˜p(x, sz, sC) = −˜p(x0, sz, sC) if kx − szk = kx0− sCk and kx − sCk = kx0− szk (A.9) We then consider two cases: z ≥ 2δ − C and 0 ≤ z < 2δ − C, corresponding to Fig. A·1 and Fig. A·2, respectively.
𝐴
𝑏𝑥
𝑦
𝑧
𝐶𝛿
𝐴
𝑎Ω
0(𝑠
0)
Ω
1(𝑠
𝑧)
Ω
1(𝑠
𝐶)
Figure A·1: The sensing ranges of agents 0 and 1 where sC = (C, 0), sz = (z, 0)
and z ≥ 2δ − C.
𝑧
𝐶 0x
x
𝐴𝑎 𝐴𝑎1 𝐴𝑎2 𝐴𝑎2′ 𝐶 + 𝑧 2Figure A·2: Subsets of Aa when z < 2δ − C. Aa = Aa1∪ Aa2 where Aa1 is the
green shape. A0a2 (the blue-line green-filled shape) and Aa2 (the blue-line write-
1. If z ≥ 2δ − C, then for any point x ∈ Aa we have xx < (z + C)/2. Using P 1,
we get ˜p(x, sz, sC) > 0. It follows that H(sC) − H(sz) in (A.8) is positive since
all integrands are positive: H(sC) − H(sz) =R Z Aa p(x, s0)˜p(x, sz, sC)dx + R Z Ab\Aa p0(x, s0)p1(x, sz)dx > 0 (A.10)
2. If 0 ≤ z < 2δ − C, then we divide the set Aa into two subsets Aa1 = {x|xx ≤
(z + C)/2, x ∈ Aa} and Aa2 = {x|xx > (z + C)/2, x ∈ Aa}. In the set Aa1, we
can find a subset A0a2 = {(z + C − xx, xy), (xx, xy) ∈ Aa2} which is symmetric
to Aa2 around an axis through xx = (z + C)/2 (see Fig. A·2). Then, for any
point x ∈ Aa2, there exists a point x0 ∈ A0a2 such that kx − szk = kx0− sCk and
kx − sCk = kx0− szk. Using P 2, we obtain ˜p(x, sz, sC) = −˜p(x0, sz, sC). Hence,
Z
Aa2∪A0a2
p(x, s0)˜p(x, sz, sC)dx = 0 (A.11)
Let A1a1 = Aa1 \ A0a2, therefore Aa = Aa2 ∪ A1a1 ∪ A0a2. Accordingly, (A.8) is
positive H(sC) − H(sz) =R Z Aa2∪A0a2 p(x, s0)˜p(x, sz, sC)dx + R Z A1 a1 p(x, s0)˜p(x, sz, sC)dx +R Z Ab\Aa p(x, s0)p(x, sz)dx > 0 (A.12)
since the first term is zero due to (A.11), the second term is positive by P 1 and the third term is positive because of the positive integrand. Thus, in both cases (A.10) and (A.12) yield H(sC) > H(sz), i.e., any vector (s0, s1) such that ks0− s1k = C is
Ω0(𝑠0) Ω1(𝑠1) 0 𝐶 𝑥 𝑦 𝛿 𝑥′ Ω𝑖(𝑠𝑖) Ω𝑗(𝑠𝑗) 𝛿 𝑦′ 𝑠𝑖 𝑠𝑗
Figure A·3: Two Cartesian coordinate systems x − y and x0− y0
Lemma 2 establishes the fact that if there are only two agents in the feasible space, the optimal solution is obtained when the two agents are located at a distance of C from each other. Using the definition of Aa, we define MC as
MC = R
Z
Aa
p(x, s0)p(x, sC)dx (A.13)
and obtain a final lemma:
Lemma 4 For agents i and j, if ksi− sjk = C, then M2(si, sj) = MC.
Proof: We establish a Cartesian coordinate system where the original point is si,
the x-axis is in the same line as sj − si, as shown in Fig. A·3. Then, it immediately
follows that the result of the integration for M2(si, sj) = MC.
We prove the result by induction and the use of Lemmas 1-3. When N = 1, by Lemma 2, the optimal formation is obtained by connecting the two agents with the distance between them being C. Next, we assume that when N = k the optimal formation Gk(s) is a tree with the distance between the connected agents being C.
Without loss of generality, these k agents are labeled 1, ..., k.
Ω(Ei) = Ωai∩ Ωbi where (ai, bi) = Ei ∈ E, i = 1, . . . , k − 1. In addition, it is impossible
for more than two agents to be connected to each other because there is no cycle in the tree, which implies that Ωi1 ∩ . . . ∩ Ωip = ∅ for any ip ∈ {0, ...k} and p > 2.
When N = k +1, the optimal formationGk+1(s) is obtained by connecting the new
agent toGk(s) with the distance between connected agents being C. Assume that the
new agent is labeled h = k+1. Agent h may establish connection with p (p ≥ 1) agents at the same time (examples are shown in Figs. A·4-A·5). Accordingly, we denote the position of agent h as sph if it connects to p agents and let sp = (s0, . . . , sk, sph). Next,
we will show that H(s1) > H(sp) for any p ≥ 1.
𝑠1 𝑠ℎ1
Figure A·4: Agent h connects to agent 1
𝑠"#
𝑠#
𝑠$
Figure A·5: Agent h connects to
agents 1 and 2
Similar to M2(si, sj), we define a function
M3(si, sj, sh) = R
Z
Ω(Ei)∩Ωh
pi(x, sai)pj(x, sbi)ph(x, sh)
and we can write the objective function H(sp) for N + 1 agents as follows:
H(sp) = (k + 2)M − X m,n6=h M2(sm, sn) − p X i=1 M2(si, s p h) + p−1 X i=1 M3(sai, sbi, s p h) (A.14)
where (ai, bi) = Ei. For p = 1, 2, H(s1) and H(s2) are H(s1) = (k + 2)M − X m,n6=h M2(sm, sn) − M2(s1, s1h). H(s2) = (k + 2)M − X m,n6=h M2(sm, sn) − M2(s1, s2h) − M2(s2, s2h) + M3(sai, sbi, s 2 h)
In Figs. A·4-A·5, a1 = 1 and a2 = 2. Note that M2(s1, sh1) = M2(s1, s2h) = MC due to
the fact that ks1− s1hk = ks1− s2hk = C and invoking Lemma 3. Therefore,
H(s1) − H(s2) =M2(s2, sh2) − M3(s1, s2, s2h) =R Z Ω1∩Ω2∩Ωh p1(x, si)ph(x, s2h)[1 − p2(x, sj)]dx +R Z Ω1∩Ωh\Ω1∩Ω2∩Ωh p1(x, s1)ph(x, s2h)dx > 0 (A.15)
since both integrands are positive. Next, we obtain H(s1) − H(sp):
H(s1) − H(sp) = p−1 X i=1 (M2(si+1, sph) − M3(sai, sbi, s p h)) =(p − 1) (H(s1) − H(s2)) (A.16)
This is true due to the fact that ksi− sphk = C for i = 1, . . . , p − 1, thus M2(si+1, sph) =
MC using Lemma 3, and M3(sai, sbi, s
p
h) = M3(s1, s2, s p h).
We conclude that if agent h = k + 1 is connected to p agents of the tree, then H(s1) − H(sp) = (p − 1)(H(s1) − H(s2)) > 0. In other words, the optimal solution is
obtained when the newly added agent is connected to a single agent and the resulting formation Gk+1(s) is still a tree. Moreover, the distance between agent h and the
agent it is connected to, say j, is C, which can be proved with the same argument as that used in Lemma 2, i.e., we can perturb sh from djh = C to djh < C and show
that djh = C is the optimal solution. In addition, by Lemma 3, connecting h to any
feasible agent j results in the same objective function value.
Corollary For an optimal formation with N + 1 agents, the objective function is H(s) = (N + 1)M − N MC where M and MC are as defined in Lemma 1 and (A.13).
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