From (18) and (19), we could find that the GDOP of two-dimensional TOA-based cooperative localization systems is affected by the azimuth of the CNs, whereas the GDOP of the AOA-based is relying on both the azimuth and the distance between the PT and the CN. Theoretically, the deployment of cooperative nodes is a multiple parameter optimization problem with high complexity, especially for the large-scale CNs instances. In order to simplify the multiple parameter optimization problem, the OPMPD approach employs stepwise strategy to obtain the optimal position of each CN step by step just as eigenvalue-based approach. Whereas, OPMPD approach has three major advantages. First, OPMPD approach is deduced on the basis of partial differentiation which provides a more explicit scheme. Since there is no need to calculate the generalized eigenvalue and eigenvector of FIM, unnecessary computation is avoided. Second, inertia dependence factor is introduced to indicate the relationship among the optimal position of each CN, which suggests a discipline of the deployment. Third, OPMPD approach is suitable
for both two-dimensional TOA-based cooperative localization system and AOA-based cooperative localization system.

### Objective function based on stepwise strategy

In OPMPD approach, the optimal position of the CN to be assigned depends on not only the geometry of the PT and the GN but also the location of the CN already deployed since each CN is placed in sequence. To begin with, assume the first CN shall be placed, the GDOP of the TOA-based cooperative system is found to be

$$ G D O{P}_{\mathrm{C}\hbox{-} \mathrm{TOA}}^1=\sqrt{\frac{N+1}{{\displaystyle \sum_{n=1}^N{ \sin}^2\left({\beta}_1-{\alpha}_n\right)}+{\displaystyle \sum_{n=1}^{N-1}{\displaystyle \sum_{m= n+1}^N{ \sin}^2\left({\alpha}_n-{\alpha}_m\right)}}}} $$

(20)

Obviously, the current GDOP is only relevant to *β*
_{1} since *α*
_{1}, *α*
_{2}, …, *α*
_{
N
} is known. Considering that the minimization of GDOP is
equivalent
to the maximization of \( {\displaystyle {\sum}_{n=1}^N{ \sin}^2\left({\beta}_1-{\alpha}_i\right)} \), we extract the component relevant to *β*
_{1} and define the initial observation function as

$$ Q\left({\beta}_1\right)={\displaystyle {\sum}_{n=1}^N{ \sin}^2\left({\beta}_1-{\alpha}_n\right)} $$

(21)

Accordingly, if the former (*m*−1) CNs have been placed, the observation function with respect to the *m*th CN will be updated to

$$ Q\left({\beta}_m\right)={\displaystyle {\sum}_{n=1}^N{ \sin}^2\left({\beta}_m-{\alpha}_n\right)}+{\displaystyle {\sum}_{j=1}^{m-1}{ \sin}^2\left({\beta}_m-{\beta}_{\mathrm{O}, j,\mathrm{T}}\right)} $$

(22)

where *β*
_{O,j,T} denotes the best azimuth of the *j*th CN that has been acquired.

Similarly, the GDOP of the AOA cooperative system can be expressed as

$$ G D O{P}_{\mathrm{C}\hbox{-} \mathrm{AOA}}^1=\sqrt{\frac{r_1^{\hbox{-} 2}+{\displaystyle \sum_{n=1}^N{d_n}^{\hbox{-} 2}}}{r_1^{-2}{\displaystyle \sum_{n=1}^N{d_n}^{\hbox{-} 2}{ \sin}^2\left({\beta}_1-{\alpha}_n\right)}+{\displaystyle \sum_{n=1}^{N-1}{\displaystyle \sum_{m= n+1}^N{d_n}^{\hbox{-} 2}{d_m}^{\hbox{-} 2}{ \sin}^2\left({\alpha}_n-{\alpha}_m\right)}}}} $$

(23)

Suppose each *r*
_{
m
} is a fixed value, then we have

$$ Q\left({\beta}_1\right)={\displaystyle {\sum}_{n=1}^N{d_n}^{\hbox{-} 2}{ \sin}^2\left({\beta}_1-{\alpha}_n\right)} $$

(24)

$$ Q\left({\beta}_m\right)={\displaystyle {\sum}_{n=1}^N{d_n}^{\hbox{-} 2}{ \sin}^2\left({\beta}_m-{\alpha}_n\right)}+{\displaystyle {\sum}_{j=1}^{m-1}{r_j}^{\hbox{-} 2}{ \sin}^2\left({\beta}_m-{\beta}_{\mathrm{O}, j,\mathrm{T}}\right)} $$

(25)

In addition, we particularly discuss the effect of *r*
_{
m
} in section 4.3 to fully analyze the performance of GDOP of AOA-based cooperative localization system.

### OPMPD approach in TOA-based cooperative localization system

The core concept of OPMPD approach is to get the extremum of *Q*(*β*
_{
m
}) by calculating the partial derivative with respect to *β*
_{
m
}. Above all, we built the first-order partial differential equations to search the arrest points. For TOA-based cooperative localization systems, let ∂*Q*(*β*
_{1})/∂*β*
_{1} = 0 and define \( {\beta}_{1, T}^1 \) and \( {\beta}_{1,\mathrm{T}}^2 \) as the arrest points of *Q*(*β*
_{1}), then we have

$$ \left\{\begin{array}{l} \sin 2{\beta}_{1, T}^1{\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}= \cos 2{\beta}_{1, T}^1{\displaystyle {\sum}_{n=1}^N \sin 2{\alpha}_n}\\ {} \sin 2{\beta}_{1, T}^2{\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}= \cos 2{\beta}_{1, T}^2{\displaystyle {\sum}_{n=1}^N \sin 2{\alpha}_n}\end{array}\right. $$

(26)

Suppose \( {\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}\ne 0 \), \( \cos 2{\beta}_{1, T}^1\ne 0 \), and \( \cos 2{\beta}_{1, T}^2\ne 0 \), (26) can be refreshed according to the equal ratios theorem

$$ \frac{ \sin 2{\beta}_{1, T}^1}{ \cos 2{\beta}_{1, T}^1}=\frac{ \sin 2{\beta}_{1, T}^2}{ \cos 2{\beta}_{1, T}^2}=\frac{{\displaystyle {\sum}_{n=1}^N \sin 2{\alpha}_n}}{{\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}} $$

(27)

Assume \( {\beta}_{1, T}^2>{\beta}_{1, T}^1 \), then

$$ \begin{array}{l}{\beta}_{1, T}^1=\frac{1}{2}\times \arctan \left(\frac{{\displaystyle {\sum}_{n=1}^N \sin 2{\alpha}_n}}{{\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}}\right)\in \left[-\frac{\pi}{4},\frac{\pi}{4}\right]\\ {}{\beta}_{1, T}^2={\beta}_{1, T}^1+\frac{1}{2}\pi \in \left[\frac{\pi}{4},\frac{3\pi}{4}\right]\end{array} $$

(28)

For easy of depiction and distinction, we term \( {\beta}_{1, T}^1 \) and \( {\beta}_{1, T}^2 \) in (28) as positive azimuth factor and negative azimuth factor of the first CN, respectively.

Substitute \( {\beta}_{1, T}^1 \) and \( {\beta}_{1, T}^2 \) into the second-order partial derivative
*Q*″ respectively to find which azimuth factor would obtain the maximum of observation function, then we have

$$ {Q}^{{\prime\prime}}\left({\beta}_{1, T}^1\right)=2\left|\overset{\rightharpoonup }{A_{\mathrm{T}}}\right|\left|\overset{\rightharpoonup }{B_{1,\mathrm{T}}^1}\right| \cos {\gamma}_1,{Q}^{{\prime\prime}}\left({\beta}_{1, T}^2\right)=2\left|\overset{\rightharpoonup }{A_{\mathrm{T}}}\right|\left|\overset{\rightharpoonup }{B_{1,\mathrm{T}}^2}\right| \cos {\gamma}_2 $$

(29)

with

$$ \begin{array}{l}\overset{\rightharpoonup }{A_{\mathrm{T}}}=\Re + j\kappa ={\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}+ j{\displaystyle {\sum}_{n=1}^N \sin 2{\alpha}_n},\\ {}\overset{\rightharpoonup }{B_{1,\mathrm{T}}^1}= cos2{\beta}_{1, T}^1+ j sin2{\beta}_{1, T}^1,\\ {}\overset{\rightharpoonup }{B_{1,\mathrm{T}}^2}= cos2{\beta}_{1, T}^2+ j sin2{\beta}_{1, T}^2\end{array} $$

(30)

where \( \overset{\rightharpoonup }{A_{\mathrm{T}}} \) is the azimuth vector of system, and *ℜ* is the cosine azimuth coefficient. \( \overset{\rightharpoonup }{B_{1,\mathrm{T}}^1} \) and \( \overset{\rightharpoonup }{B_{1,\mathrm{T}}^2} \) are defined as the positive azimuth vector and negative azimuth vector of the first CN, *j* denotes the imaginary unit, *γ*
_{1} denotes the angle between \( \overset{\rightharpoonup }{A_{\mathrm{T}}} \) and \( \overset{\rightharpoonup }{B_{1,\mathrm{T}}^1} \),and *γ*
_{2} denotes the angle between \( \overset{\rightharpoonup }{A_{\mathrm{T}}} \) and \( \overset{\rightharpoonup }{B_{1,\mathrm{T}}^2} \).

From (27), (28), (29) and (30), we could draw three useful conclusions. First, the value of cos(*γ*
_{1}) is either 1 or −1. So does cos(*γ*
_{2}). Second, the direction of \( \overset{\rightharpoonup }{B_{1,\mathrm{T}}^1} \) is opposite to that of \( \overset{\rightharpoonup }{B_{1,\mathrm{T}}^2} \). Third, if *ℜ* < 0, the azimuth of \( \overset{\rightharpoonup }{A_{\mathrm{T}}} \) is confined from *π*/2 to 3*π*/2, thereby leading to \( G D O{P}_{\mathrm{TOA}}\left({\beta}_{1, T}^1\right) \) achieving the minimum value with cos *γ*
_{1} = − 1 and \( {Q}^{{\prime\prime}}\left({\beta}_{1, T}^1\right)<0 \). Conversely, if *ℜ* > 0, the azimuth of \( \overset{\rightharpoonup }{A_{\mathrm{T}}} \) is confined from − *π*/2 to *π*/2, which means that \( G D O{P}_{\mathrm{TOA}}\left({\beta}_{1,\mathrm{T}}^2\right) \) would get the minimum value with cos *γ*
_{2} = − 1 and \( {Q}^{{\prime\prime}}\left({\beta}_{1,\mathrm{T}}^2\right)<0 \).

Consequently, a deployment mechanism of the first CN can be summarized as

$$ {\beta}_{\mathrm{O},1,\mathrm{T}}=\left\{\begin{array}{cc}\hfill \begin{array}{l} random\\ {}\kern0.8em {\beta}_{1,\mathrm{T}}^1\end{array}\hfill & \hfill \begin{array}{l} if\mathit{\Re}=0\\ {} if\mathit{\Re}<0\end{array}\hfill \\ {}\hfill {\beta}_{1,\mathrm{T}}^2\hfill & \hfill if\mathit{\Re}>0\hfill \end{array}\right. $$

(31)

where *β*
_{O,1,T} represents the optimal azimuth corresponding to the least GDOP and \( \overset{\rightharpoonup }{B_{\mathrm{O},1,\mathrm{T}}}= cos2{\beta}_{\mathrm{O},1,\mathrm{T}}+ jsin2{\beta}_{\mathrm{O},1,\mathrm{T}} \) represents the optimal azimuth vector of the first CN.

After the first CN is arranged at the optimum position, both the azimuth vector of system and the objective function will be refreshed to

$$ \overset{\rightharpoonup }{A_{\mathrm{T}}}=\Re + j\kappa =\left( cos2{\beta}_{\mathrm{O},1,\mathrm{T}}+{\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}\right)+ j\left( \sin 2{\beta}_{\mathrm{O},1,\mathrm{T}}+{\displaystyle {\sum}_{n=1}^N \sin 2{\alpha}_n}\right) $$

(32)

$$ Q\left({\beta}_2\right)={\displaystyle {\sum}_{n=1}^N{ \sin}^2\left({\beta}_2-{\alpha}_n\right)}+{ \sin}^2\left({\beta}_2-{\beta}_{\mathrm{O},1,\mathrm{T}}\right) $$

(33)

Define \( {\beta}_{2,\mathrm{T}}^1 \) and \( {\beta}_{2,\mathrm{T}}^2 \) as the arrest points of *Q*(*β*
_{2}), then we have

$$ {\beta}_{2,\mathrm{T}}^1={\beta}_{1,\mathrm{T}}^1,\kern0.8em {\beta}_{2,\mathrm{T}}^2={\beta}_{1,\mathrm{T}}^2 $$

(34)

since

$$ \frac{\mathrm{s} in2{\beta}_{2,\mathrm{T}}^1}{ \cos 2{\beta}_{2,\mathrm{T}}^1}=\frac{ \sin 2{\beta}_{2,\mathrm{T}}^2}{ \cos 2{\beta}_{2,\mathrm{T}}^2}=\frac{{\displaystyle {\sum}_{n=1}^N \sin 2{\alpha}_n+ \sin 2{\beta}_{\mathrm{O},1,\mathrm{T}}}}{{\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n+ \cos 2{\beta}_{\mathrm{O},1,\mathrm{T}}}}=\frac{{\displaystyle {\sum}_{n=1}^N \sin 2{\alpha}_n}}{{\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}} $$

(35)

The optimal azimuth of the second CN can be determined as *ℜ* updated to \( {\displaystyle {\sum}_{n=1}^N \cos 2{\alpha}_n}+ \cos 2{\beta}_{\mathrm{O},1,\mathrm{T}} \). From (35), it is clear that either the positive azimuth factor or the negative azimuth factor of each successive CN equals to that of the first CN. Hence, we define \( {\beta}_{\mathrm{T}}^1 \) and \( {\beta}_{\mathrm{T}}^2 \) as the positive azimuth factor and negative azimuth factor of the system where \( {\beta}_{\mathrm{T}}^1={\beta}_{1,\mathrm{T}}^1 \), \( {\beta}_{\mathrm{T}}^2={\beta}_{1,\mathrm{T}}^2 \).

Suppose the absolute
value of initial cosine azimuth coefficient is larger than 2, the absolute
value of cosine azimuth coefficient refreshed will decrease as the CN is introduced since the directions of \( \overset{\rightharpoonup }{B_{\mathrm{O},1,\mathrm{T}}} \) and \( \overset{\rightharpoonup }{B_{\mathrm{O},2,\mathrm{T}}} \) are opposite to that of initial \( \overset{\rightharpoonup }{A_T} \). That is

$$ \left| \cos 2{\beta}_{\mathrm{O},1,\mathrm{T}}+ \cos 2{\beta}_{\mathrm{O},2,\mathrm{T}}+{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}\right|<\left| \cos 2{\beta}_{\mathrm{O},1,\mathrm{T}}+{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}\right|<\left|{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}\right| $$

(36)

Moreover, if initial cosine azimuth coefficient is even greater, there exists a theoretical quantity of the CNs termed as *Z*
_{T} satisfying

$$ {\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+1,\mathrm{T}}\ne {\beta}_{\mathrm{O},{Z}_{\mathrm{T}},\mathrm{T}}\dots ={\beta}_{\mathrm{O},2,\mathrm{T}}={\beta}_{\mathrm{O},1,\mathrm{T}} $$

(37)

which means that the optimal azimuth of the former *Z*
_{T} CNs equal to *β*
_{O,1,T} rather than \( {\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+1,\mathrm{T}} \).

For simplicity, we define *Z*
_{T} as inertia dependency factor and estimate it by

$$ \left\{\begin{array}{l}\left(\left({Z}_{\mathrm{T}}-1\right)\cdot \cos 2{\beta}_{\mathrm{O},1,\mathrm{T}}+{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}\right){\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}>0\\ {}\left({Z}_{\mathrm{T}} \cos 2{\beta}_{\mathrm{O},1,\mathrm{T}}+{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}\right){\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}<0\end{array}\right. $$

(38)

From (38), we have

$$ \left\{\begin{array}{l}1+\left({Z}_{\mathrm{T}}-1\right) V>0\\ {}{Z}_{\mathrm{T}} V+1<0\end{array}\right. $$

(39)

with

$$ V=\frac{ \sin 2{\beta}_{\mathrm{O},1, T}}{{\displaystyle \sum_{n=1}^N sin2{\alpha}_n}}=\frac{ \cos 2{\beta}_{\mathrm{O},1, T}}{{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}} $$

(40)

Hence,

$$ {Z}_{\mathrm{T}}=\left\lceil \sqrt{{\left({\displaystyle \sum_{n=1}^N \sin 2{\alpha}_n}\right)}^2+{\left({\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}\right)}^2}\right\rceil $$

(41)

where ⌈⌉ denotes of the operation of rounding up.

The effect of the magnitude of *M* on the deployment is also taken into consideration in our research. Assume the initial value of *ℜ* is less than zero. If *Z*
_{T} ≥ *M*, the relationship among the optimal azimuth of each CN is found to be

$$ {\beta}_{\mathrm{O}, M,\mathrm{T}}\dots ={\beta}_{\mathrm{O},2,\mathrm{T}}={\beta}_{\mathrm{O},1,\mathrm{T}}={\beta}_{\mathrm{T}}^1 $$

(42)

On the contrary, if *Z*
_{T} < *M*, it is necessary to explore the optimal azimuth of CN whose index is larger than *Z*
_{T}. For the (*Z*
_{T} + 1)th CN to be placed, *ℜ* is updated to

$$ \Re ={Z}_{\mathrm{T}}\cdot \cos 2{\beta}_{\mathrm{T}}^1+{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n} $$

(43)

Since \( \left({Z}_{\mathrm{T}} \cos 2{\beta}_{\mathrm{T}}^1+{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}\right)<0 \), we have

$$ {\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+1,\mathrm{T}}={\beta}_{\mathrm{T}}^2 $$

(44)

Then, for the (*Z*
_{T} + 2)th CN to be placed, we have

$$ \Re ={Z}_{\mathrm{T}}\cdot \cos 2{\beta}_{\mathrm{T}}^1+{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n}+ \cos 2{\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+1,\mathrm{T}}=\left({Z}_{\mathrm{T}}-1\right)\cdot \cos 2{\beta}_{\mathrm{T}}^1+{\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n} $$

(45)

So

$$ {\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+2,\mathrm{T}}={\beta}_{\mathrm{T}}^1 $$

(46)

Consequently, the relationship among the optimal azimuth of each CN is found to be

$$ \begin{array}{l}{\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+2 K,\mathrm{T}}=\dots ={\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+4,\mathrm{T}}={\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+2,\mathrm{T}}={\beta}_{\mathrm{O},{Z}_{\mathrm{T}},\mathrm{T}}\dots ={\beta}_{\mathrm{O},2,\mathrm{T}}={\beta}_{\mathrm{O},1,\mathrm{T}}={\beta}_{\mathrm{T}}^1\\ {}{\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+2 K-1,\mathrm{T}}=\dots ={\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+3,\mathrm{T}}={\beta}_{\mathrm{O},{Z}_{\mathrm{T}}+1,\mathrm{T}}={\beta}_{\mathrm{T}}^2\end{array} $$

(47)

with

$$ K=\left\lfloor 0.5\left( M+1-{Z}_{\mathrm{T}}\right)\right\rfloor $$

(48)

where ⌊⌋ denotes of the coperation of rounding down. Similar conclusion can also be derived in the case of the initial value of *ℜ*
equal
or
greater than zero.

Hence, a solution to the optimal deployment of cooperative nodes in two-dimensional TOA-based localization system aiming at the lowest GDOP can be summarized as follows.

Step 1: According to the spatial relationship between the GN and PT, establish the initial observation function and calculate \( {\beta}_{\mathrm{T}}^1 \), \( {\beta}_{\mathrm{T}}^2 \), *Z*
_{T} and the initial *ℜ*.

Step 2: Determine the optimal azimuth of the first CN from \( {\beta}_{\mathrm{T}}^1 \) and \( {\beta}_{\mathrm{T}}^2 \), according to the polarity of *ℜ*.

Step 3: If *Z*
_{T} < *M*, deploy the whole CNs in the same azimuth as the first CN assigned.

Step 4: If *Z*
_{T} ≥ *M*, deploy the former *Z*
_{T} CNs in the same azimuth as the first CN assigned while place the (*Z*
_{T}
*+*1)th CN in the azimuth vertical to that of the first CN. Then, \( {\beta}_{\mathrm{T}}^1 \) and \( {\beta}_{\mathrm{T}}^2 \) are alternatively chosen as the optimal azimuth of the rest CNs.

### OPMPD approach in AOA-based cooperative localization system

As mentioned in the initial portion of the “Section 4,” in two-dimensional AOA-based localization system, both the azimuth and distance of each CN exert
an
influence on the performance of the GDOP.

First of all, we analyze the influence of *β*
_{1} on the value of \( G D O{P}_{\mathrm{C}\hbox{-} \mathrm{AOA}}^1 \) depicted in (23) assuming each *r*
_{
m
} is a fixed value. With the help of a method similar to that in TOA-based system, the positive azimuth factor and negative azimuth factor of the system can be derived as

$$ \begin{array}{l}{\beta}_{\mathrm{A}}^1={\beta}_{1,\mathrm{A}}^1=0.5\times \arctan \left({\displaystyle {\sum}_{n=1}^N{d_n}^{\hbox{-} 2} \sin 2{\alpha}_n}/{\displaystyle {\sum}_{n=1}^N{d_n}^{\hbox{-} 2} \cos 2{\alpha}_n}\right)\\ {}{\beta}_{\mathrm{A}}^2={\beta}_{1,\mathrm{A}}^2={\beta}_{1,\mathrm{A}}^1+0.5\pi \end{array} $$

(49)

where \( {\beta}_{1,\mathrm{A}}^1 \) and \( {\beta}_{1,\mathrm{A}}^2 \) denote the arrest points of the observation function depicted in (24). The azimuth vector of system can also be obtained by

$$ \overset{\rightharpoonup }{A_{\mathrm{A}}}=\Re + j\kappa ={\displaystyle {\sum}_{n=1}^N{d_n}^{\hbox{-} 2} \cos 2{\alpha}_n}+ j{\displaystyle {\sum}_{n=1}^N{d_n}^{\hbox{-} 2} \sin 2{\alpha}_n} $$

(50)

Accordingly, the optimal azimuth of the first CN is updated to

$$ {\beta}_{\mathrm{O},1,\mathrm{A}}=\left\{\begin{array}{cc}\hfill \begin{array}{l} random\\ {}\kern1em {\beta}_{\mathrm{A}}^1\end{array}\hfill & \hfill \begin{array}{l} if\mathit{\Re}=0\\ {} if\mathit{\Re}<0\end{array}\hfill \\ {}\hfill {\beta}_{\mathrm{A}}^2\hfill & \hfill if\mathit{\Re}>0\hfill \end{array}\right. $$

(51)

where *β*
_{O,1,A} represents the optimal azimuth corresponding to the least GDOP.

Secondly, we further discuss the influence of *r*
_{1} on \( G D O{P}_{\mathrm{AOA}}^1 \). Suppose the scope of *r*
_{
m
} follows

$$ {r}_{1, \min}\le {r}_m\le {r}_{1, \max } $$

(52)

where *r*
_{1,min} and *r*
_{1,max} denote the minimum and maximum. For simplicity, let

$$ \begin{array}{l} C={r}_1^{\hbox{-} 2},\\ {} D={\displaystyle \sum_{n=1}^N{d_n}^{\hbox{-} 2}},\\ {} E={\displaystyle \sum_{n=1}^N{d_n}^{\hbox{-} 2}{ \sin}^2\left({\beta}_1-{\alpha}_n\right)},\\ {} F={\displaystyle \sum_{n=1}^{N-1}{\displaystyle \sum_{m= n+1}^N{d_n}^{\hbox{-} 2}{d_m}^{\hbox{-} 2}{ \sin}^2\left({\alpha}_n-{\alpha}_m\right)}}\end{array} $$

(53)

The first-order derivative of \( G D O{P}_{\mathrm{AOA}}^1 \) with respect to *r*
_{1} is expressed as follows:

$$ \frac{\partial GDO{P}_{\mathrm{AOA}}^1\left({r}_1\right)}{\partial {r}_1}=\frac{1}{2 GDO{P}_{\mathrm{AOA}}^1}\frac{-2\left( F- DE\right)}{r_1^3{\left( F+ CE\right)}^2} $$

(54)

Here, we pay our attention to the polarity of (*F* − *DE*) to investigate the monotone property of \( G D O{P}_{\mathrm{AOA}}^1 \). Based on (53), we have

$$ \begin{array}{l}2 E- D=-\left( \cos 2\beta {\displaystyle \sum_{n=1}^N \cos 2{\alpha}_n{d_n}^{\hbox{-} 2}}+ \sin 2\beta {\displaystyle \sum_{n=1}^N \sin 2{\alpha}_n{d_n}^{\hbox{-} 2}}\right)\\ {}\kern3.6em =\sqrt{\left({\displaystyle \sum_{n=1}^N{d_n}^{\hbox{-} 4}}\right)+2{\displaystyle \sum_{n=1}^{N-1}{\displaystyle \sum_{m= i+1}^N{d_n}^{\hbox{-} 2}{d_m}^{\hbox{-} 2}\left(1-2 si{n}^2\left({\alpha}_n-{\alpha}_m\right)\right)}}}\end{array} $$

(55)

$$ \begin{array}{l}{D}^2-4 F={\left({\displaystyle \sum_{n=1}^N{d_n}^{\hbox{-} 2}}\right)}^2-4{\displaystyle \sum_{n=1}^{N-1}{\displaystyle \sum_{m= n+1}^N{d_n}^{\hbox{-} 2}{d_m}^{\hbox{-} 2}{ \sin}^2\left({\alpha}_n-{\alpha}_m\right)}}\\ {}\kern5.5em =\left({\displaystyle \sum_{n=1}^N{d_n}^{\hbox{-} 4}}\right)+2{\displaystyle \sum_{n=1}^{N-1}{\displaystyle \sum_{m= n+1}^N{d_n}^{\hbox{-} 2}{d_m}^{\hbox{-} 2}\left(1-2 si{n}^2\left({\alpha}_n-{\alpha}_m\right)\right)}}\end{array} $$

(56)

$$ 2 F-{D}^2=-\left({\displaystyle \sum_{i=1}^p{d_i}^{\hbox{-} 4}}\right)+2{\displaystyle \sum_{i=1}^{p-1}{\displaystyle \sum_{j= i+1}^p{d_i}^{\hbox{-} 2}{d_j}^{\hbox{-} 2}\left( si{n}^2\left({\alpha}_j-{\alpha}_i\right)-1\right)}}<0 $$

(57)

From (55), (56), and (57), we have

$$ \sqrt{D^2-4 F}=2 E- D\ge 0\ge \frac{2 F-{D}^2}{D} $$

(58)

Futhermore, we have

Obviously, \( G D O{P}_{\mathrm{AOA}}^1 \) is an increasing function with respect to *r*
_{1}, which means *r*
_{1,min} is the optimal distance of the first CN. Similar conclusion is available for the other CN. After the former *m* CN have been placed, we have

$$ \Re ={\displaystyle {\sum}_{j=1}^{m-1}{r}_{j, \min}^{-2} \cos 2{\beta}_{\mathrm{O}, j,\mathrm{A}}}+{\displaystyle {\sum}_{n=1}^N{d_n}^{\hbox{-} 2} \cos 2{\alpha}_n} $$

(60)

Then the optimal azimuth of the (*m* + 1)th CN to be placed can be determined by

$$ {\beta}_{\mathrm{O}, m+1,\mathrm{A}}=\left\{\begin{array}{cc}\hfill \begin{array}{l} random\\ {}\kern1em {\beta}_{\mathrm{A}}^1\end{array}\hfill & \hfill \begin{array}{l} if\mathit{\Re}=0\\ {} if\mathit{\Re}<0\end{array}\hfill \\ {}\hfill {\beta}_{\mathrm{A}}^2\hfill & \hfill if\mathit{\Re}>0\hfill \end{array}\right. $$

(61)

Particularly, if *r*
_{1,min} = *r*
_{2,min} = … = *r*
_{
M,min} = *r*
_{min}, the inertia dependency factor can also be calculated by

$$ \left\{\begin{array}{l}\left(\left({Z}_{\mathrm{A}}-1\right)\cdot {r}_{\min}^{-2} \cos 2{\beta}_{\mathrm{O},1,\mathrm{A}}+{\displaystyle \sum_{n=1}^N{d}_n^{-2} \cos 2{\alpha}_n}\right){\displaystyle \sum_{n=1}^N{d}_n^{-2} \cos 2{\alpha}_n}>0\\ {}\left({Z}_{\mathrm{A}}{r}_{\min}^{-2} \cos 2{\beta}_{\mathrm{O},1,\mathrm{A}}+{\displaystyle \sum_{n=1}^N{d}_n^{-2} \cos 2{\alpha}_n}\right){\displaystyle \sum_{n=1}^N{d}_n^{-2} \cos 2{\alpha}_n}<0\end{array}\right. $$

(62)

so

$$ {Z}_{\mathrm{A}}=\left\lceil \sqrt{{\left({\displaystyle \sum_{n=1}^N{r}_{\min}^2{d}_n^{-2} \sin 2{\alpha}_n}\right)}^2+{\left({\displaystyle \sum_{n=1}^N{r}_{\min}^2{d}_n^{-2} \cos 2{\alpha}_n}\right)}^2}\right\rceil $$

(63)

Finally, the solution to the optimal deployment of cooperative nodes in two-dimensional AOA-based localization system can be generalized as follows.

Step 1: According to the spatial relationship between the GN and PT, establish the initial observation function and calculate \( {\beta}_{\mathrm{A}}^1 \), \( {\beta}_{\mathrm{A}}^2 \), and the initial *ℜ*.

Step 2: Choose the lower
bound of each *r*
_{
m
} as the the optimal distance for each CN.

Step 3: Determine the optimal azimuth of the first CN from \( {\beta}_{\mathrm{A}}^1 \) and \( {\beta}_{\mathrm{A}}^2 \), according to the polarity of *ℜ*.

Step 4: If *r*
_{1,min} = *r*
_{2,min} = … = *r*
_{
M,min} = *r*
_{min} is not satisfied, determine the optimal azimuth of the other CN according to the polarity of *ℜ* updated.

Step 5: If *r*
_{1,min} = *r*
_{2,min} = … = *r*
_{
M,min} = *r*
_{min} is satisfied, calculate *Z*
_{A}. If *Z*
_{A} < *M*, deploy the whole CNs in the same azimuth as the first CN assigned. If *Z*
_{A} ≥ *M*, deploy the former *Z*
_{A} CNs in the same azimuth as the first CN assigned while place the (*Z*
_{A}
*+*1)th CN in the azimuth vertical to that of the first CN. Then, \( {\beta}_{\mathrm{A}}^1 \) and \( {\beta}_{\mathrm{A}}^2 \) are alternatively chosen as the optimal azimuth of the rest CNs.