Parabolic Spiral Search Plan for a Randomly Located Target in the Plane

Abstract

This paper addresses the problem of searching for a located target in the plane by using one searcher starting its motion from the point (𝑋₀,𝑌₀). The searcher moves along parabolic spiral curve. The position of the target has a known distribution. We show that the distance between the target position and the searcher starting point depends on the number of revolutions, where the complete revolution is done when 𝑡=2𝜋. Furthermore, we study this technique in the one-dimensional case (i.e., when the searcher moves with linear search technique). It is desired to get the expected value of the time for detecting the target. Illustrative examples are given to demonstrate the applicability of this technique assuming circular normal distributed estimates of the target position.

1. Introduction

Search problem dates back to World War II, and the works of Koopman [1–3] and Stone [5, 6] offer a classic treatise of this area from an operational research perspective. Many variants and extensions of this problem, in a wide variety of directions, have been presented in both the statistical and operations research literature since Koopman [1] solved this problem for the unidimensional case. This problem has been discussed under some specific hypotheses by Richardson [4]. However, as pointed out by Koopman [2, 3], there is so much complexity in real search and rescue missions that any statistical model can only reflect part of the real-life situation and ours are no exception. On search theory in general, Stone [5] has given a good account of various results presently available, with some informative examples, and also has provided a rigorous mathematical treatment of the subject, for both discrete and continuous cases. On the other hand, Stone [6] provided an overview of different areas in the development of search theory, which could be designated as classical, mathematical, algorithmic, and dynamic. Also, exhaustive surveys of works realized on this topic have been given by Iida [7] and Benkoski et al. [8].

Studying of searching problem in the plane had found considerable interest among researchers due to its wide applications in our life. Feinerman et al. [9] introduced some search trajectories to find the desert Ants Nearby Treasure. They used multiple searchers without communication between them in the plane. Edelsbrunner and Maurer [10] obtained the optimal solutions for the postoffice problem. The certain point location problems in two dimensions are derived via geometric transforms from an optimal solution for the search problem in three dimensions: find the first point hit by a rotating or sweeping plane. Mohamed et al. [11, 12] obtained more interesting search plans that give the minimum expected value of the cost for detecting the lost target. This problem has been discussed when the located target has symmetric and asymmetric distributions and with less information available to the searchers. In both papers, they desired to minimize the expected time for detecting the target. El-Hadidy [13] introduced a new search plan in the plane which is divided into identical cells and the searcher moves along spiral with line segment curve. He found the optimal value of the arcs that the searcher should do and finds the target with minimum cost. Also, Mohamed and El-Hadidy [14] discussed the existence of the search plan that finds the conditionally deterministic target motion. They have shown the existence of the optimal search plan and found the necessary conditions that make the expected value of the first meeting time finite. When the target starts its motion as parabolic spiral from a random point in the plane, Mohamed and El-Hadidy [15] studied this problem such that no time information about the target’s position is available to the searcher. The searcher starts its motion from the origin. They formulated a search model and found the conditions that make this search plan finite. Recently, El-Hadidy [16] formulated another search model for a helix target motion in the space by using a team of three searchers. He studied the existence of the optimal search plan and found it.

In this paper, the search for a randomly located target in the plane by a single searcher is studied. The target's position is unknown, but the searcher knows its probability distribution. The searcher moves along parabolic spiral curve as in Figure 1, starting its motion from the point (𝑋₀,𝑌₀). Furthermore, we study this technique in the one-dimensional case (i.e., when the searcher moves with linear search technique but here the searcher moves with speed depending on time). This problem is more effective and more applicable to real-world search scenarios such as searching for underwater target by firing with 91RTE2 missile system; see website http://www.youtube.com/watch?v=EgQk-FEGV-Y and also the electrons movement in a cyclotron under the influence of a constant magnetic field. The purpose here is to obtain the expected value of the time for detecting the target, assuming circular normal distributed estimates of its position.

Figure 1 

The search path of the searcher.

This paper is organized as follows. In Section 2 we formulate the problem. The expected value of the time is given when the target position has circular normal distribution in Section 3. One-dimensional case that rides to the generalized linear search problem is discussed in Section 4. Finally, the paper concludes with a discussion of the results and directions for future research.

2. Modeling of Search Problem and Formulation

Here, assumptions of a parabolic spiral search plan for a randomly located target are described and the problem is mathematically formulated as an allocation of searching effort which is the expected value of the time for detecting the target.

Let 𝑋,𝑌 be two independent random variables that represent the position of the target. The searcher is assumed to move according to the model [17] 𝑥(𝑡)=𝑋₀+𝑡²(𝑡),𝑦(𝑡)cos=𝑌₀+𝑡²sin(𝑡), (1) where (𝑋₀,𝑌₀) is the starting point of the searcher and 𝑡∈{0}∪ℝ⁺, where ℝ is the set of real numbers. Equations (1) show that the searcher moves along parabolic spiral curve as in Figure 1.

The searching process is continuous time and continuous space. We consider that the searcher will make a complete revolution when returns the 𝑥^′-axis; see Figure 1. At time 𝑡, the searcher moves along a circle with centre (𝑋₀,𝑌₀) and radius 𝑡² given by 𝑥(𝑡)−𝑋₀²+𝑦(𝑡)−𝑌₀²=𝑡⁴. (2)

From (1) we have 𝑑𝑥(𝑡)=̇(𝑡)𝑑𝑡𝑥=2𝑡cos−𝑡²(𝑡),(𝑡)sin𝑑𝑦=̇(𝑡)𝑑𝑡𝑦=2𝑡sin+𝑡²cos(𝑡). (3)

Thus, the speed of the searcher will be given as 𝑣=𝑥̇²+𝑦̇²=√4𝑡²+𝑡⁴=𝑡√𝑡²+4. (4)

It is clear that the speed will vanish at 𝑡=0 only, and 𝑣 depends on 𝑡. We obtain the time that the searcher will take to detect the target in a complete revolution by considering that the searcher path will intersect the line 𝑦(𝑡)=𝑚𝑥(𝑡)+𝑐 at time 𝑡, where 𝑚 is the slope of the line and 𝑐 is the absolute constant. Using (1) we get 𝑐+𝑚𝑋₀+𝑚𝑡²cos(𝑡)=𝑌₀+𝑡²sin(𝑡), but at 𝑡=0 we have 𝑌₀=𝑚𝑋₀+𝑐 then 𝑚𝑡²cos(𝑡)=𝑡²sin(𝑡). Consequently, 𝑡=tan⁻¹𝑚, but at 𝑚=0 we have 𝑡=0,2𝜋,4𝜋,…, (i.e., the complete revolutions will be done at the time values 0,2𝜋,4𝜋,…).

In addition, the area that the searcher will search through making the complete revolution is given in the following.

After time 𝑡, in the first revolution the searcher will arrive to the point (𝑋₀+𝑡²cos(𝑡),𝑌₀+𝑡²sin(𝑡)) and make an angle 𝑡 with 𝑥^′-axis; see Figure 2.

(a)

(b)

Figure 2 

(a) The position of the searcher at time 𝑡 and 𝑡+2𝜋. (b) The area at time 𝑑𝑡.

With the previous understanding of using polar coordinates, the area that is done by the first revolution is given by 𝐴₁=∫₀^2𝜋∫^𝑡2₀𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡, where 𝑔(𝑟,𝑡) is the probability density function of the target position. And the area between the first and second revolution is given by 𝐴₂=∫₀^2𝜋∫^{(𝑡+2𝜋)2}_𝑡2𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡. Similarly, the area between the (𝑖−1)th and 𝑖th revolution is given by 𝐴_𝑖=∫₀^2𝜋∫^{(𝑡+2𝑖𝜋)2}_{(𝑡+2(𝑖−1)𝜋)2}𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡.

In this work, the search space is a standard Euclidean 2-space 𝐸 with points designated by ordered pairs (𝑥,𝑦). We aim now to evaluate 𝐷, that is, the expected value of the time for detecting the target.

Theorem 1. If the searcher moves along a parabolic spiral curve starting its motion from random point (𝑋₀,𝑌₀), then the expected value of detecting the target is given by 𝐷≤2𝜋₀^2𝜋𝑡2₀𝑔()+𝑟,𝑡𝑟𝑑𝑟𝑑𝑡_∞𝑖=1𝑖₀^{2𝜋(𝑡+2𝑖𝜋)2}_{(()𝜋)𝑡+2𝑖−12}𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡. (5)

Proof. From our hypothesis we can see that 𝐷≤2𝜋₀^2𝜋𝑡2₀𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡+4𝜋₀^{2𝜋(𝑡+2𝜋)2}_𝑡2𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡+6𝜋₀^{2𝜋(𝑡+4𝜋)2}_{()𝑡+2𝜋2}𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡+⋯=2𝜋₀^2𝜋𝑡2₀𝑔()+𝑟,𝑡𝑟𝑑𝑟𝑑𝑡_∞𝑖=1𝑖₀^{2𝜋(𝑡+2𝑖𝜋)2}_{(()𝜋)𝑡+2𝑖−12}𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡. (6)

If we consider that the target has been detected after 𝑛-revolutions, 𝑛>1, then we have 𝐷≤2𝜋₀^2𝜋𝑡2₀𝑔()+𝑟,𝑡𝑟𝑑𝑟𝑑𝑡_𝑛𝑖=1𝑖₀^{2𝜋(𝑡+2𝑖𝜋)2}_{(()𝜋)𝑡+2𝑖−12}𝑔(𝑟,𝑡)𝑟𝑑𝑟𝑑𝑡. (7) It is noticed from (7) that 𝐷 depends on 𝑛. Hence, there exists a relationship between the random distance 𝐻 (the distance between the target position (𝑋,𝑌) and the searcher starting point (𝑋₀,𝑌₀)) and the number of revolutions 𝑛.

Theorem 2. Assume that the searcher made 𝑛 complete revolutions; then 𝐿 (the distance from (𝑋₀,𝑌₀) to the searcher) is given by (𝑡+2𝑛𝜋)²; 𝑛 is a natural number.

Proof. The jump in the first revolution is given by (𝑡²−0) and the jump in the second revolution is given by (𝑡+2𝜋)²−𝑡². In addition, the jump in the last revolution is given by (𝑡+2𝑛𝜋)²−(𝑡+2(𝑛−1)𝜋)². Then, the distance from (𝑋₀,𝑌₀) to the searcher is given by 𝐿=𝑡²−0+(𝑡+2𝜋)²−𝑡²+(𝑡+4𝜋)²−(𝑡+2𝜋)²+⋯+(𝑡+2𝑛𝜋)²−(𝑡+2(𝑛−1)𝜋)²=𝑡²+4𝜋²+4𝑡𝜋+12𝜋²+4𝑡𝜋+20𝜋²+4𝑡𝜋+28𝜋²+4𝑡𝜋+⋯+(8𝑛−4)𝜋²+4𝑡𝜋=𝑡²+4𝑛𝑡𝜋+4𝜋²[1+3+5+7+⋯+(2𝑛−1)]=𝑡²+4𝑛𝑡𝜋+4𝜋²𝑛²=(𝑡+2𝑛𝜋)². (8)

3. The Case of a Target Position Given by Circular Normal Distribution

Stone et al. [18] studied more interesting problem; that is, on June 2009, an Airbus 330–200 with 228 passengers and crew of the Air France Flight 447 is disappeared over the South Atlantic during a night flight from Rio de Janeiro to Paris. An international air and surface search effort recovered the first wreckage on June 6 five and one half days after the accident. More than 1000 pieces of the aircraft and 50 bodies were recovered and their positions logged. A French submarine as well as French and American research teams searched acoustically for the underwater locator beacons on each of the two flight recorder's (black boxes) for 30 days from June 10 to July 10, 2009, with no results. They described this problem and analyzed the results of this analysis. The analysis shows that all impact points are contained within a 20 nautical mile radius circle from the point at which the emergency situation began. The results of this analysis are represented by a second distribution which is circular normal with center at the last known position. One of the important advantages of circular normal distribution is that they are sensitive to shifts in the centering of the searching process.

Thus, we assume that the position of the target has a bivariate normal distribution with parameters 𝜎₁ and 𝜎₂. Let (𝑋,𝑌) give the initial target’s actual position. Then 𝑋 is normally distributed with mean 0 and standard deviation 𝜎₁. In addition, 𝑋 is independent of 𝑌, which is normally distributed with mean 0 and standard deviation 𝜎₂. Let 𝑓=1𝑥,𝑦2𝜋𝜎₁𝜎₂−1exp2𝑥²𝜎²₁+𝑦²𝜎²₂,for(𝑋,𝑌)∈ℚ (9) be the probability density function of the bivariate normal distribution, where ℚ denotes a standard Euclidean 2-space. Thus, the distribution of error in the navigation system yields 𝑓 as given in (10) for the density of the target distribution. If 𝜎₁=𝜎₂=𝜎, then (9) becomes 𝑓=1𝑥,𝑦2𝜋𝜎²−𝑥exp²+𝑦²2𝜎²,for(𝑋,𝑌)∈ℚ, (10) and the target distribution is called circular normal. Consequently, (7) becomes −1𝐷≤2𝜋𝜋₀^2𝜋𝑡2₀exp−𝑟²2𝜎²𝑑−𝑟²2𝜎²+𝑑𝑡_𝑛𝑖=1−12𝑖𝜋₀^{2𝜋(𝑡+2𝑖𝜋)2}_{(()𝜋)𝑡+2𝑖−1}²exp−𝑟²2𝜎²𝑑−𝑟²2𝜎²𝑑𝑡=−2₀^2𝜋exp−𝑟²2𝜎^2𝑡2₀+𝑑𝑡_𝑛𝑖=1𝑖₀^2𝜋exp−𝑟²2𝜎^{2(𝑡+2𝑖𝜋)2}_{(()𝜋)𝑡+2𝑖−12}𝑑𝑡=2₀^2𝜋1−exp−𝑡⁴2𝜎²+𝑑𝑡_𝑛𝑖=1𝑖₀^2𝜋−(()𝜋)exp𝑡+2𝑖−1⁴2𝜎²−()−exp𝑡+2𝑖𝜋⁴2𝜎²𝑑𝑡. (11) Using Mathematica 7 programme, we get Γ𝐷≤22𝜋+1/4,8𝜋⁴/𝜎²−Γ(1/4)22^3/41/𝜎^21/4+𝜋2^𝑛_𝑖=1𝑖)3(𝑖−1Ei4,8((𝑖−1)𝜋)⁴𝜎²3−2𝑖Ei4,8(𝑖𝜋)⁴𝜎²+()31+𝑖Ei4,8((𝑖+1)𝜋)⁴𝜎², (12) where Ei gives the exponential integral function.

Example 3. Suppose that the target position (black boxes, etc.) follows the circular normal distribution with probability density function (10). For many values of 𝜎 and 𝑛=5,10,15,…,45,50, we get the expected value of the time for detecting the target as in Table 1.
It appears from numerical calculations that the value of 𝐷 has small increments with decreasing the value of the parameter 𝜎 as in Figure 3.
Naturally, 𝜎 and 𝐷 depend on the target position. This means that 𝐷 does not depend only on 𝜎 but also depends on 𝑛, where from Theorem 2 we have 𝑛=(√𝐿−𝑡)/2𝜋.

Table 1 

Values of 𝐷 given the values of 𝜎 and 𝑛=5,10,15,…,45,50.

(a)

(b)

Figure 3 

The curves represent the relation between 𝐷, 𝜎, and 𝑛. In the part (a) Scatter plot of 𝜎 versus 𝐷 and in the part (b) Surface Plot of 𝑛 versus 𝐷, 𝜎.

4. One-Dimensional Case

In one-dimensional case, the searcher moves with the model 𝑥(𝑡)=𝑋₀+𝑡²(𝑡),𝑦(𝑡)cos=𝑌₀. (13) This model is produced from the projection of model (1) on 𝑥-axis; that is, Π₁(𝑥(𝑡),𝑦(𝑡))=𝑥(𝑡)=𝑋₀+𝑡²cos(𝑡). Consequently, the speed of the searcher will be given as 𝑣=𝑥̇=2𝑡cos(𝑡)−𝑡²sin(𝑡). (14) At 𝑥̇=0, we have 𝑡(2cos(𝑡)−𝑡sin(𝑡))=0, which leads to 𝑡=2cot(𝑡), 𝑡≠0. We choose only positive values of 𝑡. From (13), the parabolic spiral search will ride to the linear search problem. The method of linear search is an important methodology to find a lost target on the real line. The position of the target is represented by the random variable 𝑋 where 𝑋 has a probability density function 𝑤(𝑥). The searcher moves continuously along the real line in both directions of the starting point 𝑎₀. The searcher would conduct his search in the following manner (see Figure 4): start at 𝑎₀=0 and go to the left (right) as far as 𝑎₁. Then, turn back to explore the right (left) part of 𝑎₀=0 as far as 𝑎₂. Retrace the steps again to explore the left (right) part of 𝑎₁ as far as 𝑎₃, and so forth.

Figure 4 

The generalized linear search path.

The searcher starts from any point on the real line with uniform velocity and tries to find the target in minimal expected time. It is assumed that the searcher can change the direction of its motion without any loss of time. The target can be detected only if the searcher reaches the target. This problem has been studied extensively in many variations, mostly by A. Beck and M. Beck [19, 20], Reyniers [21, 22], and Balkhi [23, 24]. On the other hand, when the target moves on the real line according to a known random process, the searcher moves continuously along the line in both directions of the starting point until the target is met. In an earlier work, this problem has been studied by Mohamed [25], El-Rayes et al. [26], Washburn [27], and Stone [28]. Recently, Mohamed et al. [29, 30] discussed this problem when the target moves randomly on one of real lines. They formulated a search model and found the conditions under which the expected value of the first meeting time between one of the searchers and the target is finite. Furthermore, they have shown the existence of the optimal search plan that minimized the expected value of the first meeting time and found it.

The problem studied here is more applicable than the linear search problem because the speed of the searcher depends on 𝑡 as in (14). Equations (11) and (12) show that the searcher moves on the line with the linear search technique as in Figure 5.

Figure 5 

The search path of the searcher in one-dimensional case with speed 𝑣=2𝑡cos(𝑡)−𝑡2sin(𝑡).

In this case, to get the time values of the turning points as in Figure 4, we let 𝑦=𝑡 and we use the numerical method to solve the system (𝑡).𝑦=𝑡,𝑦=2cot (15) For simplicity, let 𝑋₀=𝑌₀=0. Thus, the roots of this system are the intersection points of the line 𝑦=𝑡 and the equation 𝑦=2cot(𝑡) as in Figure 6.

Figure 6 

The turning point in one-dimensional case.

Figure 6 gives an illustration of the roots (the time of the turning points) through the time interval ]0,𝑇]. From (14) the distances ℎ_𝑖 that the searcher does through the times 𝑡_𝑖, 𝑖=1,2,… are given by ℎ_𝑖=𝑡_𝑖2𝑡_𝑖cos𝑡_𝑖−𝑡²_𝑖sin𝑡_𝑖=𝑡²_𝑖2cos𝑡_𝑖−𝑡_𝑖sin𝑡_𝑖. (16)

Now, we can easily obtain the expected value of the time for detecting the target, where the target position has known distribution.

Theorem 4. If the target location has known distribution with probability density function 𝑤(𝑥), then the expected value of the time for detecting the target is given by 𝐷≤^∞_{𝑖=1𝑖𝑖𝑠𝑜𝑑𝑑}𝑡_𝑖^ℎ𝑖_ℎ𝑖−1𝑤(𝑥)𝑑𝑥+^∞_{𝑖=2𝑖𝑖𝑠𝑒𝑣𝑒𝑛}𝑡_𝑖^ℎ𝑖_ℎ𝑖−2𝑤(𝑥)𝑑𝑥. (17)

Proof. If the target lies in ]0,ℎ₁], then 𝐷≤𝑡₁∫^ℎ1₀𝑤(𝑥)𝑑𝑥, and if it lies in ]0,ℎ₂], then 𝐷≤𝑡₂∫^ℎ2₀𝑤(𝑥)𝑑𝑥. Also, if the target lies in ]ℎ₁,ℎ₃], then 𝐷≤𝑡₃∫^ℎ3_ℎ1𝑤(𝑥)𝑑𝑥 and if it lies in ]ℎ₂,ℎ₄] then 𝐷≤𝑡₄∫^ℎ4_ℎ2𝑤(𝑥)𝑑𝑥, and so forth. Consequently, we have 𝐷≤𝑡₁^ℎ1₀𝑤(𝑥)𝑑𝑥+𝑡₂^ℎ2₀𝑤(𝑥)𝑑𝑥+𝑡₃^ℎ3_ℎ1𝑤(𝑥)𝑑𝑥+𝑡₄^ℎ4_ℎ2𝑤(𝑥)=𝑑𝑥+⋯_{∞𝑖𝑖=1isodd}𝑡_𝑖^ℎ𝑖_ℎ𝑖−1𝑤(𝑥)𝑑𝑥+^∞_{𝑖𝑖=2iseven}𝑡_𝑖^ℎ𝑖_ℎ𝑖−2𝑤(𝑥)𝑑𝑥. (18)

Example 5. Let the previous illustrative Example 3 and make a projection on 𝑥-axis, then (10) becomes 𝑓=1𝑥,𝑦√2𝜋𝜎²−𝑥exp²2𝜎²,for−∞≤𝑥≤∞. (19) In addition, we let the target be detected through the time interval ]0,10]; then by using Figure 6 and (16) we can get the time values of the turning points and its corresponding distances as in Table 2.
For many values of 𝜎 we can get the values of 𝐷 as in Table 3.
It is clear that from numerical calculations the value of 𝐷 increases with increasing the value of the parameter 𝜎 as in Figure 7.

Table 2 

The time values of the turning points through the time interval ]0,10] and its corresponding distances.

Table 3 

Values of 𝐷 given the values of 𝜎.

Figure 7 

The relation between 𝐷 and 𝜎 in one-dimensional case.

5. Conclusion and Future Work

A parabolic spiral search plan for a randomly located target in the plane has been presented, where the target position is given by independent random variables 𝑋, 𝑌. We get the expected value of the time for detecting the target in Theorem 1. In Theorem 2, we show that the distance between the target position and the searcher starting point depends on the number of revolutions, where the complete revolution is done when 𝑡=2𝜋.

To study this problem in one-dimensional case, we make a projection of the searcher path on 𝑥-axis. We find that the case of one-dimension of this problem rides to the generalized linear search problem. In Theorem 4, we can easily get the expected value of the time for detecting the target after knowing the values of the turning points, where the target position is given by a random variable 𝑋. The effectiveness of this technique is illustrated using numerical examples assuming circular normal distributed estimates of the target position.

In future research, it seems that the proposed model will be extendible to the multiple searchers case by considering the combinations of movement of multiple targets.