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      Stefka Fidanova1 & Hussain Saleh2

ACO for GPS Surveying      

Ant Colony Optimisation for Scheduling the Surveying Activities of Satellite Positioning Networks

 Stefka Fidanova1 & Hussain Saleh2

 1Institut of Parallel Processing, Bulgarian Academy of Science, 1113 Sofia, Bulgaria

 E-mail:stefka@parallel.bas.bg

 2Institut de Recherches Interdisciplinaires et de D´┐Żveloppements

 en Intelligence Artificielle, IRIDIA, CP 194/6, Universit´┐Ż Libre de Bruxelles

 E-mail: hsaleh@ulb.ac.be

 
 
Abstract. This paper introduces several approaches based on ant colony optimization for efficiently scheduling the surveying activities of designing satellite surveying networks. These proposed approaches use a set of agents called ants that co-operate to iteratively construct potential observation schedules. Within the context of satellite surveying, a positioning network can be defined as a set of points which are coordinated by placing receivers on these points to determine sessions between these points. The problem is to search for the best order in which these sessions can be observed to give the best possible schedule. The same problem arise in Mobile Phone Surveying networks. Several case studies have been used to experimentally assess the performance of the proposed approaches in terms of solution quality and computational effort.

 
1 Introduction

The continuing research on naturally occurring social systems offers the prospect of creating artificial systems that generate practical solutions to many Combinatorial Optimisation Problems (COPs). Metaheuristic techniques have evolved rapidly in an attempt to find good solutions to these problems within a desired time frame [17]. These metaheuristics (e.g., simulated annealing, tabu search, genetic algorithms and ant colony optimisation, etc) are appealing because they attempt to solve complex optimisation problems by incorporating processes which are observed at work in real life [2,4]. For example, in the case of ants, using their simple individual interactions mediated by pheromones (one ant follows the chemical scent of another), they can collectively determine the shortest route from their nest to a food source without using visual cues [1]. When applied to satellite surveying, these techniques can assist surveyors in creating a better observation schedule for designing the whole positioning network [7]. 

In this paper, the application of the Ant Colony Optimization (ACO) is proposed to optimise the use of satellite systems in designing surveying networks. These systems are designed to provide instantaneous position, velocity and time information almost anywhere on the globe at any time and with any weather. Present usage of the satellite systems includes civil engineering projects and deformation analysis, land surveying, geographic information systems and mapping, general navigation (aircraft, offshore, fleet tracking), and personal navigation (hiking, boating, hunting), etc. [12]. The most widely known satellite navigation systems are: the American Global Positioning System (GPS), the Russian GLObal Navigation Satellite System (GLONASS), and the forthcoming European Satellite Navigation System (GALILEO). Here, it is the use of GPS to establish positioning networks that is being investigated. A GPS positioning network involves the use of a number of receivers to co-ordinate a large number of geomatic points located on the ground. In practice this means determining how each GPS receiver should be moved between points to be surveyed in an efficient manner taking into account several important parameters. These parameters are personnel availability, locations of points to be co-ordinated, receivers and satellites to be used, sessions to be observed, lengths of session observations and the receivers schedule, etc [15]. An important part of a GPS survey is to determine the best practical plan for carrying out the survey and to find the best possible alternative in case of any unpredicted fault during the survey. The paper is organised as follows.  Section 2 presents the formulation of the GPS positioning network problem as a combinatorial optimisation problem. Section 3 explains the stages of applying ACO metaheuristic for optimising the GPS network problem. Sections 4, 5, and 6 outline several instances of ACO that adopt different pheromone updating criteria based on some diversification strategies. These strategies guide the search to areas in the solution space which have not yet been explored and forces ants to search for better solutions. Section 7 introduces the proposed GPS-ACO metaheuristic technique. Section 8 provides results of computational experiments, while conclusions and further work are summarized in Section 9.

2 Formulation the GPS Network problem

The main purpose of surveying is to determine the position of arbitrary geometric points (a,b,c,d,e)  on the ground with respect to other known points (A,B). These positions form the basic network for geomatic information which supports all the environmental and engineering activities in any country. In GPS surveying, after completing the reconnaissance stage for an area to be surveyed, GPS receivers are used to create a GPS network covering the whole area. The aim of this stage is to define beforehand the required number and locations of the points (control stations within the concept of the GPS surveying) to be surveyed. These stations are fixed points on the ground and are located by an experienced surveyor according to the nature of the land and the requirements of the survey. The observation stage starts by setting up GPS receivers on these stations to record satellites signals for a period of time. The immediate outcome of the GPS observations is a vector (a session within the concept of the GPS surveying) between these two stations (e.g., the vector ab between station a and station b).  To obtain a potential observation schedule, these sessions have to be observed efficiently.  When the observation time of session ab is completed, the receivers will be moved (according to the potential schedule) to other stations for similar tasks till the whole network is completely surveyed. The receiver informs the surveyor when s/he has to move it to another station for further measurements using a set of geometric information depicted on the screen of the receiver. At the end of the survey, a GPS network of connected sessions is established between control stations. The survey ends by the computational stage in which all the observed sessions will be downloaded from the receiver memory into the LapTop. A special software will be used (e.g., Ski) for processing the data and producing an electronic map depicted on the screen of the computer [16].

Currently, an experienced GPS surveyor creates the observation schedule manually using intuition and experience on a day-to-day basis. Metaheuristic techniques allow the formulation of an efficient strategy for providing a complete solution to design a GPS surveying network and to effectively reduce the total cost of carrying out the whole survey [18]. This total cost represents the sum of the cost of moving receivers between control stations. A number of receivers (X, Y, etc) are placed at stations  (a, b, c, d, etc) to be co-ordinated by determining sessions (ab, ac, dc, etc) between these stations. The problem is to search for the best order in which to consecutively observe these sessions to have the best schedule at minimum cost, i.e., 

where

C(V)    : the total cost of a feasible schedule V (N, R, U).

Sp              : the route of the receiver p in a schedule;

N          : the set of stations N={1,…,n}; 

R          : the set of receivers R={1,...,r};  

U          : the set of sessions U={1,...,u};  

3 Ant Colony Optimization

Ants live together in colonies and they use chemical cues called pheromones to provide a sophisticated communication system. An isolated ant moves essentially at random but an ant encountering a previously laid pheromone will detect it and decide to follow it with high probability and thereby reinforce it with a further quantity of pheromone. The repetition of the above mechanism represents the collective behaviour of a real ant colony which is a form of autocatalytic behaviour where the more the ants follow a trail, the more attractive that trail becomes. The above behaviour of real ants has inspired ACO which has proved to be an effective metaheuristic technique for solving many complex COPs [3, 5, 9, 11, 14]. This technique uses a colony of artificial ants that behaves as cooperative agents in a mathematical space where they are allowed to search and reinforce pathways (solutions) in order to find the optimal ones. The features of artificial ants are: having some memory, not being completely blind and the process time is discrete [8]. In the proposed GPS-ACO technique an initialisation phase takes place during which ants are positioned on different nodes (sessions) with empty tabu lists and initial pheromone distributed equally on paths connecting these sessions. Ants update the level of pheromone while they are constructing their schedules by iteratively adding new sessions to the current partial schedule. At each time step, ants compute a set of feasible moves and select the best one according to some probabilistic rules based on the heuristic information and pheromone level. The higher value of the pheromone and the heuristic information, the more profitable is to select this move and resume the search. The selected node is putted in the tabu list related to the ant to prevent to be chosen again. Heuristic information represents the nearer sessions around the current session, while pheromone level “memory” of each path represents the usability of this path in the past to find good schedules. At the end of each iteration, the tabu list for each ant will be full and the obtained cheapest schedule is computed and memorized. For the following iteration, tabu lists will be emptied ready for use and the pheromone level will be updated. This process is repeated till the number of iterations (stopping criteria) has been reached. In more details, the proposed GPS-ACO technique constructs the cheapest observation schedule for a GPS network using the following two stages:

3.1 Schedule Construction Stage

After each move, an ant leave a pheromone trail on the connecting path to be collected by other ants to compute the transition probabilities. Starting from the initial session i, an explorer ant m chooses probabilistically session j to observe next using the following transition rule: 

  (1)

where

(i,j) : the intensity measure of the pheromone deposited by each ant on the path (i,j). The intensity changes during the run of the program.

: the intensity control parameter.

(i,j) : the visibility measure of the quality of the path (i,j). This visibility, which remains constant during the run of the program, is determined by (i,j)=1/l(ij), where l(ij) is the cost of move from session i to the session j.

: the visibility control parameter.

Sm(i) : the set of sessions that remain to be observed by ant m positioned at session i.

Equation 1 shows that the quality of the path (i,j) is proportional to its shortness and to the highest amount of pheromone deposited on it (i.e., the selection probability is proportional to path quality).

3.2 Pheromone Updating Stage

Ants change the pheromone level on the paths between sessions using the following updating rule:

(2)

where

: the trail evaporation parameter.

(i,j)  : the pheromone level.   

The amount of deposited pheromone is the mechanism by which ants communicate to share information about good paths. Stagnation may occur during the pheromone updating and this can be happened when the pheromone level is significantly different between paths connecting the observed schedule. This means that some of these paths have received higher amount of pheromone more than other and an ant will continuously select these paths and neglect the others. In this situation, ants keep constructing the same schedule over and over again and the exploration of the search stops. Stagnation can be avoided by influencing the probability for choosing the next path which depends directly on the pheromone level. To make better use of the pheromone and exploit the search space of a schedule more effectively, several ideas based on the pheromone control strategy have been implemented, tested and analysed. Some of these ideas are: additional pheromone trail limits, smoothing of the pheromone trails, re-initialization of the pheromone trial and additional reinforcement of the pheromone, etc. In the following section, different approaches based on these ideas have been proposed and implemented to effectively diverse the search space and select the best possible observation schedule for a GPS network.

4 Ant Colony System

Ant Colony System (ACS) differs from the other ACO instances due to its strategy of constructing an observation schedule [10]. This strategy can be categorized in three step. An ant positioned on session i selects the session j to observe by applying the following equation: 

(3)

where

I : a random variable selected according to the probability given by Equation 1.

q : a uniformly distributed random number to determine the relative importance of exploitation versus exploration q[0,..,1].

q0 : a threshold parameter and the smaller q0 the higher the probability to make a random choice (0  q0  1).

In each step of building a schedule, an ant located at session i samples the parameter q to move to session j. Using Equation 3, an ant selects the best path to reach the next session when (q q0) (exploitation). Otherwise, the ant will probabilistically choose the next session to be observed using Equation 2 with a bias toward the best possible path (biased exploration). While ants build their schedules, at the same time they locally update the pheromone level of the visited paths using the local updating rule as follows:

     (4)

Where

  : a persistence of the trail and the term (1- ) can be interpreted as trail evaporation.

0: the initial pheromone level which is assumed to be a small positive constant distributed equally on all the paths of the network since the start of the survey.

    The aim of the local updating rule is to make better use of the pheromone information by dynamically changing the desirability of paths. Using this rule, ants will search in wide neighbourhood of the best previous schedule. When all ants have completed their schedule, the pheromone level is updated by applying the global updating rule only on the paths that belong to the best found schedule since the beginning as follows:

   (5)

 

  (6)

where

   : a pheromone decay parameter.

Cm : the cost of the best schedule performed from the beginning by ant m.

 

This rule is intended to provide a greater amount of pheromone on the paths of the best schedule, thus intensifying the search around this schedule. In other words, only the best ant that took the shortest route is allowed to deposit pheromone.

5 MAX-MIN Ant System

The strategy of the MAX-MIN Ant System (MMAS) states that if the amount of the pheromone has a finite upper bound max and a positive lower bound min, then ACO converges to the optimal solution [21].  The main features of MMAS algorithm for obtaining an improved performance on the basic ACO metaheuristic are as follows:

Deep exploitation to the search space of the best found schedule by allowing a single ant to add pheromone after each iteration. This ant may be the one which found the best schedule in the current iteration (iteration-best ant) or the one which found the best schedule from the beginning (global-best ant).

Wide exploration to the search space of the best found schedule by initialising the pheromone trails to max. Thus, in the next iteration only the paths that belong to the best schedule will receive pheromone, while the pheromone values of the other paths are only evaporated.

As shown from the above, the aim of using only one schedule is to make the paths of the best found schedule receive large reinforcements

6 ACO Algorithm with Additional Reinforcement

The function of ACO algorithm with Additional Reinforcement (ACO-AR) is to force ants to search for a better schedule by divorcing and exploring the search space while keeping the best found schedule [13]. This can be carried out by adding extra pheromone on the unused paths during the previous iterations and this will force ants to diverse the search space and choose other new directions without repeating any bad experience and trapping again in local optimality of the schedule search space. The modified pheromone update rule in ACO-AR is given as follows:

(7)

where 

q1  0 : a reinforcement parameter;

max : asymptotically the maximal value of the pheromone. 

This condition (q1 1 - ρ) must be satisfied to fulfil the above criteria of the ACO-AR technique for updating the pheromone.

7 Implementation of the GPS-ACO Technique

It is important when implementing the proposed technique to provide suitable and carefully chosen structural and control components according to the size and type of the applied GPS network. The structural elements determine the procedure in which the GPS network problem is modelled in order to fit into the ACO framework as follows: Actual cost matrix: This matrix C[i,j] represents the travelling time between the end of observing session i and the beginning of observing session j. The size of this matrix is dependent on the required number of sessions to survey the whole network [6]. The initial quantity of pheromone: The initial quantity of pheromone 0, which is equivalent to the intensity of trail at time (0), should be set to an arbitrarily small value (0  0  1).  During the run of the program, this initial pheromone matrix is updated at each iteration and the final outcome is the general pheromone matrix [i,j]. This general matrix provides important information about the strong and weak paths which help and direct the surveyors to select next sessions to be observed. Tabu list: The tabu list is a data structure which is correlated to each ant in order to prevent ants from observing a session more than once. In this list, the sessions already observed are memorized up to time t and the ant mth is forbidden to observe them again before it has completed surveying the whole network. After each iterations, the tabu list is emptied and then ants are free again to choose their ways starting from a random initial session.

The control parameters govern the workings of the GPS-ACO technique itself and are mainly concerned with pheromone information. The intensity control parameter (  0): This parameter controls the relative weight of pheromone trail intensity ij during the selection process of the following sessions to be observed. If =0 (ants do not communicate), the nearest sessions are more likely to be selected by ants. On the other hand, high values of (ants communicate) means that the trail is very important and therefore current ants tend to choose paths previously chosen by other ants. The visibility control parameter (  0): The function of parameter is to put more or less emphasis on distance versus pheromone. This parameter has positive impact on the schedule quality and this can be seen in adjusting the relative importance of visibility ij when evaluating the cost of a path in the schedule. If =0, only pheromone intensity is functioning and this will lead to the rapid selection of schedules that may not be optimal. The evaporation control parameters (0 (, )  1): These parameters are used to adjust the magnitude of the pheromone laid by an ant when constructing its schedule and therefore is called the trail resistance. In case of ACS, the parameter is used only in the local update step when some pheromones are removed from the paths of the current schedule to diversify the search. The lower the faster the information gathered in previous iterations is forgotten. On the other hand, must be set to a value less than 1 to avoid unlimited accumulation of trail. The evaporation global control parameter has the same influence as the evaporation local control parameter and is used in the global update step of the ACS technique and in the both MMAS and ACO-AR techniques. The stopping criteria: Several terminating criteria were adopted and the simplest one is to terminate the process after a pre-defined number of iterations. Each iteration of the GPS-ACO process requires two steps to create a schedule (ant cycle); construction of a schedule and updating the level of pheromone. Another stopping criterion is to terminate the search when no further improvement can be observed for a fixed number of iterations.

8 Computational Results

The effectiveness of the proposed GPS-ACO technique has been evaluated using different types and sizes of GPS networks observed in Malta and the Seychelles. The Malta GPS network with a triangular-type consists of 38 sessions connecting 25 stations. The actual operating schedule for this network with cost of 1405 minutes was manually generated using the intuition and experience of the surveyors [19]. By implementing the proposed technique, the overall cost 1405 minutes was reduced to 895 minutes using ACS, 915 minutes using MMAS, and 875 minutes using ACO-AR as shown in Table 1. The Seychelles network with a linear-type consists of 71 sessions connecting 67 stations. The actual operating schedule for this network with cost of 994 minutes was manually generated using the intuition and experience of the surveyors [20]. By implementing the proposed technique, the overall cost 994 minutes was reduced to 853 minutes using ACS, 865 minutes using MMAS, and 859 minutes using ACO-AR as shown in Table 1. The above results indicate that the proposed GPS-ACO technique consistently produced better schedules and this was measured using the relative reduction of the cost with respect to the cost of the actual operational schedule, i.e., 

Where

COS : The cost of the Operational Schedule created by a surveyor.

CACO : The cost of the computed schedule created by GPS-ACO metaheuristic technique.

RRC : Relative Reduction of the Cost.  

Table1. Computational results for ACO-GPS techniques 


Technique Malta RRC Seychelles RRC
OS 1405 0 994 0
ACS 895 36.3 853 14.19
MMAS 915 34.88 865 12.98
ACO-AR 875 37.72 859 13.88
 

Different control parameter settings were carried out to investigate the impact of these parameters on schedule quality and computational effort as shown in Tables 1. When the value of one parameter was tested, the others were set at their default value, namely =0.3, =0.3, =0.3. The initial amount of pheromone 0 was set to a fixed value 0=0.0005 on all the paths, while the number of iterations was set to 200. With regards to the parameter , better results were obtained with smaller values when =1. On other hand, larger values of  tend to bias the search process toward elite schedules and lead to parameter convergence. With regards to the parameter , better results were obtained with either =1 or =2 and this indicates that the incorporating pheromone into the move cost has generally a positive impact on solution quality. For the parameter , best results were found with =0.4 and this indicates that a significant level of diversification is desirable. With regards to the parameter , the best results were obtained when   = 0.7 for Malta network and = 0.6 for the Seychelles network.

9 Conclusion

In this paper, the influence of pheromone updating strategy of the ACO metaheuristic has been investigated and achieved good results for the GPS positioning network problem with a static nature. For future work, dynamic optimization will be searched for improving the use of space technology in other real life applications (e.g., GPS ambiguity resolution, precise orbit determination of low earth orbiters, modeling atmospheric effects, etc). These applications require powerful dynamic optimization tools that account for the uncertainty present in a changing environment. This will provide the state of the art and latest research on how dynamic metaheuristic algorithms may be applied to effectively and efficiently solve and optimize this kind of complex problems. 

Acknowledgments

This research was supported by Marie Curie Program (MERG-CT-2004-51-714).

References

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