An Algorithm Inspired by the Deterministic Annealing Approach to Avoid Local Minima in Artificial Potential Fields Nara Strappa Facchinetti Doria∗† , Eduardo Oliveira Freire‡ and Jo˜ao Carlos Basilio§ ∗ Coordenadoria
de Eletrˆonica - Instituto Federal de Sergipe, Aracaju, Sergipe, Brazil - Universidade Federal de Sergipe, Aracaju, Sergipe, Brazil Email:
[email protected] ‡ PROEE - Universidade Federal de Sergipe, Aracaju, Sergipe, Brazil Email:
[email protected] § PEE/COPPE - Universidade Federal do Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, Brazil Email:
[email protected] † PROEE
Abstract—Local minima are still an important unsolved problem for artificial potential field approaches. In order to overcome this problem, we propose in this paper a new method inspired by the deterministic annealing approach that can be classified as LMA, since it prevents the robot from being trapped in a local minimum point, and allows the robot to continue its trajectory towards the final destination. In order to show the effectiveness of the proposed algorithm, simulations are carried out to compare the proposed approach with the following methods: i) the artificial potential field with no mechanism to deal with local minimum; ii) an LME method based on the simulated annealing approach; iii) another LME method also based on deterministic annealing.
I.
I NTRODUCTION
The Artificial Potential Fields method, proposed by Khatib [1], is an approach used to drive a robot to its goal while avoiding obstacles. The method consists in minimizing a cost function which is given by the sum of attractive and repulsive artificial potential fields of the operation environment. The attractive potential is generated by the goal while the repulsive potentials are due the obstacles. Such cost function is known as the potential function and the gradient descent of this function is the force used to drive the robot towards its final destination while avoiding obstacles. In [1], the potentials that must be used for manipulators are described. In mobile robot context, the analysis may me done in an analogous way, and the total potential field function is given as: Uart (x) = Uxd (x) + UO (x),
(1)
where Uxd (x) and UO (x) are the potentials due to the obstacles and the goal, respectively, and x is the vector of coordinates of the point under consideration. Notice that the resulting potential field is the sum of two components: attraction (Uo (x)) and repulsion(Uxd (x)). From Eq. (1) it is possible to define that: F ∗ = Fx∗d + FO∗ , c 978-1-4799-2722-7/13/$31.00 2013 IEEE
(2)
where Fx∗d is the attractive force that drives the robot to the goal xd , given by −k(x−xd ), and FO∗ is the resulting repulsive force due to the obstacles close to the robot, being expressed as: Fx∗d = −grad[Uxd (x)], FO∗ = −grad[UO (x)],
(3)
As the robot approaches an obstacle, the repulsive potential field increases, approaching to infinity at the vicinity of the obstacle. On the other hand, to avoid undesirable disturbances, the potential field due to the obstacle should be limited to a given region of influence, being insignificant outside of it. In order to do so, Khatib [1] proposes a repulsive potential function whose influence is restricted to a certain distance of the obstacle, being given as: UO (x) =
1 1 2 η( ρ
− 0
1 2 ρO )
if ρ ≤ ρO if ρ > ρO ,
(4)
where ρO is the largest distance of influence of the potential field and ρ is the distance to the obstacle. The major problem concerning the Artificial Potential Field approach is the possibility that the robot may be driven to a point where the potential function has a local minimum, and the force that guides the robot, which is given by the gradient descent of the potential function, is zero, preventing the robot from reaching its goal. This problem is so common and so important that several approaches where developed to deal with. They where classified in three categories, as follows: •
Local Minimum Removal (LMR)
•
Local Minimum Avoidance (LMA)
•
Local Minimum Escape (LME)
A comparative study between these three approaches can be seen in [2], in which the main differences between them are highlighted.
The LMR approaches (like the one presented in [3]) are able to completely solve the local minimum problem when applied to certain types of obstacles. However they require high computational cost for real time navigation and also some initial knowledge of the environment. The LMA approaches also need some initial knowledge about the environment and are generally implemented in a hierarchical scheme. For instance, in the approach described in [4], the artificial potential field method is a low level layer designed to perform the obstacle avoidance task while a higher level layer is used to establish intermediate goals that should be achieved in order to reach the final destination. In the LME approaches, like those presented in [5] and [6], the robot may get stuck in a local minimum point. When this situation occurs, some escape mechanism is triggered, driving the robot away from the local minimum, so it may be able to search for the final goal again. The major drawback of the LME approaches is exactly the fact that they cannot prevent the occurrences of local minima, resulting in less efficient paths until the final goal. However, due to their low computational cost and the fact that no previous information about the environment is needed, the LME approaches are still commonly used. Local minima are still an important unsolved problem for artificial potential field approaches. In this context, this paper introduces a new method inspired by the deterministic annealing approach that can be classified as LMA, since it prevents the robot from being trapped in a local minimum point, and allows the robot to continue its trajectory towards the final destination. In order to show the proposed method effectiveness, a comparative study has been carried out between the proposed method and the following methods: i) the artificial potential fields with no mechanism to deal with local minima (APF method); ii) an LME method based on the simulated annealing approach (LME-SA method); iii) an LME method inspired by the deterministic annealing approach (LME-DA method). The APF method (artificial potential fields with no mechanism to deal with local minima problem), has been considered to show the existence of local minimum points in the proposed environments. The LME-SA method, which is an approach presented in [5] and [6] (a LME method based on simulated annealing), was considered for comparison since as the deterministic annealing [7], it was derived from simulated annealing. Finally, the LME-DA method has also been considered for comparison since it is an intermediate step in the development of the algorithm proposed here. It is important to remark that the approaches considered in this paper can deal with unknown environments, and in the experiments carried out to perform the comparative analysis no previous information about the environment was available. This paper is organized as follows. The necessary theoretical background is briefly presented in Section II. The proposed LMA method inspired by the deterministic annealing approach (LMA-DA method) is described in Section III. The simulation results and a comparative analysis are presented in Section IV. Finally, conclusions are drawn in Section V. II.
T HEORETICAL BACKGROUND
The LMA method proposed in this paper is inspired by the deterministic annealing approach, which is, strictly speaking,
related to the simulated annealing approach. So, in order to allow a better understanding of the proposed approach and the results that will be presented, a brief explanation of the simulated annealing and of the deterministic annealing will be provided in this section. In addition, a brief description of the LME-SA approach proposed by Janabi-Sharifi and Vinke [8] will also be presented. A. Simulated Annealing (SA) The simulated annealing is a global optimization algorithm, proposed by Kirkpatrick et al. [9] which, under certain conditions, insures that an optimal global solution is reached through the minimization of a cost function. The simulated annealing algorithm was inspired by the Metropolis algorithm [10], which simulates the evolution of a solid to reach thermal equilibrium. At each step of this algorithm, a small modification on the position of a single atom is made, and, as a consequence, the amount of energy of the system changes. If the energy variation is negative, the movement of the atom is accepted and such a new configuration is considered as the starting point of the next step. On the other hand, if the movement of the atom increases the energy of the system, the new configuration can still be accepted, with a probability that is exponentially proportional to the temperature of the solid. In other words, for higher temperatures, there is a high probability that the movement of an atom that increases the energy of the system be accepted, and this probability decreases as the temperature of the solid also decreases. Replacing the energy of the solid with the cost function and considering a progressive and slow reduction of the temperature, the simulated annealing algorithm proposed in [9] is obtained. A set of parameters takes the place of the position of the atoms, and the simulated annealing process consists in starting the system under optimization with a high temperature and then begin to slowly reduce the temperature up to the system “freezes”, and configuration changes are no longer possible. The simulated annealing algorithm, due to the fact that it is derived from Metropolis algorithm, keeps accepting changes while the cost function is reduced, and when it increases, the new configuration can still be accepted under a certain probability directly proportional to the temperature. In contrast with other optimization approaches, that are able to move just for directions that result in the reduction of the cost function, the simulated annealing may accept a temporary increase in the cost function. This is the mechanism that enables the optimization process to escape from local minimum points in order to reach the global minimum. B. Deterministic Annealing (DA) The Deterministic Annealing, proposed by Rose [7], emerged from the concept of simulated annealing, but with noise inserted into the cost function. This noise avoids the random movement over the cost function surface, as it occurs with the simulated annealing. In this case, the noise inserted into the cost function can be seen as the variations that occur according to the temperature reduction. The cost function is parameterized by a control variable T . When T approaches infinity (T → ∞), the cost function is
generally convex and the global minimum of the function is easily identified [7]. Thus, the deterministic annealing can be seen as an approach that initially finds a global minimum for the configuration obtained with infinite temperature, which, as the temperature is progressively reduced, tends to move towards the global minimum of the system, which is reached when the temperature is equal to 0.
The modification proposed herein consists in inserting a temperature parameter in the cost function, as in the deterministic annealing approach, and so, Equations (6) and (7) are modified in order to introduce the temperature parameter, as follows:
Uatt (X, Y ) = Katt C. The LME Approach Based on Simulated Annealing The LME Approach Based on Simulated Annealing, introduced by Janabi-Sharifi and Vinke [8], consists in a direct application of the artificial potential field method to drive the robot to its goal. If the robot gets stuck at a local minimum point, an escape mechanism, based on simulated annealing, is called in an attempt to drive the robot away from the local minimum point. Basically, the escape method consists of random movements over the surface of the cost function to be minimized in order to find a location with lower potential than the potential of the local minimum point where the robot got stuck. Since temperature is a parameter of the method, for higher temperatures, points with higher potentials are accepted more easily, making it possible to escape from the attraction region of the local minimum point. As the temperature decreases, the possibility to accept a point with higher potential also decreases. Once a point with lower potential is found, the robot is directed to it and, the artificial potential field method resumes the control of the robot until it either reaches its final goal, or gets trapped by another local minimum point, which causes the escape mechanism to be called again. III.
A NEW ALGORITHM INSPIRED BY THE DETERMINISTIC ANNEALING APPROACH
We will propose, initially, in this section, an algorithm referred to as LME-DA approach, that is inspired by the deterministic annealing approach. As previously mentioned, the use of the deterministic annealing approach avoids the random movement over the cost function surface, an issue associated with the simulated annealing approach. Such advantage of deterministic annealing over simulated annealing was an inspiration to improve the LME-SA approach proposed by Janabi-Sharifi and Vinke [8]. In the sequel, we will propose another algorithm, the LMA-DA method, which was designed to overcome the potential deficiencies of the LME-SA method. Let us consider the same cost function of [1], as the artificial potential field: Utot = Uatt +
X
Urep
(5)
Uatt (X, Y ) = Katt ρ2goal
(6)
where
Urep (X, Y ) =
Krep ( ρ1o − 0
1 2 ρint )
if ρo ≤ ρint , if ρo > ρint .
(7)
Urep = Krep
1 ρo T
−
ρ2goal T
2
1 ρint
, if
ρo ≤ ρint , T
(8) (9)
The insertion of the temperature in the equations of the repulsive and attractive potentials results in an increase of the repulsion area of the obstacles and a reduction in the attraction radius of the goal. Such an effect may be observed in Figure 1.
(a) Modeled environment with low value temperature Fig. 1.
(b) Modeled environment with high value temperature
Modeled environment with different temperatures
The main differences between the LME-DA method proposed here and the LME-SA approach is that in the LME-SA, the temperature is initialized with a high value, as opposed to the LME-DA approach, in which the temperature is initialized with 1. Thus, for this value of temperature, Equations (8) and (9) are just equal to those proposed in [1] for the artificial potential fields. When the robot gets stuck at some local minimum point, the value of the temperature starts to rise until either the robot escapes from the attraction region of the local minimum point or the maximum allowable value for the temperature is reached. Then, the value of the temperature starts to be slowly decreased, until it reaches 1, and the attractive and repulsive potential equations return to be just like those proposed in [1]. In spite of the advantage of the LME-DA method to avoid random movements of the robot, as it occurs when the LME-SA approach is used [8], when the LME-DA method is used, it is first necessary that the robot be caught by a local minimum point before using the escape mechanism based on the deterministic annealing approach. Such feature produces inefficient paths, as will be presented in Section IV. The reason why inefficient paths are produced is due to the very nature of the method, which was designed to perform local minimum escape (LME), and has nothing to do with the deterministic annealing approach. In order to further improve the LME-DA approach, we modify it with the view to performing to perform local minimum avoidance (LMA) instead of local minimum escape
(LME). The resulti is an LMA method inspired by the deterministic annealing approach, here referred to as LMA-DA algorithm.
model for the robot is considered, as well as dead reckoning systematic errors. Besides the range sensors, measurements are affected by Gaussian unbiased noise.
In this LMA-DA method proposed here, the attractive and repulsive potentials are modified in the same way as in Equations (8) and (9). The difference is that now, the temperature is initialized with a high value, and is slowly and progressively reduced. This is much more like the original intuitive idea of the deterministic annealing approach. At sufficiently high temperatures, each detected obstacle has influence over the whole environment and the potential function is generally convex. Under this circumstances, the global minimum point can be easily found, and as the temperature is slowly reduced, such a point is tracked up to T = 1, when the global minimum of the original potential function is found.
All environments contains local minima that may prevent the robot from arriving at the goal point. Each environment and the results from each of the four simulations are presented in the sequel. In the pictures that represents the simulation results, the starting point is indicated by a red circle and the goal is indicated by a green circle. The distance units along both axis are in centimeters.
It is worth remarking that the approach would perform better if the environment was entirely known, but, as previously mentioned, this is not the case for the experiments considered in this paper. For unknown environments it is not possible to take into account the influence of an obstacle in the potential function if it has not been detected yet by the sensing system of the robot. Thus, depending on the environment configuration, it is possible that before the robot reaches its ultimate goal, the temperature has a value small enough to permit formation of local minimum points, which can trap the robot. To deal with this situation the solution adopted here was to escape from such local minimum points by rising the temperature again, in a similar fashion as it was made in the LME-DA method. Once the robot is out of the region of influence of the local minimum point, the temperature is again slowly reduced until T = 1. IV.
B. Environment E1 The first simulated environment has an U-shaped obstacle that is a classic configuration for a local minimum. The start and destination points have coordinates (400, 700) cm and (800, 200) cm, respectively, and the dimensions of the environment are 1000cm × 1000cm. When considering the APF method without any escape mechanism (Figure 2(a)), the robot gets stuck at the local minimum point inside the U-shaped obstacle. The other three approaches were able to drive the robot towards the destination point, however some differences between them should be noticed. Comparing the two LME approaches, LME-SA and LME-DA, it is possible to notice that the latter (Figure 2(c)) produces a much smoother path than the former (Figure 2(b)). As shown in Figure 2(d), the LMA-DA algorithm proposed in the paper does not even enter the U-shaped obstacle, avoiding the local minimum point instead of escaping from it, as was the case with the two LME approaches.
S IMULATION RESULTS
A. The Experiments Three experiments will be presented in this section to illustrate the performance of the LMA-DA method introduced in the paper. The same experiments will be carried out with the following methods: •
the APF approach, considered here in order to show that the robot tends to get stuck in local minimum points and to indicate where such a point is in each environment;
•
the LME-SA method [8], considered here for comparison because the deterministic annealing was derived from the simulated annealing;
•
the LME-DA approach, an intermediate approach that was obtained as part of the development of the algorithm proposed in this paper;
•
the LMA-DA algorithm proposed in the paper.
A comparative analysis between the results obtained for each of the four algorithms will be carried out. The experiments were conducted based on a simulator developed in Matlab. The simulated robot is a differentialdriven mobile platform, equipped with 8 range sensors, equally distributed around the robot (one sensor at each 45 degrees). The range sensors reach is about 3 meters away. A dynamic
Fig. 2.
(a) APF
(b) LME-SA
(c) LME-DA
(d) LMA-DA
Results from Environment E1
C. Environment E2 Environment E2, proposed in [6], also has a U-shaped obstacle, with the difference that in this environment the local
minimum point is strong, due to the dimensions of the obstacle. The start and destination points have coordinates (1000, 400) cm and (100, 200) cm, respectively. As seen in Figure 3(a), the APF method was not able to reach the goal and the robot got stuck inside the U-shaped obstacle. For the several sets of parameters used in the tests, the LME-SA method was also not able to reach the goal. The inability of the LME-SA method to reach the final goal may be explained by the lack of a clear methodology for the choice of parameters. Since this method has a great dependence on the parameter set, it can be said that it was not possible to find a set of values that would enable convergence; Figure 3(b) depicts one of these tests. The LME-DA approach was able to deal with the local minima problem and drove the robot towards the goal point, but, due to its nature, the robot needed to enter the U-shaped obstacle first as shown in Figure 3(c). Finally, the LMA-DA algorithm proposed here was able to lead the robot towards the goal without the need to enter the obstacle, as seen in Figure 3(d). Notice that, initially, the robot moves away from the obstacle, which is due to high initial temperature. This distance can be made smaller if a lower initial temperature is used.
Fig. 3.
(a) APF
(b) LME-SA
(c) LME-DA
(d) LMA-DA
it from reaching its goal. In such cases, as previously mentioned, the adopted solution adopted is to rise the temperature again, just like in the LME-DA approach, and once the local minimum point is overcome, the temperature is reduced slowly, until it reaches value 1. To illustrate this situation, environment E2 has been modified to environment E3. The simulation results are presented in Figure 4, considering as starting and destination, the points with coordinates (1800, 400)cm and (500, 0)cm, respectively. Once again, as shown in Figure 4(a), the APF approach was not able to drive the robot up to its final destination; the robot got stuck in the local minimum point inside the nearest U-shaped obstacle. The same happened with the LMESA approach, as shown in Figure 4(b). The reason for this to happen appears to be the inappropriate choice of sets of parameters used in the experiments). For this experiment, the LME-DA approach was also not able to lead the robot up to its final goal, as depicted in Figure 4(c). As the LME-SA method, the LME-DA approach is extremely dependent on the set of parameters used, and for this environment, it was not possible to find a set of parameters that allowed algorithm to convert. The proposed LMA-DA method was the only method able to guide the robot up to its goal, as shown in Figure 4(d). Since the system has started with a high temperature, the robot did not enter the nearest U-shaped obstacle. However, that made the robot to move moves away from the destination point a long distance. As previously mentioned, such a distance can be decreased by reducing the initial temperature of the system. As the time goes by, the temperature is slowly reduced and the robot is directed towards its final goal. However, near the final destination, there is another U-shaped obstacle, and at this moment the temperature is low enough to allow the occurrence of local minimum points, and one of them traps the robot. When this happens, the temperature must be raised again until the robot is free from the influence of this local minimum point, and starts to pursuit its final goal again, while the temperature is slowly reduced.
Results from Environment E2
D. Environment E3 When using the LMA-DA algorithm proposed here, if the initial temperature is too low, the robot may be trapped in a local minimum point inside the obstacle. This is so because no previous information on the environment is available. As the environment is initially unknown for the robot, the obstacles that have not yet been detected by the sensor system of the robot will have no influence on the artificial potential field, and depending on the configuration of the environment it will be possible that before the robot arrives to its final destination, the temperature has been sufficiently decreased to allow the formation of local minimum points in the artificial potential field, which will be able to trap the robot, therefore, preventing
Fig. 4.
(a) APF
(b) LME-SA
(c) LME-DA
(d) LMA-DA
Results from Environment E3
V.
ACKNOWLEDGMENTS
C ONCLUSIONS
In this paper a new Local Minimum Avoidance algorithm (LMA-DA) inspired by the Deterministic Annealing approach was proposed, and a comparative analysis between the proposed method and three other approaches was performed. The proposed algorithm reached the highest rate of convergence among the algorithms tested. In addition, the LMEDA and LMA-DA algorithms have produced smoother paths than the LME-SA method based on simulated annealing. This stems from the nature of the deterministic annealing, where the noise is added to the cost function through the insertion of the temperature parameter in the functions of the attractive and repulsive potentials, while in simulated annealing the search for the global minimum is made through a random walking over the artificial potential field surface, resulting in ”noisy” paths. The LMA-DA algorithm proposed here also produced better paths than the LME-DA developed as a preliminary step in the derivation of the LMA-DA method. The reason for that comes from the fact that in the LMA-DA algorithm, the robot avoids the local minimum points instead of getting stuck and then try to escape, as is the case in the LME-DA method. The convergence of LME-SA and LME-DA are affected in a great deal by parameter choices. Regarding parameter choice, the least affect method among those considered in the paper is the LMA-DA algorithm proposed here, and the most affected method is the LME-SA. The inability to find a proper set of parameters is the probable reason why the LME-SA method failed on experiments performed on environments E2 and E3, and why the LME-DA method failed on the experiment performed on environment E3. This is due to the lack of a clear methodology to chose the most appropriate set of parameters for a given situation. In this context, the development of a methodology to perform parameter selection could be proposed as a future work. The use of such a methodology would allow the selection of a suitable set of parameters and will contribute to increase the convergence rate of these three algorithms. As an example of a methodology for parameter selection, it could be mentioned the use of learning automata, an adaptive approach proposed in [11] as a way for optimizing parameters in simulated annealing.
The authors would like to thank Professors Jugurta Montalv˜ao, Elyson Carvalho and Lucas Molina for their valuable contributions during the development of this work, and CAPES and CNPq for the financial support. R EFERENCES [1]
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