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A chaotic path planning generator based on logistic map and modulo tactics
Journal Pre-proof A chaotic path planning generator based on logistic map and modulo tactics Lazaros Moysis, Eleftherios Petavratzis, Christos Volos, Hector Nistazakis, Ioannis Stouboulos
PII: DOI: Reference:
S0921-8890(19)30587-1 https://doi.org/10.1016/j.robot.2019.103377 ROBOT 103377
To appear in:
Robotics and Autonomous Systems
Received date : 23 July 2019 Revised date : 8 November 2019 Accepted date : 13 November 2019 Please cite this article as: L. Moysis, E. Petavratzis, C. Volos et al., A chaotic path planning generator based on logistic map and modulo tactics, Robotics and Autonomous Systems (2019), doi: https://doi.org/10.1016/j.robot.2019.103377. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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A chaotic path planning generator based on logistic map and modulo tactics Lazaros Moysisa,1 , Eleftherios Petavratzisa , Christos Volosa , Hector Nistazakisb , Ioannis Stouboulosa a
Laboratory of Nonlinear Systems - Circuits & Complexity, Physics Department, Aristotle University of Thessaloniki, 54124, Greece b Department of Electronics, Computers, Telecommunications and Control, Faculty of Physics, National and Kapodistrian University of Athens, Athens, 15784 Greece
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Abstract
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A simple, short and efficient chaotic path planning algorithm is proposed for autonomous mobile robots, with the aim of covering a given terrain using chaotic, unpredictable motion. The proposed technique utilizes the logistic map with a chaotic tactic that utilizes a modulo function to produce a sequence of directions for a robot that can move in eight different directions on a grid. Extensive simulations are performed, and the results show a fast and efficient scanning of the given area. In addition, the proposed algorithm is further enhanced with a pheromone inspired memory technique, with good improvements in efficiency.
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Keywords: Autonomous mobile robot, Path planning, Terrain coverage, Chaos, Logistic map 1. Introduction
Over the last 40 years, chaos theory was integrated in almost every scientific branch and has found applications in numerous fields, from engineering to cryptography, secure communications, biology, finance, electrical circuits
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Email addresses: [email protected] (Lazaros Moysis), [email protected] (Eleftherios Petavratzis), [email protected] (Christos Volos), [email protected] (Hector Nistazakis), [email protected] (Ioannis Stouboulos) 1 Corresponding author Preprint submitted to Robotics And Autonomous Systems
November 4, 2019
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and many more, see for example [1, 3] and the references therein. Over the last two decades, chaotic systems have also found applications in the fields of robotics, and especially autonomous mobile robots [4]. Such robots, that can perform tasks without human supervision, have been extensively applied for tasks that present high complexity or danger, for example rescue missions [5], fire fighting [6], terrain exploration for search of explosives or surveillance [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], domestic tasks [18], mechanic manipulators [19] and more. In many of the aforementioned applications, a highly challenging task is the design of an efficient navigation strategy [7]. Many applications, like surveillance missions for example, require a navigation strategy that presents high unpredictability. A motion that looks unpredictable and random to an outsider will be harder to interpret as suspicious, which again is a requirement for surveillance missions. Thus, there is a high need in designing navigation strategies with high unpredictability. Of course, apart from this, an optimal strategy needs to have other attributes, like efficiency and quick coverage, which are essential. Thus, over the years researchers have proposed numerous techniques for path planning that also give good coverage results [20, 21, 22]. Out of the many proposed techniques, one approach that has been proven efficient is chaotic path planning [23, 24, 25, 26]. In this approach, a chaotic system is utilized to produce a trajectory for the system. The option to use chaotic systems to generate a trajectory seems attractive due to their qualitative characteristics, which are sensitivity to initial conditions and topological transitivity. Sensitivity to initial conditions means that by slightly changing the initial value of a chaotic system, its trajectory will be completely different, a property that leads to high complexity, unpredictability, and difficulty in reproducing the same motion twice. In addition, the property of topological transitivity guarantees that as the number of steps increases, the robot will explore the complete grid under study. Exploiting the above properties of chaotic systems, many researchers have considered the use of chaotic systems in path generation. This is usually done either by replacing the linear and directional velocities of the robotic system by the states of a chosen chaotic system [27, 28, 29, 30, 31, 32, 33], or by first generating a Chaotic Pseudo Random Bit Generator (CPRBG) [34, 35], which is then used to generate the robot movement [29, 36, 37, 38, 39, 40]. The first approach is considered for differential motion robots, while the second approach is mostly used for robots moving in discrete directions, that is four or eight, although it can be used for both types. Another recent 2
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approach is the use of chaotically generating fictive obstacles for the robot to avoid, thus causing abrupt zig-zag movements as a safety measure [41]. In this work, for the generation of chaotic motion, we consider a robot moving on a grid, with the capability of moving towards in eight different directions: up, down, left, right, up-left, up-right, down-left, down-right. This 8-direction motion is available for many commercially available robots. For example, a differential motion robot, can adopt this motion, by implementing a rotation around itself and a forward linear displacement directly to the next position. The same holds for humanoid robots. Thus, the considered approach can be widely adopted to available robots, requiring only a minimum number of sensors, like sonar on infra-red devices, to recognise the space edges, or potential obstacles. In the aforementioned works that consider motion in discrete directions, the use of CPRBGs has some disadvantages. The main disadvantage of using CPRBGs to generate a motion trajectory is that it requires the combination of 2 bits to generate a move, in case the robot moves in 4 directions, and 3 bits to generate a move, in case the robot moves in 8 directions. Thus, such methods require the generation of two or three times more iterations of the chaotic system to generate the same number of movements. In addition, most CPRBGs also utilize a de-skewing technique [42] to guarantee true randomness, which involves the discarding of the pairs 00 and 11 that are generated by the system. This leads to an even higher number of required iterations for the path generator. From the above, a question that arises is whether it is possible to produce a path planning method that can avoid the use of a CPRBG and the requirement of double (or triple) number of iterations, in order to produce a chaotic trajectory that will give equal or even better coverage results with the previous methods, while also having a simpler design and an easy implementation. Thus, to succesfully address this question here, instead of choosing a CPRBG, a chaotic tactics based on the logistic map [43, 35, 44] and a modulo operator [45, 35, 46] is applied, which allows the quick generation of chaotic motion. The resulting algorithm is short in length, easy to implement in any coding language, and computationally efficient, which makes it very easily implementable to commercial robots that utilize limited memory boards and micro-controllers. In addition, from the experimental simulation results performed, it is seen that the algorithm has a satisfying coverage rate. Moreover, to further improve the efficiency of the proposed algorithm, an inverse pheromone model is applied to the algorithm. The pheromone 3
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approach [40, 47, 48, 49] is applied by memorizing the last places visited by the robot in order to avoid revisiting them, and for small pheromone levels does not require a lot of extra memory to implement. In the simulation results we see overall improvement in coverage and in the number of visits per space. Additionally, this method can also be applied to a robot moving in 4 directions. Overall, the proposed modulo based tactics gives a simple and easy to implement method for chaotic path planning, that can be used to a plethora of different scenarios, like non-square grids, labyrinth-like grids, or grids with obstacles. This method can be further experimented upon, adjusted and enhanced by using different discrete time maps [50, 51], by considering its application in differential motion robots moving continuously on a grid, or by considering other chaos related applications, like encryption. The rest of the paper is structured as follows: Section 2 presents the chaotic path planning method, for motion in eight and four directions. Section 3 considers the inclusion of pheromone for the motion. Section 4 provides a discussion on the simulation results. Section 5 concludes the paper, with a discussion on future aspects of this work. 2. A chaotic path planning method based on modulo tactics
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2.1. Motion in eight directions - modulo 8 tactics To create the chaotic path planning method, the logistic map is initially considered xi+1 = rxi (1 − xi ), i = 0, 1, ... (1) where the natural parameter r varies in the interval [0, 4]. Depending on the value of r, the Logistic map can produce three different dynamics: - For r < 1, x decays to a fixed point (x → 0). - For 1 ≤ r ≤ 3, the previous point loses its stability and another fixed point appears (x = 1/r).
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- For 3 ≤ r ≤ 4, a rich dynamical behaviour, because the system goes from a periodic trajectory to chaos, is observed.
Figure 1 presents the bifurcation diagram of x versus the parameter r of the Logistic map, from which it is also clear that the map function is surjective 4
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in the complete interval [0, 1] only at r = 4.0. Furthermore, Figure 2 depicts the diagram of the Lyapunov exponent [52], given by n
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1X ln |f 0 (xi )| n→∞ n i=1
LE = lim
(2)
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which is a quantitative measure of chaos, and a positive Lyapunov exponent indicates chaos, as a function of the parameter r.
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Figure 1: Bifurcation diagram for the Logistics map.
Figure 2: Diagram of Lyapunov exponent of the Logistics map.
Thus, here we choose r = 4 and x(0) ∈ [−1, 1], so that all the subsequent values will remain in the same interval. Based on this map, we need to 5
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generate a sequence of chaotic direction commands. For this, we apply the following modulo based tactics mod 8]
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di = [1000000xi
(3)
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where [·] denotes the integer part of the argument. Here, the value of the logistic map xi at each time instant is multiplied by 1000000 to generate a large number and improve the complexity of the map. Then, the integer part of 1000000xi mod 8 is computed. This will generate a sequence of integers ranging from 0 to 7. Each integer is then matched to a motion, as shown in Figure 3. Overall the above methodology utilizes the logistic map to generate a chaotic motion for the system using a modulo based chaotic tactic. By looking at the histogram of this tactic for 2000000 iterations as shown in Figure 4, with exact values shown in Table 1, we observe that there is a homogeneous distribution of the 8 possible movements for the robot. This verifies that the robot will avoid uneven patrolling motion, a problem that was addressed in [27]. Regarding the number of different paths that can be generated from this algorithm, it is understood that since the system is deterministic, there is a finite number that can be produced. For a choice of parameter r, there are roughly 1016 different initial conditions to be chosen, using a 16-digit accuracy. So, since the logistic map gives chaotic a behavior for 3.57 ≤ r ≤ 4 and the parameter r can also be chosen with a 16-digit accuracy, it is clear that methods of replicating and predicting the generated path by an antagonist using brute force techniques will fail. For the simulation of this chaotic motion, we consider a 100 × 100 grid, thus having 1002 spaces (or cells) for the robot to cover. The robot makes one move with each iteration of the algorithm, according to the generated command. If the generated direction is unacceptable, like moving outside the defined limits (or obstacles), then the robot remains in its place and awaits for the next command. In Figure 5, the motion of the robot is displayed, starting from position [50, 50]T , for 35000 steps. The grid coverage (C) is 74.11%. This is simply calculated by counting the number of visited cells and dividing by the grid size, which is 1002 , as given by the following formula C=
M 1 X I(i) M i=1
6
(4)
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Table 1: Distribution of values for 2000000 iterations of the algorithm.
Appearance 250139 249381 250326 249985 250294 249581 249727 250567
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Value 0 1 2 3 4 5 6 7
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where M is the grid size and I(i) represents the coverage of each cell on the grid, that is ( 1, when cell i is covered I(i) = (5) 0, when cell i is not covered
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Also in Figure 6, a colour coded version of the grid is shown based on the number of visits in each space. Similar simulations for 50000 steps are shown in Figures 7 and 8. Also in Figure 9, two paths are shown for 35000 steps each, where the starting position is the lower left [1, 1]T (blue, square marker) or the upper right [100, 100]T (red, no marker). Observing all of the aforementioned figures, it is seen that the initial position does not have a significant affect on coverage. For a thorough simulation study on the connection between the time steps considered and the grid coverage and the number of visits per space, we considered the average values for 100 simulations for varying time steps ranging from 15000 to 140000, for a 5000 value step. In each simulation, the starting value of the logistic map and the starting point of the robot are taken randomly. So overall, 2600 simulations are performed. The results are shown in Figures 12 and 13. It is seen that the coverage steadily converges to 100 % as the number of steps tends to 140000. For a 75% coverage, around 45000 steps are required, and the mean number of visits for the visited area is 6. As mentioned in the introduction, the proposed algorithm is easy to implement in any coding language, short in length, and computationally efficient, which makes it very easily implementable to commercial robots that utilize limited memory boards and micro-controllers. 7
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Figure 3: Movement directions and their corresponding integers.
Figure 4: Histogram for 2000000 iterations of the algorithm.
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2.2. Motion in four directions - modulo 4 tactics In case that the movement of the robot is limited to 4 directions only, the same approach can be performed by simply changing the modulo command to mod 4, with corresponding directions being 0-up, 1-right, 2-down, 3-left. Of course, since the movement capabilities are now limited, the coverage will be much less satisfying, as can be seen in Table 2, where the average of 100 simulations is considered for a range of iterations. The coverage is 8
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Figure 5: Grid coverage for 35000 steps.
Figure 6: Colour-coded grid coverage showing number of visits for 35000 steps.
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Figure 7: Grid coverage for 50000 steps.
Figure 8: Colour-coded grid coverage showing number of visits for 50000 steps.
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Figure 9: Grid coverage for 35000 steps starting from the lower left [1, 1]T (blue, square marker) or the upper right [100, 100]T (red, no marker) position.
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significantly lower from that of the 8 direction robot. 3. Inclusion of pheromone
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To make the proposed algorithm even more efficient, we will also include a pheromone based approach. Here, inspired from the pheromone based movement that many biological organisms use [40], the system utilizes a tail, saving the last positions visited. So, before moving to a new position, a check is performed to verify if the new position has been visited before. If not, then the movement is performed and in case the position has been visited before, the robot remains idle until the next decision. The procedure is outlined in the simplified Algorithm 1.
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Table 2: Average coverage and number of visits for 100 simulations for different number of iterations, for a robot moving in 4 directions
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Mean number of visits for visited points 4.8018 5.1668 5.5361 5.8745 6.0938 6.5605 6.8022 7.2198 7.5939 8.0040
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Number Coverage (%) of steps 15000 32.0317 20000 39.5129 25000 46.5744 30000 52.1421 35000 58.0658 40000 62.2408 45000 66.9652 50000 69.8975 55000 72.9886 60000 75.7261
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Data: F : Pheromone level. pos: a 2 × (F + 1) vector having in each column: the current position in pos(end) and the last F visited positions in pos(1:F). move: The new robot position, as generated by the proposed generator. Result: Validate if the proposed position is accepted, and update the tail if move not in pos(1:F) then % generated position is acceptable pos(1:F)=pos(2:end) % update tail pos(end)=move % set new current position else % generated position belongs in the tail pos(1:F)=pos(2:end) % update tail end Algorithm 1: Pheromone based validation for each motion For the choice of pheromone level, even a small number can yield improved results, compared to the model without a pheromone. In addition, a smaller pheromone level requires less memory, since only a small number of previous steps needs to be saved. Thus the method is still implementable in a mobile 12
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robot with short memory. A illustrative simulation for 35000 steps, starting again from [50, 50]T for pheromone level F = 4 is shown in Figures 10 and 11, where we see that the coverage is 85%. For simulation purposes, we again consider a 100 × 100 grid, thus having 2 100 spaces for the robot to cover. Again for a thorough study, we consider the average values for 100 simulations for varying time steps ranging from 15000 to 110000, for a 5000 value step. In each simulation, the starting value of the logistic map and the starting point of the robot are taken randomly, and the pheromone value is 1, 4, 8 or 12. The results are shown in Figures 12 and 13. 4. Evaluation of the experimental results
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Comparing the results of the simulations shown in Figures 12, 13, we observe that for all of the pheromone levels applied (F = 0, 1, 4, 8, 12), around 20000 iterations of the algorithm are required to cover half the region. For a 75% coverage, the number of iterations is raised to approximately 45000. By employing a pheromone based memory to the algorithm, a clear improvement to the level of coverage and also to the number of visits per visited spaces is observed. Even with a small level of pheromone, like 1-4, the average coverage can be raised up to 5%, which is very satisfying considering the small amount of memory required to implement the pheromone. Also, as the number of iterations increases, all approaches converge to 100% coverage, as can be seen in Figure 12. Thus, the highest differences can be seen for smaller values of iterations, were it is seen that the use of pheromone is more efficient. Moreover, all of the methods have exponential convergence rate as seen in Figure 12, which is confirmed using Origin. The fitting procedure resulted in an exponential convergence rate of the form y = y0 + AeR0 x
(6)
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where the values of y0 , A, R0 for each case are summarised in Table 3. Similarly, the number of visits is shown in Figure 13 to increase linearly, following a linear curve y = ax + b (7) where the values of a, b for each case are summarised in Table 4.
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It should definitely be pointed out here that non-square grids can be considered, or grids with obstacles. In these cases, it is logical to assume that the starting position of the robot, as well as the arrangement of the obstacles and the form of the grid, will affect the coverage of the area. As illustrative examples, three cases are presented. In the first, a non-square grid is considered. The robot starts its path from the lower left position, [1, 1]T and performs 31000 iterations of the algorithm. The results are shown in Figures 14, 15. In the second case, a labyrinth like environment is considered. The robot starts from position [2, 2]T and performs 20000 iterations. The results are shown in Figures 18, 19. In the third case, a square grid with obstacles is considered. The robot starts from the upper right position [100, 100]T and performs 35000 iterations of the algorithm. The results are shown in Figures 16, 17. For all of these special cases, inceasing the number of iterations leads steadily to grid coverage, thus the proposed algorithm can be proven effective under these special conditions as well.
Figure 10: Grid coverage for 35000 steps and pheromone level F = 4.
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5. Conclusions
In this work, a novel chaotic path planning technique was proposed, based on the logistic map and a modulo based tactic. The method does not require
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Figure 11: Colour-coded grid coverage showing number of visits for 35000 steps and pheromone level F = 4.
y0 99.25349 98.9143 99.0126 99.33867 99.44381
A -96.44361 -92.82999 -95.30624 -94.60022 -95.83784
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F=0 F=1 F=4 F=8 F=12
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Table 3: Exponential curve fitting values for the grid coverage.
R0 -3.26492·10−5 -3.39844·10−5 -3.78222·10−5 -3.91624·10−5 -4.02226·10−5
R-square 0.99891 0.99942 0.99926 0.99959 0.99961
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the use of a CPRBG, and is thus faster, easier to implement, and overall more efficient. From extensive experimental simulations, it is seen that the proposed method is very effective in guaranteeing path coverage, as the number of iterations increases. Moreover, to further improve the method, a pheromone based memory is applied to the algorithm, yielding improved performance. Future works will consider the thorough study of non-square grids with obstacles, the use of different discrete chaotic maps in combination with modulo tactics to generate the chaotic path, the application to fractional order systems [53, 54, 55], the cooperation of multiple robots [56], where the additional problem posed would be sharing information for the position of each robot and avoiding revisiting previous positions by allied robots, the in15
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Table 4: Linear curve fitting values for the number of visits.
R-square 0.9968 0.9979 0.9974 0.9974 0.9973
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b 2.166 1.942 1.705 1.563 1.527
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F=0 F=1 F=4 F=8 F=12
a 8.411·10−5 8.599·10−5 8.716·10−5 8.789·10−5 8.794·10−5
Figure 12: Coverage rate versus number of iterations.
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clusion of fictitious obstacles to improve unpredictability [41], the adjustment of this approach for potential application to differential motion robots moving on a continuous grid [57], the experimental implementation on a mobile robot using microcontrollers, and also the further use of similar modulo tactics to problems of encryption. Overall, it is our belief that this approach can be further implemented to numerous chaos related applications in robotics and engineering.
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Acknowledgements
The authors are thankful to the anonymous reviewers for their suggestions, that greatly improved the quality of this work.
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Figure 13: Mean number of visits per visited cells versus number of iterations.
Figure 14: Non-square grid, coverage for 31000 iterations
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Figure 15: Colour-coded grid coverage showing number of visits for 31000 iterations.
Figure 16: Inclusion of obstacles, coverage for 35000 iterations
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[39] C. K. Volos, I. M. Kyprianidis, I. N. Stouboulos, Experimental investigation on coverage performance of a chaotic autonomous mobile robot, Robotics and Autonomous Systems 61 (12) (2013) 1314–1322. [40] E. K. Petavratzis, C. K. Volos, I. N. Stouboulos, H. E. Nistazakis, K. G. Kyritsi, K. P. Valavanis, Coverage performance of a chaotic mobile robot using an inverse pheromone model, in: 2019 8th International Conference on Modern Circuits and Systems Technologies (MOCAST), IEEE, 2019, pp. 1–4.
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Lazaros Moysis received the B.Sc., M.Sc. and PhD degrees in 2011, 2013 and 2017 from the Department of Mathematics, Aristotle University of Thessaloniki, Greece. He is currently a post-doctoral researcher at the Physics Department, Aristotle University of Thessaloniki, at the Laboratory of Nonlinear Systems, Circuits and Complexity. His research interests include the theory of control systems, descriptor systems and chaotic systems, and problems like controllability and observability, observer design, synchronization and its application in secure communications and robotics.
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Eleftherios Petavratzis was born in Thessaloniki, Greece. He received the B.Sc. and M.Sc. degrees in 2010 and 2017 respectively from the Department of Mathematics, Aristotle University of Thessaloniki, Greece. He is currently a PhD student at the Physics Department, Aristotle University of Thessaloniki, as a member of the Laboratory of Nonlinear Systems, Circuits and Complexity. His research interests lie in control theory, autonomous and chaotic systems and motion planning.
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Christos Volos received the Physics Diploma, the M.Sc. in electronics, and the Ph.D. degree in chaotic electronics from the Physics Department, Aristotle University of Thessaloniki, in 1999, 2002, and 2008. He is currently an Assistant Professor with the Physics Department, Aristotle University of Thessaloniki, Greece. His current research interests include the design and study of analog and mixed signal electronic circuits, chaotic electronics and their applications (secure communications, cryptography, robotics), experimental chaotic synchronization, chaotic UWB communications, and measurement and instrumentation systems. He is a member of the Laboratory of Nonlinear Systems, Circuits and Complexity, Physics Department, Aristotle University of Thessaloniki.
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Hector Nistazakis was born in Athens, Greece. He received his B.Sc. (1997), M.Sc.(1999) and Ph.D. (2002), from the National and Kapodistrian University of Athens, Greece. Currently he is an Associate Professor in the Department of Physics in the National and Kapodistrian University of Athens. Prof. Nistazakis has authored or co-authored more than 90 journal papers, 75 conference papers, 14 book chapters and 1 book. He acts as a reviewer for several international
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Ioannis Stouboulos was born 30/11/1957, in Thessaloniki, Greece. He received the M.S. degree in electronic & telecommunications in 1982 and Ph.D. degree in Non Linear Electric Circuits from the Physics Department of the Aristotle University of Thessaloniki, Greece in 1998. He joined the Physics Department of the Aristotle University of Thessaloniki, Greece, in 1980 as a research assistant, where he was involved in projects concerning the analysis and synthesis of acoustical signals. Since 1995 he has been involved in projects concerning the study of nonlinear circuits and systems in the same department, where he serves as Associate Professor.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
PII: DOI: Reference:
S0921-8890(19)30587-1 https://doi.org/10.1016/j.robot.2019.103377 ROBOT 103377
To appear in:
Robotics and Autonomous Systems
Received date : 23 July 2019 Revised date : 8 November 2019 Accepted date : 13 November 2019 Please cite this article as: L. Moysis, E. Petavratzis, C. Volos et al., A chaotic path planning generator based on logistic map and modulo tactics, Robotics and Autonomous Systems (2019), doi: https://doi.org/10.1016/j.robot.2019.103377. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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A chaotic path planning generator based on logistic map and modulo tactics Lazaros Moysisa,1 , Eleftherios Petavratzisa , Christos Volosa , Hector Nistazakisb , Ioannis Stouboulosa a
Laboratory of Nonlinear Systems - Circuits & Complexity, Physics Department, Aristotle University of Thessaloniki, 54124, Greece b Department of Electronics, Computers, Telecommunications and Control, Faculty of Physics, National and Kapodistrian University of Athens, Athens, 15784 Greece
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Abstract
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A simple, short and efficient chaotic path planning algorithm is proposed for autonomous mobile robots, with the aim of covering a given terrain using chaotic, unpredictable motion. The proposed technique utilizes the logistic map with a chaotic tactic that utilizes a modulo function to produce a sequence of directions for a robot that can move in eight different directions on a grid. Extensive simulations are performed, and the results show a fast and efficient scanning of the given area. In addition, the proposed algorithm is further enhanced with a pheromone inspired memory technique, with good improvements in efficiency.
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Keywords: Autonomous mobile robot, Path planning, Terrain coverage, Chaos, Logistic map 1. Introduction
Over the last 40 years, chaos theory was integrated in almost every scientific branch and has found applications in numerous fields, from engineering to cryptography, secure communications, biology, finance, electrical circuits
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Email addresses: [email protected] (Lazaros Moysis), [email protected] (Eleftherios Petavratzis), [email protected] (Christos Volos), [email protected] (Hector Nistazakis), [email protected] (Ioannis Stouboulos) 1 Corresponding author Preprint submitted to Robotics And Autonomous Systems
November 4, 2019
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and many more, see for example [1, 3] and the references therein. Over the last two decades, chaotic systems have also found applications in the fields of robotics, and especially autonomous mobile robots [4]. Such robots, that can perform tasks without human supervision, have been extensively applied for tasks that present high complexity or danger, for example rescue missions [5], fire fighting [6], terrain exploration for search of explosives or surveillance [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], domestic tasks [18], mechanic manipulators [19] and more. In many of the aforementioned applications, a highly challenging task is the design of an efficient navigation strategy [7]. Many applications, like surveillance missions for example, require a navigation strategy that presents high unpredictability. A motion that looks unpredictable and random to an outsider will be harder to interpret as suspicious, which again is a requirement for surveillance missions. Thus, there is a high need in designing navigation strategies with high unpredictability. Of course, apart from this, an optimal strategy needs to have other attributes, like efficiency and quick coverage, which are essential. Thus, over the years researchers have proposed numerous techniques for path planning that also give good coverage results [20, 21, 22]. Out of the many proposed techniques, one approach that has been proven efficient is chaotic path planning [23, 24, 25, 26]. In this approach, a chaotic system is utilized to produce a trajectory for the system. The option to use chaotic systems to generate a trajectory seems attractive due to their qualitative characteristics, which are sensitivity to initial conditions and topological transitivity. Sensitivity to initial conditions means that by slightly changing the initial value of a chaotic system, its trajectory will be completely different, a property that leads to high complexity, unpredictability, and difficulty in reproducing the same motion twice. In addition, the property of topological transitivity guarantees that as the number of steps increases, the robot will explore the complete grid under study. Exploiting the above properties of chaotic systems, many researchers have considered the use of chaotic systems in path generation. This is usually done either by replacing the linear and directional velocities of the robotic system by the states of a chosen chaotic system [27, 28, 29, 30, 31, 32, 33], or by first generating a Chaotic Pseudo Random Bit Generator (CPRBG) [34, 35], which is then used to generate the robot movement [29, 36, 37, 38, 39, 40]. The first approach is considered for differential motion robots, while the second approach is mostly used for robots moving in discrete directions, that is four or eight, although it can be used for both types. Another recent 2
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approach is the use of chaotically generating fictive obstacles for the robot to avoid, thus causing abrupt zig-zag movements as a safety measure [41]. In this work, for the generation of chaotic motion, we consider a robot moving on a grid, with the capability of moving towards in eight different directions: up, down, left, right, up-left, up-right, down-left, down-right. This 8-direction motion is available for many commercially available robots. For example, a differential motion robot, can adopt this motion, by implementing a rotation around itself and a forward linear displacement directly to the next position. The same holds for humanoid robots. Thus, the considered approach can be widely adopted to available robots, requiring only a minimum number of sensors, like sonar on infra-red devices, to recognise the space edges, or potential obstacles. In the aforementioned works that consider motion in discrete directions, the use of CPRBGs has some disadvantages. The main disadvantage of using CPRBGs to generate a motion trajectory is that it requires the combination of 2 bits to generate a move, in case the robot moves in 4 directions, and 3 bits to generate a move, in case the robot moves in 8 directions. Thus, such methods require the generation of two or three times more iterations of the chaotic system to generate the same number of movements. In addition, most CPRBGs also utilize a de-skewing technique [42] to guarantee true randomness, which involves the discarding of the pairs 00 and 11 that are generated by the system. This leads to an even higher number of required iterations for the path generator. From the above, a question that arises is whether it is possible to produce a path planning method that can avoid the use of a CPRBG and the requirement of double (or triple) number of iterations, in order to produce a chaotic trajectory that will give equal or even better coverage results with the previous methods, while also having a simpler design and an easy implementation. Thus, to succesfully address this question here, instead of choosing a CPRBG, a chaotic tactics based on the logistic map [43, 35, 44] and a modulo operator [45, 35, 46] is applied, which allows the quick generation of chaotic motion. The resulting algorithm is short in length, easy to implement in any coding language, and computationally efficient, which makes it very easily implementable to commercial robots that utilize limited memory boards and micro-controllers. In addition, from the experimental simulation results performed, it is seen that the algorithm has a satisfying coverage rate. Moreover, to further improve the efficiency of the proposed algorithm, an inverse pheromone model is applied to the algorithm. The pheromone 3
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approach [40, 47, 48, 49] is applied by memorizing the last places visited by the robot in order to avoid revisiting them, and for small pheromone levels does not require a lot of extra memory to implement. In the simulation results we see overall improvement in coverage and in the number of visits per space. Additionally, this method can also be applied to a robot moving in 4 directions. Overall, the proposed modulo based tactics gives a simple and easy to implement method for chaotic path planning, that can be used to a plethora of different scenarios, like non-square grids, labyrinth-like grids, or grids with obstacles. This method can be further experimented upon, adjusted and enhanced by using different discrete time maps [50, 51], by considering its application in differential motion robots moving continuously on a grid, or by considering other chaos related applications, like encryption. The rest of the paper is structured as follows: Section 2 presents the chaotic path planning method, for motion in eight and four directions. Section 3 considers the inclusion of pheromone for the motion. Section 4 provides a discussion on the simulation results. Section 5 concludes the paper, with a discussion on future aspects of this work. 2. A chaotic path planning method based on modulo tactics
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2.1. Motion in eight directions - modulo 8 tactics To create the chaotic path planning method, the logistic map is initially considered xi+1 = rxi (1 − xi ), i = 0, 1, ... (1) where the natural parameter r varies in the interval [0, 4]. Depending on the value of r, the Logistic map can produce three different dynamics: - For r < 1, x decays to a fixed point (x → 0). - For 1 ≤ r ≤ 3, the previous point loses its stability and another fixed point appears (x = 1/r).
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- For 3 ≤ r ≤ 4, a rich dynamical behaviour, because the system goes from a periodic trajectory to chaos, is observed.
Figure 1 presents the bifurcation diagram of x versus the parameter r of the Logistic map, from which it is also clear that the map function is surjective 4
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in the complete interval [0, 1] only at r = 4.0. Furthermore, Figure 2 depicts the diagram of the Lyapunov exponent [52], given by n
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1X ln |f 0 (xi )| n→∞ n i=1
LE = lim
(2)
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which is a quantitative measure of chaos, and a positive Lyapunov exponent indicates chaos, as a function of the parameter r.
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Figure 1: Bifurcation diagram for the Logistics map.
Figure 2: Diagram of Lyapunov exponent of the Logistics map.
Thus, here we choose r = 4 and x(0) ∈ [−1, 1], so that all the subsequent values will remain in the same interval. Based on this map, we need to 5
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generate a sequence of chaotic direction commands. For this, we apply the following modulo based tactics mod 8]
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di = [1000000xi
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where [·] denotes the integer part of the argument. Here, the value of the logistic map xi at each time instant is multiplied by 1000000 to generate a large number and improve the complexity of the map. Then, the integer part of 1000000xi mod 8 is computed. This will generate a sequence of integers ranging from 0 to 7. Each integer is then matched to a motion, as shown in Figure 3. Overall the above methodology utilizes the logistic map to generate a chaotic motion for the system using a modulo based chaotic tactic. By looking at the histogram of this tactic for 2000000 iterations as shown in Figure 4, with exact values shown in Table 1, we observe that there is a homogeneous distribution of the 8 possible movements for the robot. This verifies that the robot will avoid uneven patrolling motion, a problem that was addressed in [27]. Regarding the number of different paths that can be generated from this algorithm, it is understood that since the system is deterministic, there is a finite number that can be produced. For a choice of parameter r, there are roughly 1016 different initial conditions to be chosen, using a 16-digit accuracy. So, since the logistic map gives chaotic a behavior for 3.57 ≤ r ≤ 4 and the parameter r can also be chosen with a 16-digit accuracy, it is clear that methods of replicating and predicting the generated path by an antagonist using brute force techniques will fail. For the simulation of this chaotic motion, we consider a 100 × 100 grid, thus having 1002 spaces (or cells) for the robot to cover. The robot makes one move with each iteration of the algorithm, according to the generated command. If the generated direction is unacceptable, like moving outside the defined limits (or obstacles), then the robot remains in its place and awaits for the next command. In Figure 5, the motion of the robot is displayed, starting from position [50, 50]T , for 35000 steps. The grid coverage (C) is 74.11%. This is simply calculated by counting the number of visited cells and dividing by the grid size, which is 1002 , as given by the following formula C=
M 1 X I(i) M i=1
6
(4)
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Table 1: Distribution of values for 2000000 iterations of the algorithm.
Appearance 250139 249381 250326 249985 250294 249581 249727 250567
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Value 0 1 2 3 4 5 6 7
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where M is the grid size and I(i) represents the coverage of each cell on the grid, that is ( 1, when cell i is covered I(i) = (5) 0, when cell i is not covered
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Also in Figure 6, a colour coded version of the grid is shown based on the number of visits in each space. Similar simulations for 50000 steps are shown in Figures 7 and 8. Also in Figure 9, two paths are shown for 35000 steps each, where the starting position is the lower left [1, 1]T (blue, square marker) or the upper right [100, 100]T (red, no marker). Observing all of the aforementioned figures, it is seen that the initial position does not have a significant affect on coverage. For a thorough simulation study on the connection between the time steps considered and the grid coverage and the number of visits per space, we considered the average values for 100 simulations for varying time steps ranging from 15000 to 140000, for a 5000 value step. In each simulation, the starting value of the logistic map and the starting point of the robot are taken randomly. So overall, 2600 simulations are performed. The results are shown in Figures 12 and 13. It is seen that the coverage steadily converges to 100 % as the number of steps tends to 140000. For a 75% coverage, around 45000 steps are required, and the mean number of visits for the visited area is 6. As mentioned in the introduction, the proposed algorithm is easy to implement in any coding language, short in length, and computationally efficient, which makes it very easily implementable to commercial robots that utilize limited memory boards and micro-controllers. 7
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Figure 3: Movement directions and their corresponding integers.
Figure 4: Histogram for 2000000 iterations of the algorithm.
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2.2. Motion in four directions - modulo 4 tactics In case that the movement of the robot is limited to 4 directions only, the same approach can be performed by simply changing the modulo command to mod 4, with corresponding directions being 0-up, 1-right, 2-down, 3-left. Of course, since the movement capabilities are now limited, the coverage will be much less satisfying, as can be seen in Table 2, where the average of 100 simulations is considered for a range of iterations. The coverage is 8
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Figure 5: Grid coverage for 35000 steps.
Figure 6: Colour-coded grid coverage showing number of visits for 35000 steps.
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Figure 7: Grid coverage for 50000 steps.
Figure 8: Colour-coded grid coverage showing number of visits for 50000 steps.
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Figure 9: Grid coverage for 35000 steps starting from the lower left [1, 1]T (blue, square marker) or the upper right [100, 100]T (red, no marker) position.
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significantly lower from that of the 8 direction robot. 3. Inclusion of pheromone
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To make the proposed algorithm even more efficient, we will also include a pheromone based approach. Here, inspired from the pheromone based movement that many biological organisms use [40], the system utilizes a tail, saving the last positions visited. So, before moving to a new position, a check is performed to verify if the new position has been visited before. If not, then the movement is performed and in case the position has been visited before, the robot remains idle until the next decision. The procedure is outlined in the simplified Algorithm 1.
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Table 2: Average coverage and number of visits for 100 simulations for different number of iterations, for a robot moving in 4 directions
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Mean number of visits for visited points 4.8018 5.1668 5.5361 5.8745 6.0938 6.5605 6.8022 7.2198 7.5939 8.0040
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Number Coverage (%) of steps 15000 32.0317 20000 39.5129 25000 46.5744 30000 52.1421 35000 58.0658 40000 62.2408 45000 66.9652 50000 69.8975 55000 72.9886 60000 75.7261
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Data: F : Pheromone level. pos: a 2 × (F + 1) vector having in each column: the current position in pos(end) and the last F visited positions in pos(1:F). move: The new robot position, as generated by the proposed generator. Result: Validate if the proposed position is accepted, and update the tail if move not in pos(1:F) then % generated position is acceptable pos(1:F)=pos(2:end) % update tail pos(end)=move % set new current position else % generated position belongs in the tail pos(1:F)=pos(2:end) % update tail end Algorithm 1: Pheromone based validation for each motion For the choice of pheromone level, even a small number can yield improved results, compared to the model without a pheromone. In addition, a smaller pheromone level requires less memory, since only a small number of previous steps needs to be saved. Thus the method is still implementable in a mobile 12
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robot with short memory. A illustrative simulation for 35000 steps, starting again from [50, 50]T for pheromone level F = 4 is shown in Figures 10 and 11, where we see that the coverage is 85%. For simulation purposes, we again consider a 100 × 100 grid, thus having 2 100 spaces for the robot to cover. Again for a thorough study, we consider the average values for 100 simulations for varying time steps ranging from 15000 to 110000, for a 5000 value step. In each simulation, the starting value of the logistic map and the starting point of the robot are taken randomly, and the pheromone value is 1, 4, 8 or 12. The results are shown in Figures 12 and 13. 4. Evaluation of the experimental results
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Comparing the results of the simulations shown in Figures 12, 13, we observe that for all of the pheromone levels applied (F = 0, 1, 4, 8, 12), around 20000 iterations of the algorithm are required to cover half the region. For a 75% coverage, the number of iterations is raised to approximately 45000. By employing a pheromone based memory to the algorithm, a clear improvement to the level of coverage and also to the number of visits per visited spaces is observed. Even with a small level of pheromone, like 1-4, the average coverage can be raised up to 5%, which is very satisfying considering the small amount of memory required to implement the pheromone. Also, as the number of iterations increases, all approaches converge to 100% coverage, as can be seen in Figure 12. Thus, the highest differences can be seen for smaller values of iterations, were it is seen that the use of pheromone is more efficient. Moreover, all of the methods have exponential convergence rate as seen in Figure 12, which is confirmed using Origin. The fitting procedure resulted in an exponential convergence rate of the form y = y0 + AeR0 x
(6)
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where the values of y0 , A, R0 for each case are summarised in Table 3. Similarly, the number of visits is shown in Figure 13 to increase linearly, following a linear curve y = ax + b (7) where the values of a, b for each case are summarised in Table 4.
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It should definitely be pointed out here that non-square grids can be considered, or grids with obstacles. In these cases, it is logical to assume that the starting position of the robot, as well as the arrangement of the obstacles and the form of the grid, will affect the coverage of the area. As illustrative examples, three cases are presented. In the first, a non-square grid is considered. The robot starts its path from the lower left position, [1, 1]T and performs 31000 iterations of the algorithm. The results are shown in Figures 14, 15. In the second case, a labyrinth like environment is considered. The robot starts from position [2, 2]T and performs 20000 iterations. The results are shown in Figures 18, 19. In the third case, a square grid with obstacles is considered. The robot starts from the upper right position [100, 100]T and performs 35000 iterations of the algorithm. The results are shown in Figures 16, 17. For all of these special cases, inceasing the number of iterations leads steadily to grid coverage, thus the proposed algorithm can be proven effective under these special conditions as well.
Figure 10: Grid coverage for 35000 steps and pheromone level F = 4.
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5. Conclusions
In this work, a novel chaotic path planning technique was proposed, based on the logistic map and a modulo based tactic. The method does not require
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Figure 11: Colour-coded grid coverage showing number of visits for 35000 steps and pheromone level F = 4.
y0 99.25349 98.9143 99.0126 99.33867 99.44381
A -96.44361 -92.82999 -95.30624 -94.60022 -95.83784
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F=0 F=1 F=4 F=8 F=12
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Table 3: Exponential curve fitting values for the grid coverage.
R0 -3.26492·10−5 -3.39844·10−5 -3.78222·10−5 -3.91624·10−5 -4.02226·10−5
R-square 0.99891 0.99942 0.99926 0.99959 0.99961
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the use of a CPRBG, and is thus faster, easier to implement, and overall more efficient. From extensive experimental simulations, it is seen that the proposed method is very effective in guaranteeing path coverage, as the number of iterations increases. Moreover, to further improve the method, a pheromone based memory is applied to the algorithm, yielding improved performance. Future works will consider the thorough study of non-square grids with obstacles, the use of different discrete chaotic maps in combination with modulo tactics to generate the chaotic path, the application to fractional order systems [53, 54, 55], the cooperation of multiple robots [56], where the additional problem posed would be sharing information for the position of each robot and avoiding revisiting previous positions by allied robots, the in15
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Table 4: Linear curve fitting values for the number of visits.
R-square 0.9968 0.9979 0.9974 0.9974 0.9973
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b 2.166 1.942 1.705 1.563 1.527
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F=0 F=1 F=4 F=8 F=12
a 8.411·10−5 8.599·10−5 8.716·10−5 8.789·10−5 8.794·10−5
Figure 12: Coverage rate versus number of iterations.
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clusion of fictitious obstacles to improve unpredictability [41], the adjustment of this approach for potential application to differential motion robots moving on a continuous grid [57], the experimental implementation on a mobile robot using microcontrollers, and also the further use of similar modulo tactics to problems of encryption. Overall, it is our belief that this approach can be further implemented to numerous chaos related applications in robotics and engineering.
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Acknowledgements
The authors are thankful to the anonymous reviewers for their suggestions, that greatly improved the quality of this work.
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Figure 13: Mean number of visits per visited cells versus number of iterations.
Figure 14: Non-square grid, coverage for 31000 iterations
References
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[1] A. T. Azar, S. Vaidyanathan, Chaos modeling and control systems design, Vol. 581, Springer, 2015. [2] A. T. Azar, S. Vaidyanathan, Advances in chaos theory and intelligent control, Vol. 337, Springer, 2016.
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Figure 15: Colour-coded grid coverage showing number of visits for 31000 iterations.
Figure 16: Inclusion of obstacles, coverage for 35000 iterations
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Figure 17: Colour-coded grid coverage showing number of visits for 35000 iterations.
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Figure 18: Labyrinth-like grid, coverage for 20000 iterations
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Figure 19: Colour-coded grid coverage showing number of visits for 20000 iterations.
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Lazaros Moysis received the B.Sc., M.Sc. and PhD degrees in 2011, 2013 and 2017 from the Department of Mathematics, Aristotle University of Thessaloniki, Greece. He is currently a post-doctoral researcher at the Physics Department, Aristotle University of Thessaloniki, at the Laboratory of Nonlinear Systems, Circuits and Complexity. His research interests include the theory of control systems, descriptor systems and chaotic systems, and problems like controllability and observability, observer design, synchronization and its application in secure communications and robotics.
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Eleftherios Petavratzis was born in Thessaloniki, Greece. He received the B.Sc. and M.Sc. degrees in 2010 and 2017 respectively from the Department of Mathematics, Aristotle University of Thessaloniki, Greece. He is currently a PhD student at the Physics Department, Aristotle University of Thessaloniki, as a member of the Laboratory of Nonlinear Systems, Circuits and Complexity. His research interests lie in control theory, autonomous and chaotic systems and motion planning.
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Christos Volos received the Physics Diploma, the M.Sc. in electronics, and the Ph.D. degree in chaotic electronics from the Physics Department, Aristotle University of Thessaloniki, in 1999, 2002, and 2008. He is currently an Assistant Professor with the Physics Department, Aristotle University of Thessaloniki, Greece. His current research interests include the design and study of analog and mixed signal electronic circuits, chaotic electronics and their applications (secure communications, cryptography, robotics), experimental chaotic synchronization, chaotic UWB communications, and measurement and instrumentation systems. He is a member of the Laboratory of Nonlinear Systems, Circuits and Complexity, Physics Department, Aristotle University of Thessaloniki.
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Hector Nistazakis was born in Athens, Greece. He received his B.Sc. (1997), M.Sc.(1999) and Ph.D. (2002), from the National and Kapodistrian University of Athens, Greece. Currently he is an Associate Professor in the Department of Physics in the National and Kapodistrian University of Athens. Prof. Nistazakis has authored or co-authored more than 90 journal papers, 75 conference papers, 14 book chapters and 1 book. He acts as a reviewer for several international
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Ioannis Stouboulos was born 30/11/1957, in Thessaloniki, Greece. He received the M.S. degree in electronic & telecommunications in 1982 and Ph.D. degree in Non Linear Electric Circuits from the Physics Department of the Aristotle University of Thessaloniki, Greece in 1998. He joined the Physics Department of the Aristotle University of Thessaloniki, Greece, in 1980 as a research assistant, where he was involved in projects concerning the analysis and synthesis of acoustical signals. Since 1995 he has been involved in projects concerning the study of nonlinear circuits and systems in the same department, where he serves as Associate Professor.
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: