An Autonomous Mobile Robot Guided by a Chaotic True Random Bits Generator Ch.K. Volos, I.M. Kyprianidis, I.N. Stouboulos, S.G. Stavrinides, and A.N. Anagnostopoulos
Abstract In this work a robot’s controller, which ensures chaotic motion to an autonomous mobile robot, is presented. This new strategy, which is very useful in many robotic missions, generates an unpredictable trajectory by using a chaotic path planning generator. The proposed generator produces a trajectory, which is the result of a sequence of planned target locations. In contrary with other similar works, this one is based on a new chaotic true random bits generator, which has as a basic feature the coexistence of two different synchronization phenomena between mutually coupled identical nonlinear circuits. Simulation tests confirm that the whole robot’s workplace is covered with unpredictable way in a very satisfactory time.
1 Introduction The last two decades the subject of autonomous mobile robots, has become a topic of great interest because of its ever-increasing applications in commercial and military activities [4, 8]. In many cases the success of robot’s missions, mostly is determined by the path planning, where researchers try to find a plan for producing an unpredictable trajectory. This feature was the starting point for the use of nonlinear dynamical systems in the design of autonomous robots. Ch.K. Volos Department of Mathematics and Engineering Studies, Hellenic Army Academy, Athens, GR-16673, Greece e-mail:
[email protected] I.M. Kyprianidis I.N. Stouboulos A.N. Anagnostopoulos Physics Department, Aristotle University of Thessaloniki, GR-54124, Greece e-mail:
[email protected];
[email protected];
[email protected] S.G. Stavrinides () Kavala Institute of Technology, Department of Electrical Engineering, Kavala, Greece e-mail:
[email protected] S.G. Stavrinides et al. (eds.), Chaos and Complex Systems, DOI 10.1007/978-3-642-33914-1 45, © Springer-Verlag Berlin Heidelberg 2013
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The goal of using chaotic systems in mobile robots is achieved by designing motion controllers, which ensures the desirable chaotic behavior. The basic feature of chaotic systems, which is the high unpredictable behavior, is a necessary condition that ensures also the unpredictable scanning of the terrain. For this reason very known chaotic systems, discrete or continuous, have been used [4, 8]. In this work, a new strategy, which generates an unpredictable robot’s trajectory, is introduced. In contrary with other similar works, the proposed robot’s motion controller consists of a chaotic True Random Bits Generator (TRBG), which is based on the coexistence of two different synchronization phenomena between mutually coupled identical nonlinear circuits, the well-known complete chaotic synchronization and a recently new proposed inverse -lag synchronization. This paper is organized as follows. In Sect. 2, the basic features of chaotic systems and the synchronization phenomena, which are used in this work are described. Section 3 presents the new type of chaotic TRBG. The adopted model for the mobile robot and the analysis of the robot’s trajectory are described in Sect. 4. Finally, Sect. 5 includes the conclusion remarks.
2 Chaotic Systems and Synchronization Phenomena A nonlinear dynamical system, in order to be considered as chaotic, must fulfil three basic conditions [2]. Its periodic orbits must be dense, it must be also topologically mixing and it must be very sensitive on initial conditions. Nevertheless, the most important of the above mentioned features is the sensitivity on initial conditions. This means, that a small variation on a system’s initial conditions will produce a totally different chaotic trajectory. This is the feature, which is contributed to the desired robot’s unpredictable path planning. Also, the study of the interaction between coupled chaotic systems was a landmark in the evolution of the chaotic synchronization’s theory [3]. Recently, a new synchronization phenomenon between two mutually coupled identical nonlinear circuits has been presented [9]. This type of synchronization, which is called inverse -lag synchronization, has been observed when a mutually coupled system is in a phase locked (periodic) state, depending on the coupling factor. The equation describing this synchronization type is: x1 .t/ D x2 .t C / ; D T=2
(1)
where x1 is the signal of the first circuit while x2 is the signal of the second circuit with a time lag , which is equal to T/2, where T is the period of the signals x1 and x2 . However, the most important observation is that the inverse -lag synchronization coexists with the complete synchronization [9]. This coexistence depends on the coupling factor and the set of system’s initial conditions. As it is
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known, in complete synchronization the interaction between two identical coupled chaotic systems leads to a perfect coincidence of their chaotic trajectories, i.e. x1 .t/ D x2 .t/; as t ! 1
(2)
3 The Proposed Chaotic TRBG In this paper, the proposed chaotic TRBG consists of three blocks (Fig. 1). The first block (S1 / includes the coupled system, which is the necessary component in this TRBG. This system is based on the 3rd order autonomous nonlinear circuit which has been used for the demonstration of the inverse -lag synchronization for the first time [9]. The bidirectional or mutual coupling, between the identical nonlinear circuits, is achieved via a linear resistor. The state equations (3) describe the normalized coupled system. In this system, the first three equations describe the first of the two coupled circuits, while the other three describe the second one. 8 dx 1 ˆ ˆ dt D y1 ˆ ˆ dy 1 ˆ ˆ D z1 C .y2 y1 / ˆ dt ˆ ˆ dz < 1 D ˛ .x1 C y1 C z1 / C b f .x1 / p.t/ dt dx2 ˆ ˆ dt D y2 ˆ ˆ dy2 ˆ ˆ D z2 C .y1 y2 / ˆ dt ˆ ˆ : dz2 D ˛ .x2 C y2 C z2 / C b f .x2 / dt
(3)
The function f(x1;2 / in system’s equations (3) is a saturation function, while is the coupling coefficient, which is present in the equations of both circuits, since the coupling between them is mutual. In this work the circuit’s parameters are: ˛ D 0:5, b D 1:0 and k D 2:0, so as each coupled circuit demonstrates chaotic behavior, and also is adjusted to be equal 2, in order to show the coexistence of the two previous mentioned synchronization phenomena. Furthermore, the term p(t) in the third equation of the system (3), is the perturbation for changing the system’s initial conditions and consequently the synchronization state of the coupling system. In details, p(t) is an external source that produces a pulse train of amplitude 0.7V having a duty cycle of 4 %. The operation of the first block (S1 / of the proposed TRBG is described in detail below: This block produces the synchronization signal [x2 (t) – x1 (t)] of the coupled system which varies between the two synchronization modes (complete synchronization and inverse -lag synchronization) depending on the system’s initial conditions. In the complete synchronization mode, the synchronization signal [x2 (t) – x1 (t)] is equal to zero, while in the inverse -lag synchronization mode, the synchronization signal [x2 (t) – x1 (t)] is not equal to zero, because the signals x1 (t) and x2 (t) are inverse with a phase difference.
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Fig. 1 The chaotic true random bits generator scheme Table 1 Results of FIPS-140–2 test, for the chaotic TRBG Monobit test
Poker test
n1 D 10068 (50.34 %)
38.896
Passed
Passed
Runs test B1 D 2588 B2 D 1308 B3 D 639 B4 D 312 B5 D 120 B6 D 128 Passed
Long run test
No
Passed
The second block (S2 / of the proposed TRBG is responsible for the quantization of the two different levels of the synchronization signal [x2 (t) – x1 (t)] into “0” and “1” according to the following equation: ¢i D
0; if x2 .t/ x1 .t/ < 1V 1; if x2 .t/ x1 .t/ > 1V
(4)
In the third block (S3 / the well-known de-skewing technique [5], which their objective is to eliminate the correlation in the output of the natural sources of random bits, is implemented. This occurs by converting the bit pair “01” into an output “0”, the bit pair “10” into an output “1”, while the pairs “11” and “00” are discarded. The “randomness” of the produced bits sequence by the proposed chaotic generator is confirmed by using one of the most important statistical test suites, the FIPS-140–2 (Federal Information Processing Standards) of the NIST (National Institute of Standards and Technology) [6]. In Table 1 the results of the use of the four most well-known tests of FIPS, such as Monobit test, Poker test, Runs test, and Long run test, for the proposed chaotic TRBG, are presented. For this reason a bits sequence of length 20000 bits, which has been obtained via a numerical integration of system’s equations (3), is used.
4 The Proposed Mobile Robot A great number of works on kinematic control of chaotic robots is based on a typical differential motion with two degrees of freedom, composed by two active, parallel and independent wheels and a third passive wheel [4, 7]. The active wheels are independently controlled on velocity and rotation sense.
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In this work the robot’s trajectory, which is independent of the robot’s motion control, is produced by a sequence of coordinates (x, y). Also, the discontinuous control law, in which the robot executes two independently actions, is adopted. The first action is a rotation motion, with constant angular velocity about its own center, to point the robot straightforward to the next target coordinate, while the second action is a straight trajectory with constant velocity toward to the target. From the simulation of the proposed robot’s kinematic motion the well-known coverage rate (C) has been calculated. The coverage rate (C) is given by the following equation: CD
M 1 X I.i/ M iD1
(5)
where, I(i) is the coverage situation for each cell, in which the terrain is divided [1]. This is defined by the following equation: I.i/ D
1; when the cell i is covered 0; when the cell i is not covered
(6)
where, i D 1; 2; : : :; M. The robot’s workplace is supposed to be a square terrain with dimensions M D 30 30 D 900 in normalized unit cells. The simulation starts from the terrain’s middle cell, (x0 , y0 / D .15; 15/. Based on the terrain’s dimensions each five-digit binary number is converted to the equivalent decimal number, which represents in turn x and y coordinate number, by ignoring the last two wasteful decimal numbers 31 and 32. With this technique a sequence of 4,000 (target points) coordinates is produced. The result for the first 1,500 of the produced target points is shown in Fig. 2a. In this figure a colour scale map of the terrain’s cells versus the times of visiting is shown. There are cells, which have been visited from 0 to 8 times in those 1,500 iterations. As a conclusion we may note the uniform distribution of the visiting times on the robot’s workplace. Also, the robot, when is moving from one target point to the next, visits other terrain’s cell, possibly many times with different direction. As a consequence only a few cells of the robot’s workplace are left uncovered. In Fig. 2b, the coverage rate versus the number of target points is shown, starting from the above mentioned, initial position. From this diagram the coverage of almost the entire terrain is confirmed. Furthermore, the curve in Fig. 2b has an exponential form (7) which is confirmed by the fitting procedure with Origin. x
y D y0 C A e t
(7)
where y0 D 98:91524, A D 98:90236 and t D 960:75846 with R-Square equal to 0.999.
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Fig. 2 (a) Colour scale map of the terrain’s cells versus the times of visiting, for 1,500 planned points and (b) the coverage rate versus the number of planned points, for the robot with the proposed path-planning chaotic generator, for 4,000 planned points
5 Conclusion In this paper, a chaotic path planning generator for autonomous mobile robots is presented. This generator is based on a controller that defines the position goal in each step by imparting chaotic motion behavior. Statistical tests of the proposed chaotic generator guarantees the “randomness” of the produced bits sequence and consequently the “randomness” of the planned positions. Furthermore, validation tests based on numerical simulations of the robot’s motion control, confirm that the proposed method can obtain very satisfactory results in regard to unpredictability and fast scanning of the robot’s workplace, in comparison with other similar works [4].
References 1. Choset, S.: Coverage for robotics – a survey of recent results. Ann. Math. Artif. Intel. 31, 113– 126 (2006) 2. Hasselblatt, B., Katok, A.: A First Course in Dynamics: with a Panorama of Recent Developments. Cambridge University Press, Cambridge (2003) 3. Luo, A.C.J.: A theory for synchronization of dynamical systems. Commun. Nonlinear Sci. Numer. Simulat. 14 1901–1951 (2009) 4. Martins-Filho, L.S., Macau, E.E.N.: Patrol mobile robots and chaotic trajectories. Math. Probl. Eng. 2007, 1 (2007) 5. Menezes, A.J., Van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography CRC, West Palm Beach (1997) 6. NIST: Security requirements for cryptographic modules. FIPS PUB 140–2. http://csrc.nist.gov/ publications/fips/fips140–2/fips1402.pdf(2001) 7. Sieqwart, R., Nourbakhsh, I.R., Introduction to Autonomous Mobile Robots. MIT, Cambridge (2004)
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8. Sooraksa, P., Klomkarn, K.: No-CPU chaotic robots: from classroom to commerce. IEEE Circ. Syst. Mag. 10, 46–53 (2010) 9. Volos, Ch.K., Kyprianidis, I.M., Stouboulos, I.N., Various synchronization phenomena in bidirectionally coupled double scroll circuits. Commun. Nonlinear Sci. Numer. Simulat. 16, 3356–3366 (2011)
