Chaotic Robots - IEEE Xplore

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“No-CPU” Chaotic Robots: From Classroom to Commerce Pitikhate Sooraksa and Kitdakorn Klomkarn

© STOCKBYTE & ROBOT ROOM CLEANER: WIKIMEDIA COMMONS

Abstract For more than a decade, chaotic signals have been adopted to generate gaits for robots via simple and effective chaotic circuits within or without CPUs. The signals are used to guide a robot to navigate chaotically. This type of robots is useful in many applications ranging from engineering education such as high school science projects to military de-mining tasks alike. Computer simulations using various choices of chaotic patterns reveal that the pattern of chaotic orbit generated by Chua’s circuit is the best in the sense of wide area coverage, high availability of design documentation and easy circuit construction. Experimental results show that the chaotic robots described in this article are very promising for transforming several tasks in such devices as vacuum cleaners, lawn mowers, de-mining vehicles, and so on. This cost-effective utilization can be simply done by embedding the chaotic circuits into the target applications without CPUs. Keywords: Chaos, robot, chaotic pattern, Chua’s circuit, gait generation. Digital Object Identifier 10.1109/MCAS.2010.935740

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IEEE CIRCUITS AND SYSTEMS MAGAZINE

Introduction haotic phenomena are fascinating to many researchers in various fields such as mathematics, physics, engineering, economics, sociology, and so on. In robotics, the first chaotic mobile robot that can navigate following a chaotic pattern was proposed by Nakamura and Sekikuchi [1, 2], where the Arnold’s equation was used to generate the desired motions. Further investigations on chaotic trajectories of the same type of robots using other equations were carried out in [3–8], among others, as introduced and discussed in the next section. An objective in exploiting chaotic signals for an autonomous mobile robot is to enlarge and to take advantage of coverage areas resulting from its traveling paths. Large coverage areas are desirable for many applications such as robots designed for floorcleaning, grass-cutting, or even de-mining in military missions, not to mention other obvious benefits

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for instance interdisciplinary education in science and engineering, even simple chaotic robots are fascinated toys for children. In [4], among many other chaotic patterns, we found by simulation that Chua’s pattern is the most interesting one due to its largest coverage areas, low cost in construction, high availability in accessing documentation for circuit design, and ease for implementation [10]. It is remarked that, due to different stimulation methodologies, different numerical results may be obtained. When calculating the coverage areas, the integration subroutine in a commercial mathematical software package may be used, where different integration methods may lead to different simulation figures, as a result of the sensitivity to initial conditions of chaos. Therefore, we encourage the readers to simulate their own systems using this article only as a guideline. It is also remarked that even simulation results can provide one with a good plan for implementation as what to expect and how to cope with the technical issues. But in real-world applications, typically simulation results neglect some unforeseen dynamical factors such as internal or external friction and other work-space restrictions. Nevertheless, implementation of a lab-scale robot is suitable as a start, which can help justify the effectiveness of the employed chaotic signals and the resulting behaviors, as illustrated in [9, 11]. According to [9], a chaotic gait pattern contributes to a successful self-untapped function, while the periodic pattern cannot. For implementation on such chaotic robots to date, most gaits are generated by a microcontroller or CPU for example see [1, 2]. With motivation for creating simple stimulus-response ability, in hardware prototype view point, a low-cost chaotic robot can construct with no CPUs. We propose a chaotic mobile robot which is directly controlled by a simple autonomous chaotic circuit. The generated chaotic motion unit is implemented by using a low frequency coupled Chua’s: LFCC circuit, as an example. To demonstrate the main idea and approach, we first depict a mobile robot model and the problems encountered in mathematical modeling and analysis. We then simulate possible candidate chaotic patterns so as to investigate possible trajectories of the robots versus different choices of chaotic signals. The selected candidate chaotic circuit will then be constructed in a lab-scale and be tested. In our case, a chaotic two-wheel mobile robot is controlled independently by two DC motors, which are driven by the LFCC circuit as described in the next section. To bridge the gap between classroom de-

Y VL

V

VR

θ

y

x X Figure 1. Kinematic scheme of the two-wheel mobile robot.

sign and commercial production, the circuit is embedded into a commercial lawn mower thereby transforming it into a real chaotic robot. A Mobile Robot Model A simple kinematic model for a chaotic robot is shown in Fig. 1. The robot consists of a trajectory generator circuit, a motor control circuit, a battery, and two DC motors. The robot’s motion subsystem has one free wheel and two fix wheels controlled by the two motors, respectively. In order to obtain the navigation path, we applied chaotic signals into each motor independently so as to control the velocity of each wheel of the robot. The kinematic scheme of the mobile robot is also shown in Fig. 1, where V [m/s] is the linear velocity of the robot, VL is the velocity of the left wheel, VR is the velocity of the right wheel, L [m] is the distance between the two wheels, u [rad] is the angle describing the orientation of the robot and v [rad/s] is the angular velocity. The motion of the robot is described by x? cos u £ y? § 5 £ sin u ? u 0

0 v 0§ c d. v 1

(1)

Adopting the mirror reflection law [2], at the boundary where the robot is turned back if it is obeyed the predesigned limit, the incident angle is equal to the reflected angle. The problem for target searching by the mobile robot using chaotic signals is depicted in Fig. 2. To search for potential candidate chaotic patterns for this task, we first examine the dynamic equation of the chaotic pattern. Among other candidates, two

Pitikhate Sooraksa and Kitdakorn Klomkarn are with the School of Computer Engineering and Information Science, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang Chalongkrung Rd., Bangkok, 10520, Thailand. E-mail: [email protected].

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. . . which pattern is a good candidate to be chosen? . . . We search for one that has the largest possible coverage area through computer simulation. and of an attractor type, as shown in Figs. 3(a) and (b), respectively. dx 5 a sin z 1 c cos y dt dy 5 b sin x 1 a cos z dt dz 5 c sin y 1 b cos x, dt

Obstacle

(2)

where the parameters are x0, y0, z0 5 3 4, 3.5, 0 4 ,

Workspace

z

Figure 2. The problem of target searching through a chaotic path.

3 2 1 0 –1 –2 –3 –3

0.5 –2

–1 x

0 0

1

2

f 1 x 2 5 m1x 1 0.5 1 m0 2 m1 2 1 0 x 1 1 0 2 0 x 2 1 0 2 ,

x0, y0, z0 5 3 0.1, 0.1, 0.1 4 ,

a, b, c 5 3 0.39, 0.025, 0.6395 4 ,

y

m0, m1 5 3 28/7, 25/7 4 .

3 –0.5 (a)

Since there are many chaotic equations, which pattern is a good candidate to be chosen? Intuitively, it depends on the application and constraints at hand. In general, we search for one that has the largest possible coverage area through computer simulation.

10 z

(3)

where

15 5 0

0

–20 y

–40

–60

–60 –80 –80 (b)

–40 x

–20

0

20

Figure 3. Typical chaotic patterns: (a) Attractor: Chua’s equation (b) Non-attractor: Arnold’s equation.

popular equations used for guiding the robot’s wheels are Arnold’s and Chua’s equations, (2) and (3), which represent chaotic behaviors of a non-attractor type 48

and dx 5 a 1 y 2 x 2 f 1 x 22 dt dy 5 b1x 2 y 1 z2 dt dz 5 2cy, dt

20

–5 –10 20

a, b, c 5 3 0.27, 0.135, 0.135 4 ,


Selection of Chaotic Candidates To help justify the coverage areas, as shown in Fig. 4, an index R is defined as the ratio of the area Au, which the trajectory passes through, to the total area AT : R5

Au . AT

(4)

Among many candidates, 12 different equations out of 25, including Chua’s, Chen’s, Lorenz’s, Rossler’s and Sprott’s systems, have been inspected and selected as good candidates [4]. It may be troublesome in selecting a particular one from a large number of available candidates. In this case, we simply focus on demonstrating the FIRST QUARTER 2010

Y (m)

Au2

Q=2

Au1 Q = 1

Q=3

Q=4

Au4

Au3 X (m)

Figure 4. A trajectory in the work-space.

methodology rather than the performing details. Searching for a set of optimal parameters for Chua’s circuit for generating the best possible pattern is a tedious and very time-consuming task. For convenience, therefore, we retain their original parameters of these chaotic systems as used in the literature. Under such canonical settings, the simulations reported in [4] reveal that Chua’s circuit actually outperforms the others in the sense of wide area coverage, high availability of design documentation and easy circuit construction. Using Chua’s circuit, which has an attractor, we compare the performance with Arnold’s system, which does not have an attractor. To carry out the simulation, we assign chaotically guiding parameters by mapping variables 5 x, y, z 6 in Chua’s system equations to angles u i, i 5 1, 2; where i 5 1 corresponds to the angle for the cosine component and i 5 2 for that of the sine in the robot equation (1), respectively. The mapping is depicted in Table 1. Comparison results for five working space environments, given in [6], are simulated in the case of 3 m 3 3 m and 5 m 3 5 m work-spaces, with average performance results summarized in Table 2. Duration for run-time for simulation in this observation was 500 sec to 1000 sec for the case of 3 m 3 3 m and was 2,000 sec to 3,000 sec for the case of 5 m 3 5 m. The reason for using different work-space sizes and runtimes is to investigate various situations. More results are reported in [6]. Runge-Kutta method is used for numerical integration and the simulation is written in C language. Since the aim of this article is to envisage the exploitation of the chosen circuits, the data used in this section should be adequate. The results show that Chua’s circuit outperforms Arnold’s system in our simulations. Implementation of Chua’s Circuit We now describe the implementation of the chaotic Chua’s circuit, which is shown in Fig. 5(a). The circuit is a simple oscillator which can exhibit a rich variety of bifurcation and chaotic behaviors. The circuit consists FIRST QUARTER 2010

Table 1. Mapping chaotic parameters to robot’s kinetic variables [6]. System

u1

u2

Arnold Chua

Z Z

Z X

Table 2. Performance index R for coverage areas. System

3m33m

5m35m

Arnold Chua

77.46% 88.72%

80.17% 87.98%

of one inductor, two capacitors, one linear resistor, and one piecewise-linear resistor. The dynamic equation of the circuit is described by [10] dvc1 1 5 1 vc2 2 vc1 2 2 f 1 vc1 2 dt R dvc2 1 C2 5 1 vc1 2 vc2 2 1 iL dt R di3 L 5 2vc2, dt C1

(5)

where iL is the current through the inductor L, vc1 and vc2 are voltages across capacitors C1 and C2, respectively, the function f 1 2 is a three-segment piecewise-linear

vc1

vc2 R

iR

Chua’s Diode

C1

iL

L

C2

(a)

iR Gb Ga –Bp

vR

Bp Gb

(b)

Figure 5. (a) Chua’s circuit; (b) Piecewise-linear nonlinear resistor.


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The objective of utilizing the chaotic robot is to enhance its target-searching behaviors. a ; RGa, b ; RGb, the general equations of Chua’s circuit become

A1 +− I1 +

Z1

Z2

Z3

V1

+ V2 Z5

− + A2

−

Zin Figure 6. An inductorless Chua’s circuit architecture using resistors and capacitors to create equivalent inductance.

nonlinear resistor having characteristics shown in Fig. 5(b) and is defined by 1 f 1 vR 2 5 GbvR 1 1 Ga 2 Gb 2 1 0 vR 1 Bp 0 2 0 vR 2 Bp 0 2 , (6) 2 where Ga and Gb are the slopes in the inner and outer regions, respectively, and Bp denotes the break points of the piecewise-linear curve. For simplicity, we use the dimensionless state equation of the circuit. After a change of variables:

x;

vc1 vc2 iLR ,y; ,z; Bp Bp Bp

a;

C2 R2C2 t ,b; ,t; C1 L RC2 Rc

Q

Rc

Q S1

S2

R

C1

C2

L Chua’s Diode

Figure 7. Two coupled low-frequency Chua’s circuits.

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dx 5 a 1 y 2 x 2 f 1 x 22 dt dy 5x2y1z dt dz 5 2by , dt

–I2

Z4

−

Chua’s Diode

(7)

(8)

where 1 (9) f 1 x 2 5 bx 1 1 a 2 b 2 3 0 x 1 1 0 2 0 x 2 1 0 4 . 2 In the simulation, we actually used a low-frequency chaotic Chua’s circuit to drive the two DC motors. The originally high-valued inductor can be realized by using an equivalent inductor [13] having input impedance Zin as shown in Fig. 6. In fact, the equivalent inductance can also be realized by using Z4 5 1/sC and Z1 5 R1, Z2 5 R2, Z3 5 R3, Z5 5 R5, and is computed by using

Leq=

R1R3 R5C . R2

(10)

Implementation of LFCC Circuit Recall that the objective of utilizing the chaotic robot is to enhance its target-searching behaviors. The circuit shown in Fig. 5 may cause the robot to travel with too-smooth trajectories, while a zigzag pattern is more preferable in order to search for a target through patrolling around the restricted area. In order to enhance the searching operations, we mutually couple two identical autonomous oscillators. In other words, two low-frequency Chua’s circuits PRBS are coupled together through a linear resistor via two electronic switches, which are controlled by pseudo-random binary sequences: the PRBS signals genR erated with a polynomial of the form X4 1 X 1 1 [14]. In Fig. 7, when the states of the L C1 C2 switches S1 and S2 are ON and OFF, respectively, the states y of two Chua’s circuits are coupled together. The state equation of the coupled circuit is given by FIRST QUARTER 2010

. . . it is a meaningful application for such chaotic robots to search mines on sites. dx1 5 a 1 y1 2 x1 2 f 1 x1 22 dt dy1 5 x1 2 y1 1 z1 1 k 1 y2 2 y1 2 dt dz1 5 2by1 dt dx2 5 a 1 y2 2 x2 2 f 1 x2 22 dt dy2 5 x2 2 y2 1 z2 1 k 1 y1 2 y2 2 dt dz2 5 2by2. dt

(11.1)

where k 5 1 R/Rc 2 . On the other hand, when the states of the switches are in the other way around, namely S2 is ON and S1 is OFF, one has a coupled Chua’s circuit which is coupled between state y and state x on the left side and the right side of Fig. 7, respectively. In this case, the state equation of the coupled circuit is given by equation (11.2). dx1 5 a 1 y1 2 x1 2 f 1 x1 22 1 k 1 x2 2 x12 dt dy1 5 x1 2 y1 1 z1 dt dz1 5 2by1 dt dx2 5 a 1 y2 2 x2 2 f 1 x2 22 1 k 1 x1 2 x2 2 dt dy2 5 x2 2 y2 1 z2 dt dz2 5 2by2 . dt

(11.2)

Testing a Lab-Scale Prototype For Chua’s circuit in this experiment, the circuit parameters in Fig. 7 are as follows: Ga 5 20.76 mS, Gb 5 20.41 mS, 6BP 5 1 V, R 5 1.8 kV, C1 5 23.5 m F, C2 5 235 m F, and L 5 45 H. From a practical viewpoint, a 45-H inductor is too large for implementation of a low-frequency operation. Thus, we use (10) to obtain an equivalent one, with R1 5 R2 5 R3 5 R4 5 1 kV, R5 5 1.8 kV, Z4 5 1/sC, and C 5 23.5 mF. Note that the coupling parameter Rc is the design parameter, which should be selected the value different than that of the linear resistor R of Chua’s circuit. In other words, in our case, the de-synchronized scenario for k is not equal to 1 would be acceptable for generating a zigzag movement of the robot. FIRST QUARTER 2010

As a trajectory generator, the LFCC circuit including PRBS with the polynomial is driven by clock signals with 0.5 Hz. To record the trajectory in the experiment, the robot has a mark-pen attached at its tail, as shown in Fig. 8, which can plot a trajectory on the work-space. A 2 m 3 2 m area on a big piece of plain white paper is divided into 400 blocks, in which each block has a dimension of 10 cm 3 10 cm, and is set up to be the work-space. In this experiment, the robot starts at a corner of the work area. During motion, when the robot is approaching a wall, its reflection path is set by exactly the same as the common law of reflection–the reflected angle is equal to the incident one. To emulate the situation by role-playing in de-mining tasks, ten red marks are randomly assigned as shown in Fig. 9, each is assumed as a buried location of a personal killing mine, M14 [16]. This type of mine in the real situation can still be found in landmines today, in the rural areas along the borders between Thailand and Cambodia, Thailand and Myanmar, Thailand and Laos, as well as Thailand and Malaysia, left out from various past wars. The explosion kills many innocent people, therefore it is a meaningful application for such chaotic robots to search mines on sites. In Fig. 9, time is recorded on the map whenever the robot finds a mine, which is indicated by the time instants when the robot finds the red marks on the ground paper. We use a sport watch to count the time instants and then insert the recorded timed into a picture for presentation after the robot found all marked locations. According to our experiment,

Figure 8. A prototype of a “No-CPU” chaotic robot.


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This will be a good task to work out toward a commercialized product in the future.

14.57 min 11.21 min 11.43 min 10.18 min

19.27 min

after 30 minutes, the robot can cover 300 blocks, which is accounted for 75% of the whole area, and is able to find all red marks. As demonstrated, the labscale chaotic robot has achieved a target searching task quite successfully, implying a great potential for building real chaotic robots for real applications in the near future.

Some Other Potential Applications Guided by the study of [15], by adopting the ideas of 6.14 min “teaching with a research spirit” and modifying the 22.31 min topic and contents of “from KLC to class D amplifier” to “from Chua’s circuit to chaotic robots,” an effective learning program in electronics and electrical en3.48 min 28.11 min gineering can be developed and run in the same manner as other similar ones reported in the literature. Besides the benefits from learning a chaotic roStart 30.05 min bot in a classroom or a simulation desk, to convey the idea and experiment from the laboratory to the Figure 9. Time stamps for successful target-searching of the chaotic robot. marketplace, we have modified a commercial lawn mower by adding a DC motor to control the front wheels as shown in Fig. 10(a), in which an LFCC circuit was also embedded at the back of the mower to generate the chaotic motion pattern shown in Fig. 10(b). The field test was carried out successfully, as demonstrated by Figs. 11(a) and (b) in the front yard of an engineering building (a) (b) in our university, the King Mongkut’s Institute of Technology Figure 10. Converting a (a) lawn mower to (b) a chaotic robot. Ladkrabang, Thailand in 2004. A satisfactory result was obtained even though there was an unexpected accident when the cutting blade of the robot hit a large hard stone covered by the grass-field causing huge damage to the blade. This situation had been unforeseen in simulations beforehand. The incident helps us gain more insight in redesigning a new version of the (a) (b) machine concerning with safe Figure 11. The field test was carried out successfully in the work space shown in (a) and (b). operation and maintenance. 52


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Experimental results show that the chaotic robots described in this article are very promising for transforming several tasks in such devices as vacuum cleaners, lawn mowers, de-mining vehicles, and so on. For new and derivative versions of LFCC circuits, combined attractors of nonlinear circuits can also be implemented [7]. In the case of complex nonlinear systems, where many nonlinear circuits coupled together, an array of these essentially coupled circuits can be compactly built using VLSI as discussed in [12]. This will be a good task to work out toward a commercialized product in the future. Conclusion In this article, the chaotic robot using LFCC circuit is presented, which can be considered as a “no-CPU” robotic architecture. Simulation results reveal that Chua’s circuit is suitable for generating navigating signals to implement the robot for target-searching tasks, due to its excellent trajectory throughout a given work-area. Experimental result confirms the effectiveness of the chaotic robot in finding randomly given targets, representing an equivalent role-play for finding locations of mines buried underground in a land field. Constructing lab-scale chaotic robots, and synthesizing and analyzing Chua’s circuit for related applications can provide great activities for high-school students, as well as for undergraduate college students in engineering, which may even lead to the development of a full course in “edutainment” in nonlinear science and chaos technology with inherently interdisciplinary nature. In addition, possible commercial applications for embedding chaotic circuits into commercial machines and devices are quite possible, as verified and demonstrated by a lawn mower. As can be foreseen, chaos technology indeed has some promising and attractive applications in industrial, military, economic and even education business in the future. Acknowledgment This work is supported in part by the Thailand Research Fund under grants RSA4680007, IUG5080026, and RGJPhD/231-2547. Pitikhate Sooraksa is currently Associate Professor of Electrical Engineering at the School of Computer Engineering and Information Science, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Ladkrabang, Bangkok, Thailand. His FIRST QUARTER 2010

research interests include IT-mechatronics, development of rapid prototypes in embedded systems and computer-aided control. He received B.Ed. (Hons), M.Sc. in Physics from Srinakharinwirot University, M.S. from George Washington University (1992) and Ph.D. from University of Houston (1996), both in Electrical Engineering. Kitdakorn Klomkarn is a Ph.D. candidate in Electrical Engineering at the King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand. His main research interests are in chaos engineering and applications. He received B.Ind and M.Eng from KMITL, both in Electrical Engineering. References [1] A. Sekiguchi and Y. Nakamura, “The chaotic mobile robot,” in Proc. IEEE/RSJ Int. Conf. Intelligent Robots and System, 1999, vol. 1, pp. 172–178. [2] Y. Nakamura and A. Sekiguchi, “Chaotic mobile robot,” IEEE Trans. Robot. Automat., vol. 17, no. 6, pp. 898–904, 2001. [3] Y. C. Bae, J. W. Kim, and Y. G. Kim, “Obstacle avoidance methods in the chaotic mobile robot with some integrated chaos equations,” Int. J. Fuzzy Logic Intell. Syst., vol. 3, pp. 729–729, 2003. [4] A. Jansri, K. Klomkarn, and P. Sooraksa, “Further investigation on trajectory of chaotic guiding signals for robotics systems,” in Proc. Int. Symp. Communications and Information Technology, 2004, pp. 1166–1170. [5] Y. C. Bae, “Target searching method in the chaotic mobile robot,” in Proc. 23rd Digital Avionics Systems Conf., 2004, vol. 2, pp. 12.D.7–12.1-9. [6] A. Jansri, K. Klomkarn, and P. Sooraksa, “On comparison of attractors for chaotic mobile robots,” in Proc. 30th Annu. Conf. IEEE Industrial Electronics Society, IECON, 2004, vol. 3, pp. 2536–2541. [7] C. Chanvech, K. Klomkarn, and P. Sooraksa, “Combined chaotic attractor mobile robots,” in Proc. SICE-ICASE Int. Joint Conf., 2006, pp. 3079–3082. [8] P. Arena, L. Fortuna, M. Frasca, G. Lo Turco, L. Patané, and R. Russo, “Perception based-navigation through weak chaos control,” in Proc. IEEE 44th Conf. Decision and Control, 2005, pp. 221–222. [9] S. Steingrube, M. Timme, F. Wörgötter and P. Manoonpong, “Selforganized adaptation of a simple neural circuit enables complex robot behaviour,” Nature Phys., in press doi:10.1038/nphys1508. [10] R. N. Madan, Chua’s Circuit: A Paradigm for Chaos. Singapore: World Scientific, 1993. [11] A. Buscarino, L. Fortuna, M. Frasca, and G. Muscato, “Chaos does help motion control,” Int. J. Bifurcat. Chaos, vol. 17, no. 10, pp. 3577–3582, 2007. [12] L. Fortuna, P. Arena, D. Balya, and A. Zarandy, “Cellular neural networks: A paradigm for nonlinear spatio-temporal processing,” IEEE Circuits Syst. Mag., vol. 1, no. 4, pp. 6–21, 2001. [13] L. A. B. Torres and L. A. Aguirre, “Inductorless Chua’s circuit,” Electron. Lett., vol. 36, pp. 1915–1916, 2000. [14] V. N. Yamorik and S. N. Demidenko, Generation and Application of Pseudorandom Sequences for Random Testing. New York: Wiley, 1988. [15] C. Trullemans, L. De Vroey, S. Sobieski, and F. Labrique, “From KCL to Class D amplifier,” Circuits Syst. Mag., vol. 9, no. 1, pp. 63–74, 2009. [16] Thailand Mine Action Center. (2010, Jan. 1). Humanitarian de-mining in Thailand [Online]. Available: www.tmac.go.th/aboutus/dm01.htm


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