International Journal of Advanced Robotic Systems
ARTICLE
Applications of Chaotic Dynamics in Robotics Review Paper
Xizhe Zang1*, Sajid Iqbal1, Yanhe Zhu1, Xinyu Liu1 and Jie Zhao1 1 Harbin Institute of Technology, Harbin, Heilongjiang, China *Corresponding author(s) E-mail:
[email protected] Received 26 September 2014; Accepted 02 March 2016 DOI: 10.5772/62796 © 2016 Author(s). Licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract This article presents a summary of applications of chaos and fractals in robotics. Firstly, basic concepts of determin‐ istic chaos and fractals are discussed. Then, fundamental tools of chaos theory used for identifying and quantifying chaotic dynamics will be shared. Principal applications of chaos and fractal structures in robotics research, such as chaotic mobile robots, chaotic behaviour exhibited by mobile robots interacting with the environment, chaotic optimization algorithms, chaotic dynamics in bipedal locomotion and fractal mechanisms in modular robots will be presented. A brief survey is reported and an analysis of the reviewed publications is also presented. Keywords Bifurcation, Legged Locomotion, Chaos, Chaotic Mobile Robot, Fractal, Modular Robot, Optimiza‐ tion Methods, Swarm Intelligence
1. Introduction Deterministic chaos has been employed for developing consumer electronic products and intelligent industrial systems. Aihara described chaos engineering as broad enquiries on the technical applications of chaos [1-3]. Figure. 1(a) shows the first electrical appliance in the world developed using chaos. Figure. 1(b) illustrates that the
comfort provided by the appropriate chaos control is more than that offered by the standard control techniques. A dishwasher with a two-link nozzle represents another application of chaos theory to home appliances, Figure. 2 [4]. A key property of chaos is that simple dynamical systems can often engender complex dynamics. These systems can be implemented using simple analogue hardware [5]. Chaos and fractals (self-similar, iterated, and fine struc‐ tures having fractal dimension) are intrinsic features and behaviours in nature. During the last decade, these con‐ cepts have been transformed into practical applications. The time is ripe for reviewing the application of chaos and fractals in robotics. The robotics community is trying to emulate these natural behaviours by investigating human‐ oids, bio-robots and biologically inspired systems such as swarms. These systems confront complex problems such as noise-sensing, vibrations and robot-environment interac‐ tions leading to chaos. Similarly, during natural calamities, search and rescue robots have to navigate in a highly irregular and erratic environment in order to handle a motion planning problem. Such nonlinearities have led researchers to employ chaotic motion planning techniques for mobile robots to ensure a rapid search of the whole workspace [6]. The applications of chaos in robotics are classified into two types: Int J Adv Robot Syst, 2016, 13:60 | doi: 10.5772/62796
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Section 5 presents the discussion and the co presented in section 6.
(a)
Figure 2. Mechanism of dishwasher based on chaos [4]
Figure. 2 Mechanism of dishwasher based on chaos
(b) Figure. 1Figure Thermal vs. fan temperature swings [3].vs. 1. (a) Asensation chaotic kerosene heater (b) Thermal sensation temperature swings [3]
Chaos and fractals (self-similar, iterated and • Chaos analysis and
ordered arrangement: it is not disorder in the usual sense. The sensitive dependence implies that arbitrary close initial 2. Chaos Theory: An Overview conditions follow trajectories that move away from each other after a certain time, as shown in Figure. 3. The chaotic attractor produced Lorenz in century, phase spacethree resembles a revolution Duringbythe 20th great butterfly [10]. The butterfly effect has become the emblem relativity, quantum mechanics and chaos. Lik of chaos theory (see Figure. 4). It is the widespread name fine two revolutions, chaos annihilated the i for the sensitive dependence on initial conditions.
Newtonian Physics [7]. Chaos theory—a dynamical systems theory or theory of Chaos analysis implies the observation of chaotic behav‐ oscillations—is the study of unstable aperiodi iour in robots, whereas chaos synthesis entails the genera‐ in deterministic dynamical systems, which sho tion of artificial chaos to make different robots accomplish specific tasks [2]. In order to explore the applications of dependence on initial conditions. Ho chaos in robotics with reference to both types, this paper is deterministic law governs their nonlinear be organized as follows: section 2 provides an overview of 9]. Chaos lies within a well-ordered arrangem chaos theory. Section 3 presents chaos analysis in robotics, including (1) initial chaos research in robotic arms, (2) the Figure 3. Two trajectories that start other but diverge within disorder in close theto each usual sense. Thea sensitive d few tens of seconds [11] chaotic dynamics exhibited by mobile robots interacting implies that arbitrary close initial conditi with the environment and (3) the chaotic behaviour of Three important propertiesthat of deterministic chaosfrom are trajectories move away each ot passive dynamic bipeds. sensitive dependence on initial conditions, topological certain time, as shown in Figure. 3. The chao Chaos synthesis in robotics is presented in section 4 as the mixing (transivity) and dense periodic orbits [8]. It is application of chaotic systems for motion planning of produced by Lorenz in phase unpredictable due to sensitive dependence on space initial resembles autonomous mobile robots, the escape from local minima conditions. [10]. Due toThe topological mixing, the system cannot the embl butterfly effect has become optimization problems, and the fractal structures for decomposed into two subsystems. Furthermore, in the puzzle.in Such nonlinearities have led researchers beto theory (see Figure. 4). It is the widespread n developing versatile modular robots. Section 5 presents the middle of random behaviour it owns an element of employdiscussion chaoticand motion planning techniques for mobile the conclusion is presented in section 6. sensitive on initial conditions. regularity, i.e., unstabledependence periodic orbits (UPOs). Given the robots to ensure a rapid search of the whole workspace limited accuracy of measurement of initial conditions, a 2. Chaos Theory: An Overview [6]. long-lasting prediction of trajectory of initial points is difficult [12]. The chaotic trajectory appears to be random During the 20th revolutions The applications ofcentury, chaosthree in great robotics are occurred: classified into since it does not reveal any periodic pattern. relativity, quantum mechanics and chaos. Like the other two types: two revolutions, chaos annihilated the ideology of Newto‐ Due to determinism, chaos is predictable for the short time nian Chaos analysis and Physics [7]. Chaos theory—also called dynamical horizon but unpredictable in the long run due to sensitive systems Chaos synthesis. theory or theory of nonlinear oscillations—is the dependence on initial conditions. Deterministic chaos is of unstable aperiodic behaviour in deterministic characterized by high sensitive dependence on initial Chaos study analysis implies the observation of chaotic Figure. 3 Two trajectories that start close to each oth dynamical systems, which show sensitive dependence on inability to predict future consequences, the behaviour in robots, whereas chaos synthesis entails conditions, the diverge a few tens and of seconds [11]. initial conditions. However, a deterministic law governs fractal dimension, thewithin Lyapunov exponent so on. Chaos generation of artificial chaos to make different robots has been found to arise in a multitude of dynamical systems, their nonlinear behaviour [8, 9]. Chaos lies within a wellaccomplish specific tasks [2]. In order to explore the Three important properties of deterministic 2 Int J Adv Robot Syst, 2016, 13:60 | doi: 10.5772/62796 applications of chaos in robotics with reference to both sensitive dependence on initial conditions, types, this paper is organized as follows: section 2 mixing (transivity) and dense periodic orbi provides an overview of chaos theory. Section 3 presents unpredictable due to sensitive dependence • Chaos synthesis.
from astronomy to zoology [8, 13-15]. This phenomenon and its closely related cousin fractal have been presented as a new paradigm for understanding our world.
Figure 4. Lorenz (strange) attractor
Over the last few decades, the terms nonlinear dynamics and chaos have become known to most scientists and engineers. Nonlinearities occur in feedback processes, in the systems containing interacting subsystems and in the systems interacting with the environment. This scenario is quantitatively and qualitatively distinct from the situations in which perturbations develop linearly. It is a striking reality that simple devices, e.g., a double pendulum, and a very complex event such as weather follow the same dynamics, which can be predicted only for short time horizons [6]. Due to the availability of high-speed comput‐ ers, new analytical techniques and sophisticated experi‐ ments, it has become evident that the chaotic phenomenon is universal in nature and has across-the-board consequen‐ ces in various fields of human endeavour. 3. Chaos Analysis in Robotics 3.1 Initial chaotic dynamics research in various manipulators and robots Chaos research in robotic systems is the study of nonlinear ordinary differential equations (ODEs) that model the
system. In robotics, chaotic dynamics research is not new. Vakakis et al. investigated the nonlinear dynamics of a hopping robot, as shown in Figure. 5, which depends on passive bouncing oscillations for engendering motion [16, 17]. The leg of the hopping robot was modelled as a nonlinear spring. McCloskey and Burdick advanced this hopping robot to a two degrees of freedom (DOF) system by including forward running motion. The period-dou‐ bling cascade appeared in a bifurcation diagram (plot of dynamic variable versus the bifurcation parameter) [18]. Lankalapalli and Ghosal demonstrated that nonlinear ODEs, which describe the motion of a feedback-controlled two-revolute robot (undertaking repetitive motions), could display chaos. They presented plots of bifurcation dia‐ grams and used Lyapunov exponents for testing chaos, Figure. 6 [19, 20]. Buhler and Koditschek examined the planar juggling robot. Its task involved robot-environment interactions, which produced nonlinearities [21, 22]. Mahout et al. proved that a two-revolute joint manipulator controlled with proportional-plus-derivative (PD) law exhibited chaotic dynamics for certain values of static variables [23] (see Figure. 7). Ravishankar and Ghosal probed chaos in feedback-controlled two- and three-DOF robots. Nonlinear ODEs, which depicted the dynamics of a feedback-controlled rigid robot, demonstrated chaos for a certain range of parameters [24]. These analyses signify that chaos is intrinsic to robot dynamics. Wiener’s polynomial chaos (PC) presented a framework for separating stochastic elements of a system response from deterministic components. “Polynomial Chaos Theory” is defined as a non-sampling based technique to establish development of uncertainty in a dynamical system, when the system parameters have probabilistic uncertainty [25]. It can be employed for statistical analysis of dynamical systems since it allows probabilistic description of the uncertainty effects. Due to low computational cost, it is an efficient alternative to Monte-Carlo simulations [26]. Figure. 8 showed that PC accurately generated the shortterm probability density function (PDF) and indicated the stability of the system response.
Figure 5. a) Simplified hopping robot model and b) the return map bifurcation diagram; non-dimensional variable w and non-dimensional parameters, λ and β when β=0 [16]
Xizhe Zang, Sajid Iqbal, Yanhe Zhu, Xinyu Liu and Jie Zhao: Applications of Chaotic Dynamics in Robotics
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Figure 6. a) A 2R planar rigid robot and b) the bifurcation diagram for θ1 vs Kv (PD controller; Kp = 52) [20]
Figure 7. a) Time evolution of the state variable x3 and b) the Poincare's map for x1 and x3 [23]
Figure 8. Comparison of polynomial chaos and Monte Carlo methods [26]
The design of a commercially produced mechanical system is influenced by the manufacturing variation that affects its performance. Suitable analysis tools are required to simulate and predict the potential dynamics generated by this variation. For conducting such uncertain dynamic and static analyses for a robot manipulator as shown in Figure. 9, PC can be employed as a unifying framework [27, 28]. 4
Int J Adv Robot Syst, 2016, 13:60 | doi: 10.5772/62796
The trajectory control of redundant robotic arms is an important area of research that envisions efficient optimi‐ zation algorithms. A robotic arm is termed as a kinemati‐ cally “Redundant Manipulator”, if it possesses higher DOF than required to establish any orientation and position of the end effector. In redundant manipulators, inverse kinematics present infinite solutions so they can be config‐
ured optimally for an assigned task. Varghese et al. demonstrated that a redundant robot controlled by a feedback linearization technique could display quasiperiodicity and chaos [29]. The standard technique (e.g., closed-loop pseudoinverse control, Figure. 10) suggested for solving their kinematics resulted in chaotic joint motions with erratic arm configurations. The dimension of their dynamic response was found to be fractal [30-32].
Nehmzow and Walker argued that, instead of just relying on trial-and–error methods, REI should incorporate chaos theory, which will permit researchers to describe a robot’s behaviour quantitatively. A mobile agent interacting with its environment executes a complex operation that is controlled by sensors, actuators and the environment. Due to the noise generated by sensors and actuators this interaction is not always predictable. The behaviour of a robot is nonlinear because its dynamics are the compound result of its environment, the robot and the control pro‐ gramme (task) executed by the agent, as shown in Figure. 11. The robot’s trajectory can only be predicted for short time periods [37, 38].
Figure 9. Polynomial chaos used for measuring performance of a SCARA robot manipulator [28] Figure 11. REI can be viewed as an analogue computer, taking environmen‐ tal, morphological and task-related data as input, and “computing” behaviour as output [39]
Figure 10. Block diagram of the closed-loop inverse kinematics algorithm with the pseudoinverse [30-32]
3.2 Chaotic behaviour of Robot-environment interaction (REI) Mobile robotics, like other sciences, should progress from stand-alone existence-proofs (i.e., unauthenticated investi‐ gational results) to a research culture of objective duplica‐ tion and confirmation of experimental results. The grand challenge is to make it a more precise science. For advanc‐ ing the field of mobile robotics, a quantitative method for the analysis of REI should be developed. A mobile robot is an embodied-situated agent; its behaviour does not depend on the programming alone, but evolves from the interac‐ tion between the robot, the task and the environment. The studies in [33, 34] are of valuable information on this domain of robotics.
Nehmzow and Walker recreated the phase spaces defining robots’ behaviours and probed them using the tools of chaos theory. The trajectory of a robot embodies important features of its behaviour. They conducted experiments with various mobile robots implementing obstacle-avoidance and wall-following behaviours; see Figure. 12 and Figure. 13. They reconstructed their attractors from the time series and computed the Lyapunov exponents and correlation (fractal) dimensions of the attractors underpinning the behaviours of the agents (see Table 1) [39, 40].
Figure 12. a) The Pioneer II mobile robot b) Billiard ball (obstacle avoidance)
Schoner et al. were the forerunners who underscored the behaviour of the pioneer robot [39, 40] notion 12. of a unifying dynamical Figure a) The theoretical Pioneer language—the II mobile robot b) Billiard ball (obstacle avoidance) behaviour of the pione systems theory—for designing the autonomous robot Lyapunov exponent and fractal dimension are two main architectures. In [35], they discussed the key ideas, e.g., quantifiers for chaos. Sensitive dependence on initial Lyapunov exponent and fractal dimensionconditions are two main quantifiers for chaos. Sensitive phase space, fixed point, attractor, etc., in a tutorial form. is the most distinctive property of a chaotic also deliberated the useproperty of quantitative system. Two trajectories in phase space near each other is Smithers the most distinctive ofperform‐ a chaotic system. Two trajectories intophase space near ance measures for the behaviours of embedded-situated diverge as time increases. The Lyapunov exponent com‐ increases. The Lyapunov divergence and Lyapunov a positive Lyapunov e agents [36]. He claimed “...Behaviour exponent is not a property computes of an putes this this divergence and a positive exponent agent, it is a dynamical process constituted of the interactions chaos. the correlation of an Similarly, the correlation dimension of an represents attractor isSimilarly, a measure of itsdimension periodicity in phas between an agent and its environment.” attractor is a measure of its periodicity in phase space.
a zero correlation dimension and chaotic attractors possess higher dimensions. These qu Xizhe Zang, Sajid Iqbal, Yanhe Zhu, Xinyu Liu and Jie Zhao: 5 REI exhibits deterministic chaos. Applications of Chaotic Dynamics in Robotics
Behaviour
Environment
Set
Embedding lag h
Embedding dimension p
λ¯ (bit / s )
¯ CD
Wall following
Square arena
140601
40
4
0.1
1.4
200601
3
5
0.2
1.9
30
5
0.2
20
5
0.2
(no obstacles) Billiard ball
Square arena (no obstacles)
240601 Billiard ball
Square arena
010701
1.9 1.6
(centre barrier) 040702 Billiard ball
Square arena
050701
1.7 ≈2.4
(centre barrier) Table 1. Summary of experimental results [39]
Robot
Environment Maze
Square
Dynamic
SraightrCorridor
A
Figure 13. “Billiard ball” behaviour in square arena—entire trajectory (left) and 150 data points (right) [39, 40]
Periodic attractors have a zero correlation dimension and chaotic attractors possess higher dimensions. These quantitative measures prove that REI exhibits deterministic chaos. Nehmzow et al. analysed REI quantitatively. They estab‐ lished vital rules for grasping the interaction among the mobile agent, its task and its environment. Such analysis will allow replication and verification of the results. Like living beings, the sensory information perceived by autonomous robots manifests dynamical behaviour since these data are the outcome of REI. Odagiri et al. investigat‐ ed the question of whether the internal structure of the agent governs the dynamics of the sensory information or whether the evolved environment determines it. They collected the sensory data from a miniature mobile robot, Khepera, during uninhibited navigation and studied the dynamical behaviour of autonomous agents by analysing this information [41, 42]. Figure. 14(a) explains the experi‐ mental arrangement. In four different settings shown in Figure. 14(b), three dissimilar robots were used and the sensory data were collected. The sensory information obtained from the agents had random elements. The power spectrum for the time series had a continuous component, the maximum Lyapunov exponent was positive, the dimension was fractal (Table 2), the auto-correlation function converged to zero at the infinite time and points in Poincare maps were confined within a certain fixed space. These quantitative measures confirmed that deterministic chaos was demonstrated by the sensory data of the mobile robots. These experiments also revealed that the sensory information flow from agents depended clearly on the inner configuration of the robots. 6
Int J Adv Robot Syst, 2016, 13:60 | doi: 10.5772/62796
B
C
Min
2.35
2.40
0.81
5.54
Avg
2.41
2.49
0.87
5.70
Max
2.52
2.60
0.95
5.85
Min
2.90
2.96
0.90
5.87
Avg
3.01
3.07
0.94
5.96
Max
3.14
3.20
1.03
6.07
Min
1.86
1.88
1.10
5.23
Avg
1.92
1.95
1.16
5.29
Max
2.03
2.05
1.23
5.35
The results were averaged over 15 independent runs. Table 2. Correlation dimensions for three robots in four different environments [42]
Rold used chaotic dynamics on autonomous agents controlled by neural networks. He demonstrated that chaotic invariants are dependable estimators for quantify‐ ing agent behaviour [43]. 3.3 Chaotic behaviour in bipedal locomotion McGeer pioneered passive dynamic walking (PDW). He experimented with various PDW mechanisms such as the one shown in Figure. 15. This unpowered biped robot walks by dynamics. Its gait is generated automatically by gravity and inertia. Its two major advantages are high efficiency and human-like walking [44-47]. Human walking has become an important biometric identifier [48-50]. The study in [51] provides details on passive bipedal robots. During the last two decades, robotics engineers, biome‐ chanists and chaologists have experimented with many different passive biped models and have uncovered various kinds of chaotic behaviours: period-doubling cascade, intermittency, quasi-periodicity and crises. Some researchers have also investigated chaos control techniques in passive walking. This area certainly calls for further enquiry, particularly chaos control of passive walking. A
four different settings shown in Figure. 14(b), three dissimilar robots were used and the sensory data were
Fig
collected.
Du bio ma un do cri con cal pa of be
Figure 14. 14 a) Experimental setup b) Tested environments i)setup maze ii) squareb) iii) dynamic iv) straightenvironments corridor [42] Figure. a) Experimental Tested i) maze ii) square iii)review dynamic straight [42].chaos is being employed for motion —the deterministic comprehensive on the statusiv) of chaos research in corridor
passive dynamic walking has already been reported in [52].
planning of autonomous mobile robots in a totally un‐ known environment. The prime benefit of the proposed chaotic motion planning lies in the deterministic nature of chaotic dynamics. The motion controller for generating chaotic trajectories is designed using a chaotic dynamical system.
The sensory information obtained from the agents had random elements. The power spectrum for the time series
Nakamura and Sekiguchi proposed the first chaotic mobile robot that can navigate following a chaotic pattern. As shown in Figure. 17, they proposed the design of a control‐ ler that produces chaotic motion using the Arnold equation in a mobile robot [53, 54]. This schema does not need any trajectory planning or workspace mapping. Due to topo‐ logical transivity the chaotic mobile robot searches the entire workspace and the sensitivity to initial conditions makes the robot exceedingly unpredictable. Random walk is an alternative method for scanning a workspace without mapping, but the proposed scheme is better than random walk.
Figure 15. McGeer’s passive dynamic biped—Dynamite [44]
4. Chaos Synthesis in Robotics 4.1 Chaotic mobile robot Motion planning or path planning of mobile robots explores an approximate non-collision path consistent with a certain performance objective. This subject has attracted much attention in recent years in robotics. Without map‐ ping, path planning is a difficult task for mobile robots. Chaotic path planning can be a solution to this predicament
Jansri et al. experimentally proved that the method of combining chaotic attractors is effective for motion plan‐ ning of mobile robots [55-58]. They experimented with robot trajectories by employing various types of chaotic patterns; for example, Chua’s, Lorenz’s and Rossler’s attractors. The goal of using these chaotic signals is to enlarge the coverage area of robots. Figure. 17 shows a proposed chaotic robot that is directly controlled by a simple autonomous chaotic circuit. Computer simulations confirmed that Chua’s circuit produced the largest possible coverage area [58]. Typically, the dynamic variables of chaotic attractors are used to drive the wheels of differential-drive robots independent from each other. Hackbarth transformed chaotic trajectories to mission space of autonomous agents [59]. The main advantage of using chaotic systems such as Lorenz motion is depicted in Figure. 18, which clearly reveals that a robot searches the working space using deterministic chaos more quickly than when employing random motion. Xizhe Zang, Sajid Iqbal, Yanhe Zhu, Xinyu Liu and Jie Zhao: Applications of Chaotic Dynamics in Robotics
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nematics of a two-wheeled mobile robot b) Prototype mobile robot c) Chaotic patrol [53, 54]
perimentally proved that the method of combining chaotic attractors is effective for motion p [55-58]. They experimented with robot trajectories by employing various types of chaotic pat a’s, Lorenz’s and Rossler’s attractors. The goal of using these chaotic signals is to enlarge the ure. 17 shows a proposed chaotic robot that is directly controlled by a simple autonomous cha ulations confirmed that Chua’s circuit produced the largest possible coverage area [58]. Figure 16. a) Kinematics of a two-wheeled mobile robot b) Prototype mobile robot c) Chaotic patrol [53, 54]
s of a two-wheeled mobile robot b) Prototype mobile robot c) Chaotic patrol [53, 54]
ntally proved that the method of combining chaotic attractors is effective for m ]. They experimented with robot trajectories by employing various types of ch renz’s and Rossler’s attractors. The goal of using these chaotic signals is to enla shows a proposed chaotic robot that is directly controlled by a simple autonom ns confirmed that Chua’s circuit produced the largest possible coverage area [ Figure 17. a) Target searching through a chaotic path b) A prototype of chaotic mobile robot [58]
get searching through a chaotic path b) A prototype of chaotic mobile robot [58] Similarly, using Lorenz, Hamilton and hyper-chaos equations, Youngchul et al. suggested obstacle avoidance and target searching behaviours for a mobile agent. Figure. 19 presents trajectories of an obstacle avoidance search. Simulation results verified that this technique generated excellent chaotic trajectories [60-62].
the dynamical variables of the Lorenz system were em‐ ployed to generate the actuation commands for the minirobot, Khepera. This method resulted in an extreme chaotic motion and the fast scanning of the whole robot workspace. Similarly, they implemented motion trajectories for a mobile agent, as illustrated in Figure. 20, using the Stand‐ ard map—a dynamical system [63].
dynamic variables of chaotic attractors are used to drive the wheels of differential-drive robo rom each other. Hackbarth transformed chaotic trajectories to mission space of autonomous a antage of using chaotic systems such as Lorenz motion is depicted in Figure. 18, which clearly es the working space using deterministic chaos more quickly than when employing random
rching through a chaotic path b) A prototype of chaotic mobile robot [58]
mic variables of chaotic attractors are used to drive the wheels of differential-dr ch other. Hackbarth transformed chaoticFigure trajectories to using mission space of auton Lorenz system [60] 19. Trajectory of mobile robot rajectory length for random movements motion [59] Likewise, Volos et implemented a chaotic path planning 18, whic of using chaotic systems suchand as Lorenz Lorenz motion isal.depicted in Figure. generator for a mobile robot to cover an entire workspace a swift andquickly erratic manner.than Figure. 21 shows a Khepera, working space using deterministic chaosinYoungchul more when employing ng Lorenz, Hamilton and hyper-chaos equations, et al. suggested obstacle avoidan which is popular in the robotics community, investigating
Figure The trajectoryagent. length for random movements andpresents Lorenz motion behaviour-basedof control [64, 65]. Threeavoidance different chaoticsearch. Si aviours for a 18.mobile Figure. 19 trajectories an obstacle [59] systems were used in the chaotic generator, which pro‐ d that thisIntechnique generated excellent chaotic duced trajectories [60-62]. a double-scroll chaotic attractor as shown in Figure. reference [63], an open-loop control law was proposed
by Martins-Filho et al. to produce erratic motions such that
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Int J Adv Robot Syst, 2016, 13:60 | doi: 10.5772/62796
21(b). Table 3 highlights the comparison among the coverage rates for these systems.
ing Standard maps respectively with 1,200 points [63].
Volos et al. also presented a motion control strategy for mobile and humanoid robots using a chaotic truly random bit generator, as shown in Figure. 22 [66, 67]. Using this generator, an autonomous robot was implemented on an experimental platform, the “Magician Chassis”, as shown in Figure. 23. This technique ensured highly unpredictable robot trajectories, which appear random from an observer’s viewpoint [68-70]. Numerical simulations confirmed the efficiency of this strategy and statistical tests also ensured the randomness of the planned motions. Caihong et al. suggested a chaotic path planner based on a
emented a chaotic path planning generator for a mobile robot to cover an entire workspace in logistic map. This simple deterministic system behaved Figure. 21 shows a Khepera, which is popular in the roboticsrandomly community, investigating and traversed large workspace coverage [71]. They also presented a fusion strategy based on the Stand‐ 4, 65]. Three different chaotic systems were used in the chaotic generator, which produced a ard map for developing a chaotic path planner for mobile ctor as shown in Figure. 21(b). Table 3 highlights the comparison among the coverage rates for Figure 20. Terrain covering using Standard maps respectively with 1,200 points [63].
Figure 21. a) The mobile robot Khepera b) Experimental double-scroll
Khepera b) Experimental chaotic attractor [64]double-scroll chaotic attractor [64]
robots [72]. Curiac and Volosencu devised an improved chaotic path planning technique for mobile robots for accomplishing boundary surveillance missions [73].
This research underpins the design and testing of a chaotic controller that integrates a known chaotic equation. While performing surveillance and search tasks, chaotic trajecto‐ ries for autonomous robots possess remarkable improve‐ ment over other methods. The proposed applications of chaotic mobile robots include security patrol, cleaning and firefighting. These efforts indicate that the application of chaotic behaviour of dynamical systems for motion planning of mobile robots is a fascinating interdisciplinary research domain. 4.2 Chaos integration in optimization algorithms
The global optimization methods with a meta-heuristic or stochastic optimization character are called “evolutionary computing techniques”, such as genetic algorithm (GA), swarm intelligence (SI), ant colony optimization (ACO), Figure 22. The chaotic truly random bits generator [66] artificial bee colony algorithm (ABC), particle swarm optimization (PSO) and so on. In nature, swarms solve their problems by group intelligence; best known examples are Coverage Rate erent systems [64, 65] System colonies of social insects such as ants, bees and termites. SI Chua 11.5% systems typicallytruly consistrandom of a population a motion control strategy for mobile and humanoid robots using a chaotic bit of simple agents Lorenz 23.0% interacting locally with one another and with their envi‐ ure. 22 [66, 67]. Using this generator, an autonomous robot was implemented on an ronment. Swarm robotics is the application of SI principles VKS 40.0% “Magician Chassis”, as shown in Figure. 23. This technique ensured to robots. highly unpredictable Table 3. Coverage rate of different systems [64, 65]
nderpins the design and testing of a chaotic controller that integrates a known chaotic eq veillance and search tasks, chaotic trajectories for autonomous robots possess remarkable hods. The proposed applications of chaotic mobile robots include security patrol, cleaning ese efforts indicate that the application of chaotic behaviour of dynamical systems for mo s a fascinating interdisciplinary research domain.
ppear random from an observer’s viewpoint [68-70]. Numerical simulations confirmed the nd statistical tests also ensured the randomness of the planned motions.
Figure 23. a) Autonomous robot using the “Magician Chassis” platform b) Terrain coverage of the robot during obstacle avoidance behaviour [69]
onomous robot using the “Magician Chassis” platform b) Terrain coverage of the robot during obsta
ndom bits generator [66]
Xizhe Zang, Sajid Iqbal, Yanhe Zhu, Xinyu Liu and Jie Zhao: Applications of Chaotic Dynamics in Robotics
ntegration in optimization algorithms chaotic path planner based on a logistic map. This simple deterministic system behaved
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4.2.1 Chaotic Genetic Algorithm (CGA)
4.2.2 Chaotic Artificial Bee Colony (CABC) Algorithm
A GA is a general purpose search heuristic, which is inspired by genetics and Darwinian selection process. This meta-heuristic is used to generate solutions for optimiza‐ tion and search problems. Many important theoretical and practical problems are concerned with the selection of the optimal configuration or parameter set to achieve a certain objective. Common algorithms rarely solve such problems [74]. Since their inception, they have received a lot of attention because of their potential as optimization algo‐ rithms for complex real world problems. The most widely used technique in engineering design optimization (EDO) is GA.
Many aspects of the collective activities of social animals such as bird flocking, ant foraging and fish schooling are self-organizing, due to which the complex group behaviour evolves from the interactions of individual members, which behave in a regular fashion. The ABC algorithm simulates the foraging behaviour of honey bees. This metaheuristic algorithm has a balanced searching and manipu‐ lation capacity.
Since dynamical systems containing feedback often produce chaos, it is speculated that chaos is also a compo‐ nent of natural evolution. Determan and Foster incorpo‐ rated a simple chaotic system—the logistic equation—into a standard GA. The results confirmed that a thoughtful use of chaos in simulated evolution is advantageous [75]. GAs and chaos are techniques that are motivated by nature. Yang and Chen also presented a new strategy of chaotic mutation by combining chaos and GAs [76]. In order to improve the performance of GAs, various chaotic systems have been employed to define new operators, which were applied in these optimization algorithms. Figure. 24 compares performance by using various random and chaotic generators on the Travelling Salesman Problem (TSP) [77]. It is clearly evident from the results that chaotic dynamical systems yield the best performance.
Lin and Huang suggested a novel strategy based on the ABC algorithm hybridized with chaos [80]. This method combined the population-based searching ability of ABC with the searching behaviour of deterministic chaos. Similarly, taking into account various constraints in a complex combat field environment, the path planning of an Uninhabited Combat Air Vehicle (UCAV) is an intricate global optimum puzzle [81]. Xu et al. implemented the ABC algorithm for a UCAV; see Figure. 25. This method has drawn the attention of researchers because of its flexibility, versatility and robustness in deciphering optimization problems. However, it has discrepancies, such as the tendency to converge prematurely and the need of large iterations for the global optimal solution. For averting these problems and expediting the searching process of optimal parameters, the irregularity and ergodicity of the chaotic variable is employed. Figure. 25(b) compares the standard ABC and CABC algorithms, which verifies the efficiency of the new chaos synthesis technique. For a quadruped robot, Lao and Duan suggested a chaotic ABC approach for solving the push recovery problem. The experimental results also verified the efficacy of the proposed algorithm [82]. 4.2.3 Chaotic Optimization Algorithm (COA)
Figure 24. Comparison of performance for various random and chaotic generators [77]
Jia and Wang proposed a chaotic genetic algorithm (CGA) based on the distance-propagation algorithm with chaotic control and prediction that can efficiently predict robot motion. In a dynamic environment, this strategy resolved the path planning problem for mobile robots [78]. Gao et al. also proposed a CGA for motion planning [79]. In compar‐ ison with standard GAs, CGA is simple, effective and speedy. 10
Int J Adv Robot Syst, 2016, 13:60 | doi: 10.5772/62796
During the last two decades, scientists have taken a great interest in developing optimization techniques based on chaotic search routines, e.g., chaotic neural network, chaotic simulated annealing and chaotic search. These chaos-based techniques are more efficient than random search and can escape local minima. Bing and Weisun put forward a chaotic optimization algorithm (COA), which uses ergodicity, regularity and stochastic properties of chaos [83]. COA can solve complicated optimization problems and is faster than the aforementioned algorithms. Hongyan et al. proposed an improved COA for motion planning of mobile robots and the simulation proved that this method was fast and accurate [84]. 4.3 Fractal structures in modular robots The robotics community has keenly researched reconfig‐ urable and scalable distributed robots because they are efficient in unstructured and dynamically changing environments. While maintaining robustness, such modu‐ lar robots must be hyper-redundant so that they can adjust
of optimal parameters, the irregularity and ergodicity of the chaotic variable ndard ABC and CABC algorithms, which verifies the efficiency of the new c
Figure 26. An example of maintenance work in a storage tank by a modular robot [86] Figure 25. a) Typical UCAV battlefield model b) The evolution curves of two algorithms [81]
Shahinpoor fabricated deployable fractal mechanisms as smart structures. These devices were multi-axis, multi-finger manipulators with fractal kinematic construction [88]. Chirikjian et al. developed modular ro to any medium. Such robots haveprototypes been implemented using Shahinpoor fractal mechanisms as with self-reconfiguration ability fabricated [89, 90]. Thedeployable mechatronic modules of these so-called metamo fractal mechanisms [85]. Fukuda shown et al. developed a hetero‐ smart structures. These devices were multi-axis, multi-arm in Figure. 27. Figure. 27(b) illustrates the locomotion process. geneous modular robotic system, Figure. 26 [86]. A Selfand multi-finger manipulators with fractal kinematic Reconfigurable Robot (SRR) is a modular system with construction [88]. Chirikjian et al. developed modular dynamic reconfiguration ability. SRRs have been inspired robotic prototypes with self-reconfiguration ability [89, 90]. by multi-cellular organisms, with the idea that a huge The mechatronic modules of these so-called metamorphic number of organisms can be created from a limited number robots are shown in Figure. 27. Figure. 27(b) illustrates the of cell types. Versatility, robustness and low cost are the locomotion process. main advantages of modular robots [87]. SRRs are built Murata et al. developed symmetric mechanical modules from robotic modules, which themselves are all-inclusive named “fractum” for investigating SRRs [91-93]. Figure. robots, with on-board sources, sensors, actuators and 28(a) exhibits a 2D fractum that, as in Chirikjian et al., communication capabilities. Figure 27. a) The locomotion process of one module around another b) Various applications of Metamorphic robot [89 utilizes electromagnetism for connection. Figure. 28(b) shows the schematic details of “fractum” the fracta,for which can be SRRs [91-93]. Murata et al. developed symmetric mechanical modules named investigating connected to assume different shapes as shown in Figure. exhibits a 2D fractum that, as in Chirikjian et al., utilizes electromagnetism for connection. Figure. 28(b) s 28(c) [93]. Having both to functional and component redun‐ schematic details of the fracta, which can be connected assume different shapes as shown in Figure. 28 dancies, this identical modular robotic system is capable of both functional and component redundancies, this identical modular robotic system is capable of self-rep self-repair and self-assembly. assembly.
AV battlefield model b) The evolution curves of two algorithms [81]
t, Lao and Duan suggested a chaotic ABC approach for solving the push reco lso verified the efficacy of the proposed algorithm [82].
imization Algorithm (COA)
cades, scientists have taken a great interest in developing optimization techn s, e.g., chaotic neural network, chaotic simulated annealing and chaotic searc fficient than random search and can escape local minima. Bing and Weisun p m (COA), which uses ergodicity, regularity and stochastic properties of chaos ion problems and is faster than the aforementioned algorithms. Hongyan et otion planning of mobile robots and the simulation proved that this method w Figure 26. An example of maintenance work in a storage tank by a modular robot [86]
Shahinpoor fabricated deployable fractal mechanisms as smart structures. These devices were multi-axis, multi-arm and Figure 28. a) 2D b)2D Schematics ofunit fractum c)modular Basic Fracta reconfiguration procedure [93] Mechanical “Fractum” b) Schematics of fractum c) Basic Figure 28. a) multi-finger manipulators withwork fractal kinematic construction [88]. Chirikjian et al. developed robotic Figure 26. An example of maintenance in a storage tankMechanical by a modularunit “Fractum” Fracta reconfiguration [93] robot [86] prototypes with self-reconfiguration ability [89, 90]. The mechatronic modules of procedure these so-called metamorphic robots are Rus et al. developed two types of 3D self-reconfigurable mechanisms and analysed a three-dimensional s shown in Figure. 27. Figure. 27(b) illustrates the locomotion process. algorithm (see Figure. 29) [94, 95]. robotstwo cantypes adjustoftheir and functionality in respons RusThese et al.fractal developed 3D shape self-reconfigurable environments and tasks. Such robots can be used information is not availableshapebeforehand about a t mechanisms and when analysed a three-dimensional robustness is the prime requirement [96,algorithm 97]. Modular robotics, self-repairing forming (seefractal Figure. 29) [94,having 95]. These fractal capabilities, a popular research theme. robots can adjust their shape and functionality in response to dynamic environments and tasks. Such robots can be used when information is not available beforehand about a task and where robustness is the prime requirement [96, 97]. Modular fractal robotics, having self-repairing capa‐ bilities, are currently a popular research theme.
ures in modular robots
ty has keenly researched reconfigurable and scalable distributed robots beca ynamically changing environments. While maintaining robustness, such mod hat they can adjust to any medium. Such robots have been implemented usin eloped a heterogeneous modular robotic system, Figure. 26 [86]. A Self-Reco stem with dynamic reconfiguration ability. have structures; been inspired by multi FractalsSRRs are scale-invariant the part resembles Figure 27. a) The locomotion process of one module around another b) Various applications of Metamorphic robot [89, 90] the whole. Such identical components are being used in Murata etof al. developed symmetric mechanical named “fractum” [91-93]. Figure. uge number organisms can bemodules created from for a investigating limitedSRRs number of28(a) cell types. Xizhe Zang, Sajid Iqbal, Yanhe28(b) Zhu, Xinyu Liuthe and Jie Zhao: 11 exhibits a 2D fractum that, as in Chirikjian et al., utilizes electromagnetism for connection. Figure. shows Figure 29. a) The robotic Molecule [94] b) The physical prototype for theofCrystalline Atom [95] Applications Chaotic Dynamics in Robotics main advantages ofthemodular SRRs are built from robotic modules, schematic details of fracta, which canrobots be connected[87]. to assume different shapes as shown in Figure. 28(c) [93]. Having Fractals are scale-invariant structures; the part resembles the whole. Such identical components are used both functional and component redundancies, this identical modular robotic system is capable of self-repair and selfrobots. Scientists have been developing distributed robotic systems consisting of identical m ith on-board actuators and homogenous communication capabilities. assembly. sources, sensors, Figure 27. a) The locomotion process of one module around another b) Various applications of Metamorphic robot [89, 90]
These modular robots are capable of self-assembly and self-repair. The dream of modular robotics has ad
chanical unit “Fractum” b) Schematics of fractum c) Basic Fracta reconfiguration procedure [93]
modular robots. Scientists have been developing homoge‐
chaos synthesis (chaotification) in robotics has a share of 44
distributed roboticmechanisms systems consisting of identical papers (57.1%) of the application bibliography. Chaotifica‐ ed two types ofnous 3D self-reconfigurable and analysed a three-dimensional shape-forming mechanical units. Thesecan modular are capable of selftion is the practice of creating chaotic dynamics in a ure. 29) [94, 95]. These fractal robots adjust robots their shape and functionality in response to dynamic assembly and self-repair. The dreamisofnot modular robotics non-chaotic system. These statistics show that d tasks. Such robots can be used when information available beforehandformerly about a task and where has advanced from proof-of-concept machines to sophisti‐ capabilities, researchers are actively apursuing chaos research in robot‐ prime requirement [96, 97]. Modular fractal robotics, having self-repairing are currently cated systems. ics. In chaos analysis in the robotics category, about 20.78% theme. (16 papers) represent initial chaotic dynamics research in the various manipulators and robots. Around 14.29% (11 papers) are related to the second group—chaotic behaviour of robot-environment interaction (REI). The papers on chaotic behaviour in bipedal locomotion represent 7.79% (six). In chaos synthesis in robotics type, papers on chaotic mobile robot are 27.27% (21) of the application bibliogra‐ phy. Around 15.58% (12 papers) represent chaos integra‐ tion in optimization algorithms. The papers on fractal structures in modular robots also constitute about 14.29% Figure 29. a) The robotic Molecule [94] b) The physical prototype for the botic Molecule [94] b) The physical Crystalline Atom [95]prototype for the Crystalline Atom [95] (11) of the application bibliography. These data offers a eye view. For example, we have only hinted at chaos invariant structures; the part resembles the whole. Such identical componentsbird’s are used in modular PDW in units. this article, as it has already been have been developing homogenous distributed robotic systems consisting of research identical in mechanical 5. Discussion exhaustively reviewed bots are capable of self-assembly and self-repair. The dream of modular robotics has advanced fromin [52]. This paper offers a detailed introduction of the concept, machines to sophisticated systems. Firstly, this paper introduces the concept of chaos, which function and applications of chaos and, thus, it boosts the leads to a full understanding of chaos. The theory of chaos research and application of chaos theory to real robots. An divulges our inability to make long-term predictions about introduction to the domain of chaotic and fractal dynamics deterministic dynamical systems. Chaotic dynamics can be in robotics been presented, with a set ofofrepre‐ a detailed introduction of has the concept, functionalong and applications chaos and,explained, thus, it boosts the categorized and measured using this theory. sentative applications. These are mere samples of the open ication of chaos theory to real robots. An introduction to the domain of chaotic and fractal dynamics in scientists have found similar Chaos is universal because frontier that will have enormous impact on the future of nonlinear dynamics in dissimilar dynamical systems such robotic systems. as double pendulums and simple electronics circuits [98-101]. The growth in chaos research is highlighted by the During the last three decades, hundreds of papers have interdisciplinary nature of the field. Due to several appli‐ described thriving applications of deterministic chaos in cations in electrical appliances such as fan heaters, dish‐ robotics. This article presents representative papers that washing machines and air-conditioners, the application of cover applications of chaotic dynamics in robotics at the deterministic chaos has attracted much attention. Deter‐ time of writing. Figure 30 shows the papers on chaos ministic chaos used to be perceived as unpredictable and research in robotics on a lustrum (five years) footing. The unstable and hence worthless. In the last few decades its noteworthy fact about the rate of growth of chaos-applica‐ utility and application have been recognized. Like other tive publications in robotics (see Figure 30) is that the domains of science and technology, chaotic dynamics have number of publications reporting chaos applications has been discovered and implemented in various robotics grown nearly consistently for the last two decades. domains.
Figure 30. Chaos research in robotics papers on lustrum (five years) basis
An analysis of the reviewed publications is presented, as the aim of this work is to provide a state-of-the-art vision of what has been done in the domain of chaos research in robotics. Out of 77 reviewed papers, 33 papers (42.9%) are related to the topic of chaos analysis in robotics, whereas 12
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Next, some endeavours for uncovering the chaotic behav‐ iour of various kinds of robots were presented. This part provides basic knowledge about the common methods and processes involved in finding the evidence for the existence of chaos in robotic motion, which can help in better application of chaos to real robots. Redundant manipula‐ tors and feedback-controlled robots display chaotic and fractal dynamics and mobile robots display chaotic dynam‐ ics when they interact with the environment. Polynomial Chaos can be employed for analysing robotic arms and chaos quantifiers such as the Lyapunov exponent and correlation dimension can be employed for analysing chaotic dynamics in REI. Such quantitative analysis will also provide the basis for independent replication and verification of experimental results and will make robotics research a more precise science. Finally, a number of chaos-based control methods and algorithms were presented, which introduce the typical
application of chaos in mobile robot path planning and in improving the efficiency of optimization algorithms. Unlike other path planning methods, chaotic path planning does not require a map of the workspace and it is more efficient than the random walk algorithm. With various chaos equations, a robot could exhibit a range of motion paths. The chaotic controllers can be implemented by embedding simple chaotic circuits into the robots. Chaotic trajectories are generated using state variables of dynami‐ cal systems, which are used as input for the wheels of differential-drive robots. Chaos-based control techniques can also be employed to improve the existing optimization algorithms. The researchers have combined chaotic dynamics with major swarm-based algorithms such as ACO, ABC and PSO. By incorporating chaos, Liu et al. proposed a hybrid PSO algorithm, chaotic PSO [102]; it is a synergy of chaotic searching behaviour and the parallel population–based evolutionary behaviour of PSO. Similar‐ ly, Gong and Wang proposed a chaos ACO algorithm [103]. Bucolo et al. investigated the advantages of integrating chaos with an ACO algorithm. These chaos-based optimi‐ zation techniques are better than standard methods in terms of efficiency, searching quality and robustness. How to chaotificate a non-chaotic system by integrating the appropriate chaos equation is the key point in chaos-based control problems. In recent years, the unearthing of chaos has generated much interest amongst investigators. Deterministic chaos leads to quantitative analysis, which is the gist of science. Despite the many efforts to find evidence for chaotic dynamics in robotics, useful applications of chaos in robotics have rarely been studied. Consequently, chaos has a modest influence on most of the robotics community. We attempted to fill this gap. For instance, the association between chaos theory and the real world is a time-series analysis in terms of nonlinear dynamics. We performed nonlinear time-series analysis of normal and pathological human gaits [45] and a passive compass-gait biped gait [104]. Two chaotic invariants, Lyapunov exponent and correlation dimension, were obtained. These quantifiers demonstrated that walking is a nonlinear process. It was established that the Lyapunov exponent and the correlation dimension could predict the walking stability of a passive walking robot. By comparing the numerical simulation with the results of the nonlinear time-series method, it was found that the walking stability of the passive biped would improve gradually with a reduction in the correlation dimension values. Therefore, the parameters of the robot could be optimized by calculating the relative exponent values under various parameters. 6. Conclusion In this paper major applications of chaos theory in robotics have been reviewed. Scientists are using chaos theory for understanding the basic principles that govern the interac‐ tion among the robot, the task and the environment.
Besides the Lyapunov exponent and fractal dimension, other quantifiers should be employed for analysis, which will broaden our understanding of robot dynamics. Until now, researchers have found several routes to chaos in the gait patterns of very simple passive dynamic bipeds. The analyses of complex biped models, which are adequately closer to the behaviour of real biological systems, will provide deep insight into the origin of chaotic dynamics and bifurcation scenarios. Swarm intelligence (SI) has emerged as an interdisciplinary research area for scientists and engineers. Like Chaotic ABC, Chaotic PSO and Chaos ACO should be employed in motion planning of autono‐ mous agents. The applications of chaotic dynamics will generate efficient motion planning techniques for mobile robots. A brief survey is reported and an analysis of the reviewed publications is also presented. The final aim of robotics is the creation of intelligent autonomous robots. The dynamical system theory is the right answer for a dynamic world. From this review of papers, it is evident that deterministic chaos is an over‐ whelming idea in science and an omnipresent phenomenon in various robotic domains. Physiological systems are inspiring the control systems and the physical shapes of the robots. Scientists and engineers are striving to realize the decades-old dream of a versatile, mobile, general-purpose autonomous robot. Chaos theory, combined with other important technologies such as artificial intelligence, machine learning and nonlinear optimal control, will help realize this goal in the offing. 7. Acknowledgements This work was supported by the National Magnetic Confinement Fusion Science Program “Multi-Purpose Remote Handling System with Large-Scale Heavy Load Arm” (2012GB102004). I have greatly benefited from discussions with Muhammad Wasif, Ghulam Abbas, Muhammad Salman Riaz, and Dr. Shahid Rasheed. I express my gratitude to all. It is a short review paper so I apologize to those researchers whose contributions have not been cited. It would be difficult to pin-point all written sources that have inspired this paper. Hence, adopting Seneca's saying "Whatever is well said by anyone belongs to me" [105], I thank all the authors in the references and the anonymous referees for valuable criticism. 8. References [1] R. Katayama, Y. Kajitani, K. Kuwata, and Y. Nishida, "Developing tools and methods for applications incorporating neuro, fuzzy and chaos technology," Computers & Industrial Engineering, vol. 24, pp. 579-592, 1993. [2] K. Aihara and R. Katayama, "Chaos engineering in Japan," Communications of the ACM, vol. 38, pp. 103-107, 1995. Xizhe Zang, Sajid Iqbal, Yanhe Zhu, Xinyu Liu and Jie Zhao: Applications of Chaotic Dynamics in Robotics
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Xizhe Zang, Sajid Iqbal, Yanhe Zhu, Xinyu Liu and Jie Zhao: Applications of Chaotic Dynamics in Robotics
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