An Improved Chaotic Motion Path Planner for Autonomous Mobile Robots Based on a Logistic Map

This paper presents a chaotic motion path planner based on a Logistic Map (SCLCP) for an autonomous mobile robot to cover an unknown terrain randomly, namely entirely, unpredictably and evenly. The path planner has been improved by arcsine and arccosine transformation. A motion path planner based only on the Logistic Chaotic Map (LCP) can show chaotic behaviour, which possesses the chaotic characteristics of topological transitivity and unpredictability, but lacks better evenness. Therefore, the arcsine and arccosine transformations are used to enhance the randomness of LCP. The randomness of the followed path planner, LCP, the improved path planner SCLCP and the commonly used Random Path Planner (RP) are discussed and compared under different sets of initial conditions and different iteration rounds. Simulation results confirm that a better evenness index of SCLCP can be obtained with regard to previous works.


Introduction
The subject of this research is an autonomous mobile robot that can be used as patrol robot or robot for cleaning in closed areas. This scanning does not demand a planning trajectory of the autonomous mobile robot in this schema that knows the workspace and obstacles. It does, however, require the robot path to cover every part of the workspace. One robot with these characteristics is a chaotic autonomous mobile robot. The main characteristics of the chaotic systems are the topological transitivity and the sensitive dependence on initial conditions [1]. Due to topological transitivity, the chaotic autonomous mobile robot is guaranteed to scan the whole connected workspace. The sensitive dependence on initial conditions is a desirable characteristic for patrol robots, since the trajectory of the chaotic autonomous mobile robot is very unpredictable [2][3][4]. The interaction between autonomous mobile robotics and chaos theory has been studied only recently [5][6][7]. The key problem of this research is how to impart the chaotic motion behaviour on to the autonomous mobile robot. Some related work has been done. For instance, the integration between the robot motion system and a chaotic system, the Arnold dynamical system, is used to impart chaotic behaviour to a robot in [8]. The same principle of systems integration is adopted to generate the random motion for complete coverage path planning of the mobile robot in [3]. In [9], an open-loop control approach is proposed to produce an unpredictable trajectory so that the state variables of the Lorenz chaotic system are used to command the velocities of the robot's wheels. Other known chaotic systems, such as a Standard or Taylor-Chirikov map and a Chua circuit [10] have been used.
However, most of the above research has regarded chaotic systems as totally default random ones. This ignorance can bring about uneven patrolling terrain in some circumstances. In this paper, we propose an improved chaotic path planner SCLCP for an autonomous mobile robot with a chaotic change of the angle. We suggest that the robot's angle should be changed using rule of Logistic chaotic map and the chaotic equation has been improved to obtain good randomness characteristics to ensure that the chaotic trajectory cover the entire areas of patrolling more evenly. In order to reveal the superiority of the chaotic path planner, the common used random path planner RP, is also discussed for contrast. Simulation results illustrate the usefulness of the proposed path planner.
The paper is organized as follows. Section 2 presents a chaotic path planner LCP and its uniformity test based on the logistic map. Another commonly used random path planner RP is discussed in Section 3. Section 4 assumes arcsine and arccosine transformations to obtain an improved path planner SCLCP based on LCP. 2  test of numerical analysis is assumed in Section 5 to test the evenness of path planner LCP, RP and SCLCP. Section 6 provides numerical simulations of SCLCP. Conclusions are drawn in Section 7.

Chaotic Path Planner LCP with Logistic Map
In this paper an autonomous mobile robot moves with a constant velocity and is steered by the variables of a logistic map. By interval mapping, the angle of the robot has the same chaotic variation characteristics as the logistic map.

Discussion of the Logistic bifurcation map
A logistic map is the deterministic, discrete-time dynamical system of an iterative map. The equation can be described in the form of: Where [0,4]   , k 0,1,...,n  is the discrete time and 0 1 n x ,x ,...,x are the states of the system at different instants in time. Starting from the initial state 0 x , the repeated iteration of Equation (1) gives rise to a fully deterministic series of states known as an orbit. It is used as a model for a chaos generator. The relative simplicity of the Logistic map makes it an excellent point of entry into a consideration of the concept of chaos. Figure 1 describes the orbit of the Logistic bifurcation map when the parameter  changes among the range [0,-4] while the initial condition 0 x remains constant. Figure 1  Though the initial condition 0 x is different, the orbits all show similar chaos characteristics when parameter  changes among the range (3.57-4]. In fact, all values of 0 x among (0-1) have these properties. Especially Figure 1 shows that when  converges to 4, the k {x } orbits become increasingly chaotic and demonstrate a pseudorandom and uniform distribution among the range (0-1) [11]. Slight variations in 0 x result in dramatically different orbits, an important characteristic of chaos. So we use the Logistic equation (2) as a random number generator.

Integrated system (LCP) of the Logistic map and the robot
The autonomous mobile robot considered in this work is a typical differential motion with two degrees of freedom, composed by two active, parallel and independent wheels, and a third passive wheel with an exclusive equilibrium function [12]. The active wheels are independently controlled on velocity and sense of turning. The sensors provide short-range distances to obstacles. The resultant motion is described in terms of linear velocity v and direction  . These equations constitute a nonholonomic dynamical system. The robot kinematics model is described by (3), where x and y identify the robot position and  is the orientation.
x v cos The discrete form of the Equation (3) is: Where h is a sufficiently small step.
The variation range scope of the robot orientation angle . While n x is in the logistic map located in (0-1). In order to integrate the logistic map into the path planner of autonomous mobile robot, we define the following transformation: n n n 1 n n Then the discrete model of the robot is: n 1 n n n 1 n n n n n 1 n n x x hv cos y y hv sin Now, the angle n  of the robot is calculated iteratively depending on equation (6). So the robot path planner designed as equation (6) possesses the chaotic characteristics. Here we defined the planner as LCP (Logistic Control Planner).

Uniformity Test of LCP
Equation (6) assumes that each autonomous mobile robot works in a smooth state space without boundaries. However, real autonomous mobile robots move in space with boundaries like the walls of the surfaces of obstacles. To avoid obstacles, we can adopt the mirror mapping approach as shown in [7]. Here, we do not think about obstacle influence on the path planner.
Because n  and n x own the same chaotic characteristics and random distribution characteristics according to Equation (5), here we only discuss the random distribution characteristics of n x for convenience and then map it to n  space.
The terrain covering can be judged through a performance index. This index can be defined using terrain division on square unit cells and computing the visited cells percentage after the robot location planning [12][13][14]. Here the uniformity test of LCP is carried through by the evenness computation among each equal interval. In this section we obtain the chaotic state variables n x of LCP and measure the probability density of n x under different initial conditions and different iteration rounds. n x sequences are divided into ten same length sub-intervals between 0-1, namely 0-0.1，0.1-0.2，… and 0.9-1 respectively and the numbers k(m) (m 1,..., 10)  that lie within each sub-interval are counted and displayed. A good pseudorandom sequence should have uniformity of unit intervals.
Four sequences are generated with different sets of initial conditions and different iteration rounds. Each element of the sequences can be changed to the robot position according to Equation (3). Their uniformity distribution and the numbers k(m) are described in Figure 2   Another commonly used method for scanning unknown workspace with barriers without a planning trajectory to use an autonomous mobile robot with a random change of the angle. Here we call this random path planner a RP for an autonomous mobile robot. We need to discuss the uniform distribution of the sequence generated by RP compared with LCP to reveal the superiority of the chaotic path planner. The simulation environment and the constant velocity of the robot are the same as those of the previous work.
A Linear Congruential Generator (LCG) represents one of the oldest and best-known pseudorandom number generator algorithms. LCG is fast and requires minimal memory (typically 32 or 64 bits) to retain state. The generator is defined by the recurrence relation: Where n x is the sequence of the pseudorandom values and a, c and m are integer constants that specify the generator. The LCG defined above has a full period if and only if the parameters c, m, and a have been a suitable chosen value. Here we use LCG as the RP sequence generator. In order to compare the uniform distribution of a standard random signal of RP with the mentioned method LCP, the same analysis method is assumed.  Figure 7 show that the distribution feature has a close relationship with the iteration rounds of the random function. The greater the rounds of iteration, the better the performance of the uniform distribution behaves. This shows that the RP can meet some needs of certain application when the iteration rounds are large enough.

Improved Path Planner SCLCP
Chaotic systems have the property of being deterministic in microscopic space and behave randomly, when observed in a coarse-grained state-space. The sensitivity of chaotic maps to initial conditions makes them optimum candidates to relate minimal critical information about the input, in the output sequence. Their iterative nature makes them quickly computable and able to produce binary sequences with extremely long cycle lengths. Therefore, the chaotic scan is stochastically superior to the scan by randomness to yield a favourable nature for a patrol robot if given the best adjusting parameters.
From Section 3 we know that the sequences distribution of LCP demonstrates characteristics with two edges that are thick and a thin middle. Therefore, the optimization goal of LCP is to be able to average the distribution of the two edges. Arcsine and arccosine transformations have been assumed for n x sequence generated by LCP and their contribution to the transformation are discussed respectively. First only the arcsine transformation effect is tested: The distribution characteristics are shown in Figure 8. Figure 8 shows that the arcsine transformation smoothes the bigger edge distribution.
In a similar way, the arccosine transformation is taken:  Figure 9 shows that the transformation smoothes the smaller edge distribution. So if the arcsine and arccosine transformation are combined, the whole distribution characteristics of LCP could be enhanced. We call this improved LCP path planner SCLCP. Figure 10 shows the transformation effect. The chaotic random attributes have been changed obviously and the amount k(m) of each region is uniform and has no significant change.

Numerical Analysis
We have discussed the n x sequence's random characteristics of LCP, RP and SCLCP through simulation and graphics qualitatively. They demonstrate the difference between characteristics significantly seen from the results. The improved SCLCP produces better evenness than other path planner. Now we want to further discuss their diversity quantitatively, mainly by frequency tests. The methods test whether the diversity between the real frequency and the theoretical one that the random sequences distribute in the subinterval is significant or not. The two common frequency test methods are the K-S (Kolmogorov-Smirnov) test and 2  test. Here the 2  test is assumed [15].
The main point of the 2  test is to divide n x sequences into k same length subintervals between 0 and 1 and examine the diversity between the real frequency and the theoretical one of the random sequence's distribution in the subinterval. Define the real frequency of the ith subinterval random data as i n and the corresponding theoretical frequency of this subinterval is defined as i m . Here i m N / k  , where N is the total random sequences number. Statistic value 2  is constructed: When N is big enough, the 2