Complete Coverage Random Path Planning Based on a Novel Fractal-Fractional-Order Multi-Scroll Chaotic System

Abstract

With the increasing demands for autonomy and coverage efficiency in tasks such as security patrol and post-disaster exploration using mobile robots, achieving random, efficient, and complete coverage path planning has become a critical challenge. Traditional chaotic path planning methods, while capable of generating unpredictable trajectories, still have limitations in terms of randomness strength, traversal uniformity, and convergence coverage. To address this, this study proposes a complete-coverage random path planning method based on a novel four-dimensional fractal-fractional multi-scroll chaotic system. The main contributions of this research are as follows: First, by introducing additional state variables and fractal-fractional operators into the classical Chen system, a fractal-fractional chaotic system with a multi-scroll attractor structure is constructed. The output of this system is then mapped into robot angular velocity commands to achieve area coverage in unknown environments. Key findings include: the novel chaotic system possesses two positive Lyapunov exponents; Spectral Entropy (SE) and Complexity (CO) analyses indicate that when parameter B is fixed and the fractional order α increases, the dynamic complexity of the system significantly rises; in a 50 × 50 grid environment, the robot driven by this system achieved a coverage rate of 98.88% within 10,000 iterations, outperforming methods based on Lorenz, Chua systems, and random walks; ablation experiments further demonstrate that the combined effects of the fractal order β, fractional order α, and multi-scroll nonlinear terms are key to enhancing system complexity and coverage performance. The significance of this study lies in that it not only provides new ideas for constructing complex chaotic systems but also offers a reliable theoretical foundation and practical solution for mobile robots to perform efficient, random, and high-coverage autonomous inspection tasks in unknown regions.

1. Introduction

With the rapid advancement of artificial intelligence technology, autonomous robots are being increasingly deployed across industrial, agricultural, and service sectors. In scenarios such as forest fire prevention, patrol surveillance, and disaster relief operations, robots are required to efficiently and comprehensively cover all accessible points within a designated area [1]. The Complete Coverage Path Planning (CCPP) problem pertains to operational contexts where mobile devices must traverse all points in a target area in an efficient manner. As a subfield of industrial motion planning, resolving the CCPP problem in an optimal way carries broad significance [2].

Traditional CCPP methods mainly rely on explicit rules or mathematical models to generate repeatable and predictable paths. Conventional algorithms include boustrophedon coverage and inward-spiral coverage [3], along with their variants. Traditional CCPP methods provide simplicity, low memory requirements, and fast response times. However, due to the limited global environmental information utilized by this algorithm, its adaptability to complex environments is suboptimal. The traditional methods utilize limited global environmental information, which often leads to a high repetition rate in the generated full-coverage paths. Bio-inspired intelligent CCPP algorithms, motivated by natural collective behaviors, include genetic algorithms (GA) [4], ant colony optimization (ACO) [5], particle swarm optimization (PSO) [6], and their variants. The methods offer flexible search processes. However, these algorithms often yield paths with excessive turns, redundant points, and inefficiencies. The path planning method based on the Biologically Inspired Hybrid Optimization Algorithm (AB-WOA) integrates multiple biomimetic mechanisms such as Ant Colony Optimization, Whale Optimization Algorithm, and Artificial Potential Field. The AB-WOA method demonstrates strong obstacle avoidance capability and path smoothness in complex environments. However, the AB-WOA method still faces practical challenges, including difficulties in parameter tuning and high computational complexity [7]. In summary, balancing convergence efficiency and real-time performance remains a significant challenge in current research.

Chaotic systems are deterministic nonlinear dynamical systems whose behavior is characterized by unpredictability, boundedness, recurrence, and high sensitivity to initial conditions [8], which render them suitable for random CCPP, that is, CCRPP. To the best of our knowledge, Nakamura et al. first introduced a chaotic path planning algorithm for mobile robots [9]. Zhao et al. proposed the GSGWO algorithm, which enhances convergence stability through a golden sine strategy and mitigates the issue of chaotic trajectories caused by traditional GWO’s tendency to fall into local optima in obstacle-avoidance path planning [10]. Kvitko et al. utilized a neuron-based chaotic mapping to drive the Pseudorandom Bit Generator (PRBG) for path planning, preserving stochasticity while suppressing disordered wandering through parameter P and a memory mechanism, thereby avoiding trajectory chaos caused by Lorenz open-loop instability [11]. Li et al. used a Chebyshev map for random coverage path planning, achieving good ergodicity and uniformity, though with suboptimal randomness [12]. Paths generated using the Taylor-Chirikov map exhibited low coverage and high repetition rates [13]. Common solutions to this issue include the incorporation of a sinusoidal transformation [14] or a random number generator [15] to enhance the randomness of the chaotic map and improve distribution uniformity. Some researchers have endeavored to constructmulti-attractor chaotic systems aimed at generating paths characterized by strong randomness, high coverage, and good uniformity [16]. Sridharan et al. designed online search algorithms for unknown terrains using chaotic systems and maps with the objective of enhancing time efficiency while maintaining randomness [17]. These approaches involved an excessive number of parameters and lacked generalizability. Nevertheless, constructing chaotic attractors that exhibit both strong randomness and high uniformity remains a promising solution for effective random path planning. Multi-scroll hyperchaotic systems exhibit greater sensitivity and unpredictability [18]. Additionally, fractal-fractional descriptions integrate both fractal and fractional-order characteristics, revealing complex phenomena unattainable with single fractional operators. The integration of fractal and fractional-order characteristics has garnered significant research interest and led to applications in medicine [19], engineering [20], finance [21], and other fields.

Fractional-order chaotic systems have attracted widespread attention due to their ability to model memory effects and generate more complex dynamical behaviors than integer-order systems. These systems exhibit higher unpredictability and richer dynamic characteristics, making them highly promising tools for chaotic path planning. Feng et al. proposed the 2D-VFCQHM hyperchaotic map, which overcomes the limitations of traditional integer-order and fixed fractional-order chaotic maps by introducing a state-adaptive variable fractional-order structure, thereby achieving a broader chaotic parameter range, higher entropy complexity, and stronger sequence randomness [22]. Almatroud et al. proposed a variable-order fractional discrete memristive Duffing map, which simulates a short-memory effect by employing piecewise constant orders, thereby overcoming the computational redundancy associated with traditional long-memory fractional-order systems. This approach induces rich, coexisting hidden attractors across multiple sub-intervals, offering a more flexible and complex chaotic source for dynamics-based encryption applications [23]. Jiang et al. proposed a fractional-order Hopfield neural network based on a parametric deformed exponential ReLU memristor. By introducing the memristor into the classical HNN to enhance nonlinear coupling and further incorporating fractional calculus theory, the model achieves the coexistence of multiple dynamical behaviors ranging from periodic to hyperchaotic states, thereby providing a highly complex chaotic sequence source [24]. Biolek extended higher-order elements to the fractional-order domain and generalized duality transformations, constructing a tunable chaotic circuit system using signal flow graphs. This framework advances chaotic systems toward fractional-order, memory-aware, and dual-realizable designs, offering a unified approach for complex dynamic circuit synthesis [25]. Agrawal et al. constructed a fractional-order financial chaotic system based on Caputo derivatives, which overcomes the limitations of traditional integer-order models in describing memory effects and nonlinear dynamics of financial systems by introducing Legendre wavelets and the Adam-Bashforth method, thereby achieving more accurate market behavior simulation and stability analysis [26]. Zhang et al. constructed a class of fractional-order spiking neural networks based on Caputo derivatives, which overcomes the challenges of discontinuous pulse activation functions by introducing Filippov solutions and phase space partitioning, thereby proving the existence of multiple stable equilibrium points under fractional orders less than 1 and enhancing the robustness and reliability of neural networks for applications such as pattern recognition [27]. Yu et al. proposed a non-polynomial memristive Hopfield neural network capable of generating controllable multiscroll attractors using a memristor that satisfies the Lipschitz condition, with the objective of decoupling the number of scrolls from circuit resource consumption while preserving mathematical smoothness [28]. Furthermore, multi-attractor chaotic systems have also been investigated to enhance the randomness and ergodic uniformity of paths. By generating scrolls, such systems can produce trajectories with greater ergodicity, facilitating full coverage tasks. However, existing multi-scroll systems often suffer from complex structures, excessive parameters, or limited generalizability. To mitigate these limitations, this paper proposes a novel four-dimensional chaotic system based on the Chen system by integrating fractal-fractional calculus with a multi-scroll structure, thereby combining the advantages of both. This system not only exhibits higher complexity and entropy but also achieves better coverage in mobile robot path planning.

Building upon research on fractional-order chaotic systems, this paper aims to advance the field of mobile robot path planning. The core innovation lies in proposing a novel four-dimensional fractal-fractional multi-scroll chaotic system, whose notable feature is the dual influence of the fractal dimension and fractional order. This achievement is realized by introducing fractal-fractional operators and a multi-scroll nonlinear term into the classical Chen system. Based on the framework of reference [29], we construct a fractal-fractional multi-scroll chaotic system incorporating the Atangana–Baleanu memory kernel. The final system exhibits unique characteristics, including higher Lyapunov exponents, a multi-scroll structure, and the coexistence of multiple chaotic attractors, thereby providing greater flexibility for random coverage tasks in path planning. The main contributions of this work are summarized as follows:

(1)

The construction of a higher-dimensional hyperchaotic system by adding state variables based on Chen’s chaotic system.

(2)

The development of a fractal-fractional chaotic system featuring a multi-scroll structure is achieved by introducing the definition of fractal-fractional order and incorporating a non-linear component into the established higher-dimensional chaotic system.

(3)

The developed fractal-fractional chaotic system is applied to drive the mobile robot for area coverage, enhancing its randomness and coverage capability.

As depicted in Figure 1, the article is structured as follows: firstly, the procedure for constructing a new chaotic system is explained; secondly, the development of a chaotic robot path planner is presented; finally, simulation and ablation studies are conducted. The paper is structured as follows: Section 2 introduces the definition and solution methods of fractal-fractional chaotic systems; Section 3 focuses on the dynamic characteristics of a novel fractal-fractional chaotic system, providing in-depth discussion through theoretical derivation and numerical analysis; Section 4 applies this system to CCRPP; and Section 5 provides a summary of the paper.

2. Materials and Methods

This section will introduce the theoretical foundation, system construction methodology, path planning integration scheme, and simulation setup adopted in this paper. The simulation experiments are conducted using MATLAB 2021b, providing the necessary technical framework and parameter configuration for the subsequent algorithm implementation and performance validation.

2.1. Relevant Theoretical Foundations

Compared to classical integer-order chaotic systems, the core advantage of fractional-order chaotic systems lies in the introduction of fractional calculus, which comprehensively surpasses integer-order systems in terms of flexibility, complexity, and modeling accuracy. Additionally, the core advantage of fractal-fractional chaotic systems lies in their dual-parameter characteristic, which combines the fractional-order calculus with the fractal dimensions. The combination of fractional-order calculus and fractal dimensions enables fractal-fractional chaotic systems to comprehensively surpass single fractional-order chaotic systems in both theoretical complexity and practical application performance.

The discretization method based on the Atangana-Baleanu fractal-fractional definition as referenced in [29] is employed herein. Initially, a dynamical equation is formulated utilizing the Atangana-Baleanu fractal-fractional derivative. Subsequently, the Atangana-Baleanu integral is employed to establish a discretized solution method. A piecewise approximation strategy is implemented for the integral term within the equation, with Lagrange piecewise interpolation utilized for approximation values within each adjacent integration interval. The approximated integral is then solved to ultimately derive a fully discretized iterative computational formula. The specific approach is outlined as follows.

Definition 1 [30]. 

Assuming that $f\left(t\right)$ is differentiable on interval ($a,b$) and that $f\left(t\right)$ is a fractal differential with $\beta$. The fractal-fractional derivative of order $\alpha$ in the Caputo sense with the Mittag–Leffler kernel is given as

$${}_a{}^{FF}D_t^{\alpha ,\beta }\left(f\left(t\right)\right)={\displaystyle \frac{AB\left(\alpha \right)}{1-\alpha }}{\int }_0^t{\displaystyle \frac{df\left(\delta \right)}{d{\delta }^{\beta }}}{E}_{\alpha }(-{\displaystyle \frac{\alpha }{1-\alpha }}{(t-\delta )}^{\alpha })d\delta ,$$

(1)

where ${}_a{}^{FF}D_t^{\alpha ,\beta }$ is fractal-fractional derivative operator; $0<\alpha ,\beta \le 1$; $AB\left(\alpha \right)=1-\alpha +(\alpha /\Gamma (\alpha \left)\right)$; $t$ is the current time; $\delta$ is the dummy integration variable representing historical time instances; and $E_{\alpha }\left(\delta \right)={\sum }_{k=0}^{\infty }{\displaystyle \frac{{\omega }^k}{\Gamma (\alpha k+1)}}$.

Definition 2 [29]. 

Assuming that function f(t) is continuous on ($a,b$); then, the fractal-fractional integral of $f\left(t\right)$ with the Mittag–Leffler kernel is defined as

$$\begin{array}{cc}\hfill {{}_a{}^{FF}I}_t^{\alpha ,\beta } \left(f\left(t\right)\right)& ={\displaystyle \frac{\alpha \beta }{AB\left(\alpha \right)}}{\int }_0^t{\delta }^{\beta -1}{\left(t-\delta \right)}^{\alpha -1}f\left(\delta \right)d\delta \hfill \\ & +{\displaystyle \frac{\left(1-\alpha \right)\beta t^{\beta -1}f\left(t\right)}{AB\left(\alpha \right)}}.\hfill \end{array}$$

(2)

${}_a{}^{FF}I_t^{\alpha ,\beta }$ is fractal-fractional integral operator; $0<\alpha ,\beta \le 1$; $AB\left(\alpha \right)=1-\alpha +(\alpha /\Gamma (\alpha \left)\right)$; $t$ is the current time; and $\delta$ is the dummy integration variable representing historical time instances. According to the results in reference [29], the equation ${}^{FF}D_t^{\alpha ,\beta }X\left(t\right)=F\left(t,X\left(t\right)\right),t\in \left[t_0,T\right]$, where $X\left(0\right)$ is the initial condition vector and $X\left(0\right)=X_0$, can be converted to the following expression:

$${}^{AB}D_t^{\alpha }\left(X\left(t\right)\right)=\beta t^{\beta -1}F(t,X\left(t\right)).$$

(3)

By applying the Atangana–Baleanu (AB) integral, Equation (3) can be rewritten as

$$\begin{array}{cc}X\left(t\right)-X\left(t_0\right)& ={\displaystyle \frac{\beta t^{\beta -1}\left(1-\alpha \right)}{AB\left(\alpha \right)}}F\left(t,X\left(t\right)\right)+{\displaystyle \frac{\alpha \beta }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}}\hfill \\ & \times {\int }_0^t{\delta }^{\beta -1}{\left(t-\delta \right)}^{\alpha -1}F\left(\delta ,X\left(\delta \right)\right)d\delta .\hfill \end{array}$$

(4)

By taking $t=t_0+n\mathsf{\Delta }t (1\le n\le \mathrm{N})$ in Equation (4), the following equation is obtained:

$$\begin{array}{cc}X\left(t_n\right)=& X\left(t_0\right)+{\displaystyle \frac{\beta \left(1-\alpha \right)}{AB\left(\alpha \right)}}{t}_n^{\beta -1}F\left(t_n,X\left(t_n\right)\right)+{\displaystyle \frac{\alpha \beta }{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha \right)}}\hfill \\ & \times {\sum }_{i=0}^{n-1}{\int }_{t_i}^{t_{i+1}}{\delta }^{\beta -1}{\left(t_n-\delta \right)}^{\alpha -1}F\left(\delta ,X\left(\delta \right)\right)d\delta .\hfill \end{array}$$

(5)

From the results in Equation (5), the discretization of Equation (5) is

$$\begin{array}{cc}{X}_{n+1}=& X_0+{\displaystyle \frac{\beta t_n^{\beta -1}\left(1-\alpha \right)}{AB\left(\alpha \right)}}F\left(t_n,X_n\right)+{\displaystyle \frac{\beta (\mathsf{\Delta }t{)}^{\alpha }}{AB\left(\alpha \right)\mathsf{\Gamma }\left(\alpha +2\right)}}\times \hfill \\ & {\sum }_{j=1}^n\left[t_j^{\beta -1}F\left(x^j\right)a_{n,j}-t_{j-1}^{\beta -1}F\left(t_{j-1},X_{j-1}\right)b_{n,j}\right]. \hfill \end{array}$$

(6)

where $a_{n,j}={\left(n+1-j\right)}^{\alpha }\left(n-j+2+\alpha \right)-{\left(n-j\right)}^{\alpha }(n-j+2+2\alpha )$, $b_{n,j}={\left(n+1-j\right)}^{\alpha +1}-{\left(n-j\right)}^{\alpha }(n-j+1+\alpha )$, $n$ = $0$ to $\mathrm{N}$, $\mathsf{\Delta }t$ is the fixed step size.

Computational Considerations and Real-Time Feasibility

The fractal-fractional discretization scheme based on the Atangana-Baleanu (AB) definition employed in this work incorporates memory effects and global dependencies, which contribute to the enhanced dynamic complexity of the model. However, this dependency comes at a computational cost: at each iteration step, a weighted sum over historical states is required, leading to algorithm time and memory complexity that scales linearly with the number of iterations $n$($O\left(n\right)$).

For strictly real-time tasks demanding extremely high update rates (e.g., millisecond-level dynamic obstacle avoidance), direct implementation of the current algorithm on standard embedded processors might be challenging. However, it is crucial to note that this paper focuses on Complete Coverage Path Planning (CCPP). The core requirement for CCPP is to generate a globally optimal or near-optimal sweeping path, not instantaneous reaction. Therefore, our method is well-suited for the following application paradigms:

Offline Planning: The complete or segmental coverage path is pre-computed by an upper-level computer before the mission starts and then dispatched to the robot for execution.

Front-end Path Generation: The proposed chaotic path planner serves as a “creative engine” to produce numerous random and ergodic candidate paths, which can then be combined with other lightweight local planners or optimizers.

To provide a clearer perspective on the computational cost, we briefly compare our scheme with simpler chaotic systems. Classical integer-order chaotic maps typically require only $O\left(1\right)$ operations per iteration, as the next state depends solely on the immediate previous state. In contrast, the discretization scheme of our fractal-fractional system (Equation (6)) exhibits $O\left(n\right)$ time and space complexity due to the necessity of computing a weighted sum over the entire historical state sequence to incorporate memory effects. This inherent complexity is the fundamental source of the system’s enhanced dynamical richness and superior ergodicity, which directly translates to the high coverage performance demonstrated in our experiments. While this results in a higher per-iteration cost than memoryless systems, it remains perfectly suitable for our target application paradigms of offline or front-end path generation. For scenarios demanding strict real-time reactivity, a viable strategy would be to employ the proposed system as a high-level trajectory generator within a hierarchical framework, coupled with a lightweight local planner for instantaneous control.

2.2. System Construction

2.2.1. Four-Dimensional Chaotic System

The present study introduces improvements based on the widely adopted Chen system.

The expression of the Chen’s chaotic system [31] is given as

$$\left\{\begin{array}{c}\begin{array}{l}{\displaystyle \frac{dx}{dt}}=A\left(y-x\right)\\ {\displaystyle \frac{dy}{dt}}=\left(C-A\right)x-xz+Cy\\ {\displaystyle \frac{dz}{dt}}=xy-Bz\end{array}\end{array}\right.,$$

(7)

where $x$, $y$ and $z$ are the three state variables of the system, and $A$, $B$, $C$ represent system parameters with $A=35$, $B=3$, $C=28$.

A hyperchaotic [32] four-dimensional Chen’s chaotic system is constructed by adding state variables to the original Chen’s chaotic system. The four-dimensional chaotic system can be written as

$$\left\{\begin{array}{c}\begin{array}{l}{\displaystyle \frac{dx}{dt}}=A\left(y-x\right)+s\\ {\displaystyle \frac{dy}{dt}}=\left(C-A\right)x-xz+Cy\\ {\displaystyle \frac{dz}{dt}}=xy-Bz\\ {\displaystyle \frac{ds}{dt}}=yz+ds\end{array}\end{array}\right..$$

(8)

In Equation (8), $x, y, z, s$ are the four state vectors of the system with $A=50, B=2, C=30, d=0.3.$

2.2.2. Four-Dimensional Fractal-Fractional Multi-Scroll Hyperchaotic System

Building upon Equation (8), to construct chaotic attractors with a well-defined multi-scroll topological structure, we introduce a nonlinear term $(m_{0}z-m_1\mathrm{c}\mathrm{o}\mathrm{s}(z\left)\right)$ The design of this term leverages the periodicity of the cosine function, aiming to systematically generate a series of equilibrium points along the z-axis direction in the phase space, thereby providing a mathematical foundation for the formation of the multi-scroll structure. Furthermore, by incorporating the fractal-fractional derivative, a four-dimensional fractal-fractional chaotic system with a multi-scroll structure is developed. The proposed system is

$$\left\{\begin{array}{c}\begin{array}{l}{}^{FFP}D_{0,t}^{\alpha ,\beta }\left(x\left(t\right)\right)=A\left(y-x\right)+s\\ {}^{FFP}D_{0,t}^{\alpha ,\beta }\left(y\left(t\right)\right)=\left(C-A\right)x-x(m_{0}z-m_1\mathrm{c}\mathrm{o}\mathrm{s}(z\left)\right)+Cy\\ {}^{FFP}D_{0,t}^{\alpha ,\beta }\left(z\left(t\right)\right)=xy-Bz\\ {}^{FFP}D_{0,t}^{\alpha ,\beta }\left(s\left(t\right)\right)=yz+ds\end{array}\end{array}\right.,$$

(9)

where $x$, $y$, $z$ and $s$ represent the system state variables. $A=50$, $B=2$, $C=30$, $d=0.2$, $m_0=0.4$, and $m_1=15$ denote the system parameters, the initial values are $[0.1, 0.1, 0.2, 0.5]$. This initial value can reveal the multi-scroll structure of the system more distinctly, with the system order $\alpha =0.995$ and fractal order $\beta =1.1$.

2.3. Path Planning Integration

Construction of the Chaotic Robot Path Planner

The general mobile robot models [33] is

$$\left\{\begin{array}{c}\begin{array}{ll}{X}_{\mathrm{R}}=v_{\mathrm{R}}·\mathit{cos}\left(k_{\theta }·{\theta }_R\left(t\right)\right)& \\ Y_{\mathrm{R}}=v_{\mathrm{R}}·\mathit{sin}\left(k_{\theta }·{\theta }_R\left(t\right)\right)& \\ {\theta }_{\mathrm{R}}=\omega \left(t\right)& \end{array}\end{array}\right..$$

(10)

Here, $X_{\mathrm{R}}$ and $Y_{\mathrm{R}}$ are the horizontal and vertical coordinates of the mobile robot; $v_{\mathrm{R}}$ is the robot’s moving speed; $k_{\theta }$ is a constant used to control the magnitude of the angle change; ${\theta }_{\mathrm{R}}$ is the robot’s steering angle; $\omega \left(t\right)$ represents the robot’s angular velocity, which reflects the rate of change in the steering angle ${\theta }_{\mathrm{R}}\left(t\right)$ with respect to time.

The constructed fractal-fractional multi-scroll chaotic Equation (9) is integrated with Equation (10), as described by Equation (11). In this integration, the state variable x, one of the four state variables, is selected to represent the angular velocity $\omega \left(t\right)$ in the mobile robot Equation (10). This approach utilizes the chaotic variable x to generate the movement trajectories of mobile robot.

$$\left\{\begin{array}{c}\begin{array}{l}{}^{FFP}D_{0,t}^{\alpha ,\beta }\left(x\left(t\right)\right)=A\left(y-x\right)+s\\ {}^{FFP}D_{0,t}^{\alpha ,\beta }(y(t))=(C-A)x-x(m_{0}z-m_1\mathrm{c}\mathrm{o}\mathrm{s}(z))+Cy\\ {}^{FFP}D_{0,t}^{\alpha ,\beta }\left(z\left(t\right)\right)=xy-Bz\\ {}^{FFP}D_{0,t}^{\alpha ,\beta }\left(s\left(t\right)\right)=yz+ds\\ X_{\mathrm{R}.n}=v_{\mathrm{R}}·\cos \left(k_{\theta }·x\right)\\ Y_{\mathrm{R}.n}=v_{\mathrm{R}}·\sin \left(k_{\theta }·x\right)\end{array}\end{array}\right..$$

(11)

In the robot motion model of Equation (11), the state variable $x\left(t\right)$ of the chaotic system is used to drive the angular velocity $\omega \left(t\right)$ of the robot. To map the dimensionless state variable $x\left(t\right)$ to an angular velocity with physical units and to ensure that the robot moves within a reasonable and feasible range, we introduce a scaling factor $k_{\omega }$. Thus, the actual mapping relationship for the angular velocity is $\omega \left(t\right)=k_{\omega }\cdot x\left(t\right)$. In all simulation experiments in this paper, the scaling factor is set to $k_{\omega }=0.8$. This value is chosen based on the following considerations: (1) It limits the maximum angular velocity of the robot, ensuring smooth motion trajectories that align with the maneuverability of practical robots; (2) It guarantees that the robot exhibits sufficient ergodicity and randomness within the limited simulation area. Although the current simulation does not explicitly model kinematic constraints such as turning radius, in practical implementations, the rapid variations in the chaotic signal can be accommodated by incorporating angular rate limiters or low-pass filters to smooth the commanded angular velocity, ensuring that the generated paths are feasible for real robots with dynamic constraints.

In Equation (11), $n$ is the number of iterations, with $X_{\mathrm{R}.n}$ and $Y_{\mathrm{R}.n}$ denoting the horizontal and vertical coordinates of the mobile robot at the n-th iteration, respectively. Subsequently, the Fractal-fractional-order solution method proposed in Ref. [29] is employed to solve Equation (11). Given the initial values ${(X}_{R.0},Y_{R.0}, x_0,y_0,z_0,s_0)=(0, 0, 0.1, 0.1, 0.2, 0.5)$ the movement path of the mobile robot can be determined. Here, $X_{\mathrm{R}.0}$ and $Y_{\mathrm{R}.0}$ represent its initial horizontal and vertical coordinates, respectively, while $x_0,y_0,z_0,s_0$ are the initial values of the four state variables for Equation (9). The values of other parameters should remain consistent with those in Equation (9).

2.4. Simulation Setup

2.4.1. Parameter Settings

In this section, we adopt the baseline parameter values from Equation (9) as follows:

$$A=50, B=2, C=30, d=0.2, m_0=0.4, m_1=15, \alpha =0.995, \beta =1.1.$$

All simulations were performed in MATLAB R2019a using ode45.

2.4.2. Mirror Reflection Method

In the evaluation metrics for mobile robot path coverage, the grid method is a commonly used approach. It is primarily employed to measure the efficiency with which a mobile robot traverses a designated area. The fundamental principle of this method involves partitioning the area into multiple grids, with grid size being adjustable according to practical application requirements. Its formula is described by

$$C={\displaystyle \frac{N_{covered}}{N}}\times 100\%.$$

(12)

Here, $C$ represents the coverage rate of the mobile robot; $N_{covered}$ is the number of covered grid cells, and $N$ is the total number of grid cells. In the coverage rate calculation, grid cells occupied by obstacles are excluded from the total number of cells $N$, and the robot path crossing such cells is not counted as covered.

For obstacles located within the mobile area boundary and its interior, the mirror reflection method is adopted for avoidance. This approach simulates the law of optical reflection (where the angle of incidence equals the angle of reflection) to map paths into the designated area, thereby ensuring continuous robot motion within the space. The specific methodology is shown in Figure 2. In this context, $m_i$ is the robot’s current position; $m_k$ is the next iterative position; $m_j$ is the intersection point of the robot trajectory and the boundary. The line segment $\mathrm{M}\mathrm{N}$ represents the boundary, while $m_y$ is a point that exceeds the boundary. According to the principle of mirror reflection, during the robot’s movement, a collision occurs at $m_j$, which is then reflected to $m_k$.

To evaluate the complexity of the chaotic path planning method proposed in this paper, Shannon entropy is introduced for path complexity analysis. By calculating the entropy value of a path, its unpredictability can be quantitatively assessed. The specific calculation formula is defined as follows:

$$H=-{\sum }_{i=1}^N P_i\mathrm{l}\mathrm{o}\mathrm{g}\left(P_i\right),$$

(13)

where $P_i$ represents the probability of a path point falling into the ith grid cell; and $N$ is the total number of grid cells covered by the path. A higher entropy value indicates a more uniform distribution of path points, greater path complexity, and increased path unpredictability. Therefore, this method is better suited for application scenarios that require in-depth exploration of the environment.

3. Results and Analysis

This section will perform a dynamical characteristics analysis of the proposed fractal-fractional multi-scroll chaotic system and evaluate its coverage performance in mobile robot path planning. Through sensitivity tests, comparative experiments, and ablation studies, the effectiveness and superiority of the proposed method will be rigorously validated.

3.1. Chaotic System Characteristics

Equation (8) is simulated under the initial values with $x_0=0.1$, $y_0=0.1$, $z_0=0.1$, and $s_0=0.1$. The system underwent $\mathrm{15,000}$ iterations. Figure 3 shows the phase space structure. As shown in Figure 3, the new four-dimensional system incorporating state variables still exhibits a clear chaotic phase structure.

To quantitatively assess the dynamic characteristics of the constructed fractal-fractional chaotic system, this paper employs numerical methods to compute the system’s Lyapunov exponent spectrum. Given that the long-range memory effect in fractal-fractional systems makes exact analytical calculations extremely difficult, we adopt a numerical estimation approach based on the linearized approximation of continuous systems.

Specifically, during each iteration step, the system state variables are updated using the fractal-fractional discrete integration formula Equation (6). Meanwhile, based on the current system state, we compute the Jacobian matrix and numerically integrate the variational equations via the fourth-order Runge-Kutta method to evolve four orthogonal perturbation vectors. These perturbation vectors are periodically subjected to Gram-Schmidt orthogonalization and normalization, and the cumulative growth rates along each orthogonal direction are recorded. Finally, the Lyapunov exponents are obtained by dividing the cumulative growth rates by the total integration time.

However, the calculation of the Lyapunov exponent spectrum shows that the four Lyapunov exponents for Equation (8) are ${\lambda }_1=0.23686$, ${\lambda }_2=-0.47341$, ${\lambda }_3=-0.465376$, and ${\lambda }_4=-20.9785$, as presented in Figure 4.

For Equation (9), a simulation experiment is conducted using the initial parameters $x_0=0.1$, $y_0=0.1$, $z_0=0.2$, $s_0=0.5$. The total number of iterations performed in this experiment is set to 10,000. The computed Lyapunov exponents for Equation (9) are ${\lambda }_1=7.52144$, ${\lambda }_2=0.0880422$, ${\lambda }_3=-3.93453$, ${\lambda }_4=-25.3554$. The corresponding Lyapunov exponent spectrum is shown in Figure 5.

As depicted in Figure 5, Equation (9) exhibits greater Lyapunov exponents compared to Equation (8). A larger Lyapunov exponent value indicates a faster rate of chaotic trajectory separation. The system state becomes completely different in a shorter time. Any small initial error will be greatly amplified. The unpredictability of the system is enhanced.

The phase space structures of Equation (9) are shown in Figure 6. As depicted in Figure 6, after the introduction of the nonlinear term, the system exhibits a multi-scroll structure, which becomes more pronounced with an increasing number of iterations. Multi-scroll [34] structures and fractal-fractional operators can impart enhanced complexity to chaotic systems.

The bifurcation diagrams corresponding to various parameters is presented in Figure 7. The bifurcation diagram visually demonstrates that the system is indeed in a chaotic state near the selected parameters. By observing the bifurcation diagram, parameter values that generate strong chaotic behavior can be selected based on evidence, rather than through blind trial and error. In path planning, the selected parameter points should be located in a wide chaotic band to avoid accidentally falling into a periodic window.

Chaotic systems possess ergodicity, meaning they can visit every point within the chaotic region within a finite time. For the given chaotic system time series, the ergodic region is a $1\times 1$ area on the $x-s$ plane. After $1500$ iterations of Equation (9), the distribution of its time series is shown in Figure 8.

The ergodic distribution diagram visually illustrates which regions in the phase space are frequently visited by system trajectories and which regions are less frequently visited. In path planning, this corresponds to which areas are covered repeatedly and which areas are under-covered. An ideal and efficient coverage path requires a distribution that is as uniform as possible. As can be seen from Figure 8, the distribution diagram shows consistent color and density of points throughout the chaotic region, indicating that the system exhibits excellent ergodicity.

In the analysis of chaotic systems, spectral entropy ($SE$) and complexity ($CO$) are two important quantitative metrics used to characterize the dynamic behavior, complexity, and unpredictability of these systems [35]. $SE$ is calculated by first transforming the time series signal into the frequency domain to decompose it into energy components across various frequencies. Subsequently, entropy is calculated based on the uniformity of this energy distribution. In contrast, $CO$ is calculated by extracting and removing periodic or stable components from the signal, thereby retaining only the residual irregular fluctuations. The randomness or unpredictability of these residual fluctuations is then statistically evaluated to indirectly assess chaotic characteristics. For parameters ($A$, $C$, $d$, $m_0$, $m_1$) = ($50$, $30$, $0.3$, $0.4$, $15$), with parameter $B$ $\in$ ($-0.5$, $2.5$) and $\mathsf{\alpha }$ $\mathrm{d}\in$ ($0.95$, $1$), Figure 9a shows the $SE$ complexity while Figure 9b depicts the $CO$ complexity; it is evident that when parameter $B$ is fixed, the system complexity progressively increases as the fractional order rises.

When the parameter plane is examined within the same interval, differences arising from distinct initial conditions can indicate the presence of multistability. Multistability is a typical phenomenon in nonlinear dynamical systems, referring to the scenario where, under identical system parameters, different initial conditions lead the system to converge to distinct steady-state behaviors. In chaotic systems, this implies that multiple attractors can coexist, each corresponding to a unique dynamical state. The asymptotic behavior of systems that support multistability depends on the set of initial conditions of the state variables. Consequently, in certain systems, the same region of the parameter plane may exhibit chaotic behavior for specific initial conditions while displaying periodic behavior for others. This characteristic is particularly significant in path planning, as it enables the same chaotic system to generate diverse trajectories, thereby enhancing the randomness and adaptability of the paths.

As depicted in Figure 10, Figure 10 shows the basins of attraction associated with the coexisting chaotic and periodic attractors in phase space when the parameter A = 54. In each basin, the black regions represent the initial conditions that cause the system to converge to the periodic attractor in phase space, while the red regions represent the initial conditions that lead the system to converge to the chaotic attractor in phase space. It should be clarified that the basin of attraction is inherently four-dimensional and is therefore defined by the set of initial conditions $(x_0,y_0,z_0,s_0)$. Consequently, Figure 10 presents the projection of this four-dimensional basin onto the $(x_0,y_0)$ initial conditions, which are constructed in this example by setting $z_0$ = 0.2 and $s_0$ = 0.5. As long as an initial condition point $(x_0,y_0)$ is chosen within the red region, the system will tend towards a chaotic attractor in phase space; whereas if selected within the black region, the system will be guided towards the periodic attractor.

Figure 11 illustrates the coexistence of periodic and chaotic attractors in the phase space of system (9). All results correspond to the parameters A = 54, B = 2, C = 30, d = −0.2, m0 = 0.4, m1 = 15, α = 0.995, β = 1.1. The initial conditions of the system are set to $(x_0,y_0,z_0,s_0)$ = (0, 0, 0.2, 0.5). In the figure, the red region represents initial conditions that lead the system to converge to a chaotic state, while the black region corresponds to periodic states. Furthermore, Figure 11 shows the coexisting attractor structures of the system when typical initial points are selected within these two basins of attraction. Specifically, the red trajectory corresponds to a chaotic attractor, and the black trajectory corresponds to a periodic attractor. This phenomenon indicates that even when the system parameters remain unchanged, the system can exhibit distinctly different dynamical behaviors solely by varying the initial conditions. Such a characteristic provides an additional degree of freedom in path planning, allowing different motion modes to be selected according to task requirements under the same chaotic system structure, thereby enhancing the flexibility and adaptability of trajectory generation.

3.2. Sensitivity Test

The constructed chaotic system demonstrates sensitivity to initial conditions, whereby even minor alterations in its initial values can significantly affect the robot’s movement trajectory. This characteristic of the chaotic system effectively meets the requirement for randomness in mobile robot path planning. Under identical settings, two distinct sets of initial values for the chaotic system were defined ($x$, $y$, $z$, $s$) = ($0.1$, $0.1$, $0.1$, $0.1$) and ($x$, $y$, $z$, $s$) = ($0.1$, $0.1$, $0.1$, $0.2$). The number of iterations was consistently set at 5000 for both cases. The resulting system trajectories are shown in Figure 12.

As observed in Figure 12, slight modifications to initial values generate markedly divergent trajectories after finite iterations. The greater the alteration in initial values, the more significant the trajectory divergence becomes. This property facilitates the generation of stochastic and unpredictable surveillance paths for mobile robots.

3.3. Path Planning Performance

When the initial values ($X_{\mathrm{R}.0}{,Y}_{\mathrm{R}.0},x_0,y_0,z_0,s_{0)}$ = ($0$, $0$, $0.1$, $0.1$, $0.2$, $0.5$) are set and the grid size is configured to a uniform $5 \times 5$ cells layout, the initial value was selected based on empirical tests showing that it produces the most uniform trajectory distribution and the highest coverage rate among candidate points. Simulation experiments for the mobile robot were conducted in MATLAB R2021b for $5000$ and $\mathrm{10,000}$ iterations respectively. The resulting robot trajectories within the $50$ × $50$ area are shown in Figure 13. It is evident from Figure 13 that, with increasing iterations, the robot’s movement trajectory can cover more areas, resulting in an increased number of covered trajectories.

To verify the advantages of the proposed fractal-fractional multi-scroll chaotic system in mobile robot path planning, a comparison is performed with the methods from references [18,35], and a random walk approach. The random walk method is implemented as follows: initially, the robot randomly selects a direction for linear motion; subsequently, at fixed time intervals $0.02 \mathrm{s}$ randomly turns to a new direction for the next movement. The trajectories generated by mobile robot path planning using the random walk method within a $50$ × $50$ area over $1000$ and $5000$ movements ar shown in Figure 14. As shown in Figure 14, the robot under the random walk method achieves progressively greater coverage of target areas as the iteration count increases.

The coverage rates calculated for the mobile robot driven by the fractal-fractional chaotic system, along with those derived from other methods, are shown in

Table 1. Comparison with Other Methods.

Model	System	Coverage Rate	Path Entropy	Iteration Count
Classical Chaotic	Lorenz	$0.3296$	$5.4941$	$1\times 10^4$
System	Chua	$0.6048$	$4.0524$	$1\times 10^4$
Random Walk	$\mathrm{N}\mathrm{A}$	$0.922$	$4.8761$	$1\times 10^4$
Reference [18]	Hyperchaotic Synchronization Control	$0.949$	$\mathrm{N}\mathrm{A}$	$1\times 10^4$
Reference [35]	Fractional-Order Chaotic System	$0.927$	8.97	$8\times 10^4$
This work	$x$	$0.9776$	5.5754	$1\times 10^4$
This work	$y$	$0.9888$	5.7118	$1\times 10^4$
This work	$z$	$0.7280$	5.1953	$1\times 10^4$
This work	$s$	$0.9504$	5.6097	$1\times 10^4$

. The proposed fractal-fractional multi-scroll chaotic system achieves a superior area coverage rate compared to the other methods, meeting the requirements for mobile robot coverage tasks more effectively.

As part of the While the sensitivity to initial conditions is a fundamental characteristic of chaotic systems that contributes to path randomness, for practical coverage tasks, it is crucial that high performance remains consistent across different initializations, ensuring the method’s robustness for daily deployment. To address this, we conducted a comprehensive robustness test. In this experiment, the four state variables $(x_0,y_0,z_0,s_0)$ of the proposed fractal-fractional multi-scroll chaotic system were independently and randomly sampled from a uniform distribution within the range $\left[-10,10\right]$ for each trial. For each set of randomly generated initial conditions, the integrated chaotic path planner was used to drive the mobile robot for $\mathrm{10,000}$ iterations within the same $50\times 50$ grid environment. This procedure was repeated for 100 independent trials.

As shown in Figure 15, the initial values of the state variables $(x_0,y_0,z_0,s_0)$ for the constructed fractal-fractional multi-scroll chaotic system were randomly and independently sampled from the uniform interval [−10, 10] in each trial. The mobile robot was then driven by the system within the 50 × 50 grid area for $\mathrm{10,000}$ iterations per run. Over 100 such randomized trials, the average coverage rate reached 97.072%. Although coverage occasionally fell below 90% in rare outliers, the vast majority of trials achieved rates above 95%. Critically, even under random initializations, the coverage performance consistently surpasses that of all comparative methods listed in Table 1. These results confirm that the high coverage efficacy is a robust property of our system, ensuring reliable performance in practical deployment despite its inherent sensitivity to initial conditions.

3.4. Ablation Study Analysis

As part of the evaluation of the proposed system, A comprehensive ablation study was conducted to validate the contribution of each component. This section compares: $\left(\mathrm{A}\right)$ the performance of the full model with three simplified variants; $\left(\mathrm{B}\right)$ an integer-order version; $\left(\mathrm{C}\right)$ a version without the multi-scroll nonlinear term; $\left(\mathrm{D}\right)$ a version without the fractal dimension. All other parameters and initial conditions remained consistent with the previous experiments. The simulation results are shown in Figure 16, and the initial values, parameters, and the corresponding coverage rates are shown in

Table 2. Fractal-Fractional Multi-Scroll and its degenerate subsystems parameters and coverage rate.

System	$\mathit{\alpha }$	$\mathit{\beta }$	${\mathit{x}}_0$	${\mathit{y}}_0$	${\mathit{z}}_0$	${\mathit{w}}_0$	Lyapunov	Coverage Rate	Iteration Count
$A$	$0.995$	$1.1$	$0.1$	$0.1$	$0.2$	$0.5$	$7.5214$	$0.9888$	$1\times 10^4$
$B$	$1$	$1$	$0.1$	$0.1$	$0.2$	$0.5$	$3.617$	$0.968$	$1\times 10^4$
$C$	$0.995$	$1.1$	$0.1$	$0.1$	$0.2$	$0.5$	$0.7909$	$0.9712$	$1\times 10^4$
$D$	$0.995$	$1$	$0.1$	$0.1$	$0.2$	$0.5$	$3.0867$	$0.9744$	$1\times 10^4$

.

As shown in Figure 16a, with the same number of iterations, the fractal-fractional-order multi-scroll system $A$ proposed in this paper is capable of forming a relatively dense multi-scroll structure. Figure 16b illustrates the chaotic system $B$, which is derived from Equation (9) by replacing the fractional order with an integer order. The integer-order chaotic system generates fewer scroll layers along the z-axis under identical iteration conditions. Figure 16c presents system $C$, obtained by removing the nonlinear term from Equation (9), where no distinct scroll structure or layered phenomenon emerges. Figure 16d shows the structure of system $D$, configured by setting the fractal dimension to be $1$ in Equation (9), demonstrating that the fractal dimension exerts a considerable influence on the chaotic system’s behavior.

As shown in Table 2, the Lyapunov exponents of systems $B$, $C$, and $D$ are all lower than those of the proposed system $A$. Correspondingly, the coverage rates achieved by mobile robots driven by systems $B$, $C$, and $D$ are also inferior to those achieved under system $A$. These ablation studies demonstrate that the synergistic interplay among the fractal-fractional order, the nonlinear term and the fractal dimension is crucial for enhancing the complexity of the chaotic system and achieving high coverage in path planning.

4. Discussion

The experimental results demonstrate that the dual modulation of the fractal order and the fractional order exerts a distinct and sensitive influence on the system’s dynamics: they systematically elevate the system’s largest Lyapunov exponent, thereby accelerating the divergence rate of trajectories and enhancing the system’s unpredictability. Simultaneously, the synergistic effect between the introduced fractal dimension and the multi-scroll nonlinear term significantly enriches the system’s phase space structure. This directly corresponds to a more uniform ergodic distribution of path points within the coverage area, as evidenced by the achieved 98.88% coverage rate and the high path entropy value.

5. Conclusions

This study successfully proposed and validated a novel four-dimensional fractal-fractional multi-scroll chaotic system for CCPP in mobile robots. The core conclusions are as follows:

(1)

A hyperchaotic system is developed by augmenting the classical Chen system with an additional state variable, integrating fractal-fractional calculus and a cosine-based nonlinear term. This integration yields a system with a complex multi-scroll attractor structure, two positive Lyapunov exponents, and higher spectral entropy, indicating significantly enhanced dynamic complexity and unpredictability.

(2)

The system is embedded as the dynamic driver for the mobile robot’s kinematic model, with its chaotic state variable controlling the robot’s angular velocity. This effectively translates intricate mathematical dynamics into practical, unpredictable motion.

(3)

Extensive simulations validate the superiority of our method. In a 50 × 50 grid environment, it achieved a 98.88% coverage rate within 10,000 iterations, significantly outperforming benchmark methods based on the Lorenz system, Chua’s circuit, and random walks. Robustness tests across 100 randomized trials confirmed an average coverage of 97.072%, demonstrating that high performance is a consistent property of the system, not dependent on specific initializations. Ablation studies further clarified that the synergistic interplay of the fractal dimension, fractional order, and multi-scroll terms is crucial for generating this high-coverage, high-entropy behavior.

The implications of this research are twofold. Theoretically, it provides a viable framework for constructing highly complex chaotic systems. Practically, it offers a reliable solution for autonomous robots requiring efficient, random, and thorough inspections in scenarios such as security patrols or post-disaster exploration. Future work will focus on integrating real-time obstacle avoidance mechanisms in dynamic environments and optimizing the computational efficiency of the fractal-fractional discretization scheme for real-time control applications. Additionally, extending this approach to multi-robot systems via chaotic synchronization represents a promising direction for enabling coordinated and unpredictable swarm coverage strategies.