Hexagonal Fuzzy Numbers Improve Mobile Robot Tracking Control

Researchers have developed a novel adaptive fuzzy sliding mode controller (AFSMC) designed to enhance the trajectory tracking accuracy of omnidirectional mobile robots operating in dynamic and uncertain environments. Published in the journal Discover Robotics, the study introduces a control strategy that integrates a sigmoid-based sliding mode switching function with an integral compensator to achieve fast convergence while effectively mitigating chattering. The core innovation lies in a fuzzy gain adaptation mechanism based on hexagonal fuzzy numbers (HFNs), which provides flexible nonlinear shaping of membership functions. This approach supports multi-objective tuning that balances robustness, responsiveness, and the smoothness of the control action, addressing persistent challenges in mobile robotics control.

The primary objective of the research is to design a controller capable of maintaining high tracking accuracy and smooth control performance despite parameter uncertainties, sensor noise, and external disturbances. The proposed AFSMC architecture combines equivalent and switching control laws to maintain resilience against these factors. By embedding HFN-based adaptation within the sliding mode framework, the method offers a robust and efficient solution for high-precision trajectory tracking in unpredictable real-world robotic applications, with potential extensions to autonomous navigation under soft-sensing and unstructured terrain conditions.

The study focuses on an omnidirectional mobile robot featuring a symmetrical four-wheel structure, where each wheel is driven by an independent DC servo motor. The control system is divided into modules for map construction, path planning, path tracking, and underlying control. The kinematic model describes the robot’s motion using global and local coordinate systems, defining a virtual path between the robot’s center of mass and a preview point. This virtual path is constructed in real time using polynomial trajectory planning, incorporating boundary conditions such as curvature and yaw rate to ensure smooth convergence to the desired trajectory.

The mathematical formulation of the control system includes a DC motor model that accounts for the electrical and mechanical dynamics of the drive system. The motor torque is expressed as a function of the current in the rotor coils, while the equivalent moment of inertia considers both rotor and load dynamics. Viscous friction forces are modeled using known and experimentally determined coefficients. Newton’s second law is applied to model the rotational dynamics of the robot’s drive shaft, linking motor input voltage to rotor coil current through a differential equation that includes inductance, resistance, and back electromotive force constants.

The HFN-based adaptive gain computation is a central component of the proposed method. Each HFN is defined by six ordered parameters that determine the rising, plateau, and falling regions of the membership function. These parameters allow for asymmetric and nonlinear shapes, enabling the fuzzy sets to capture variations between small and large errors more effectively than standard triangular or trapezoidal functions. The adaptation uses a Mamdani-style OR-combination to aggregate the membership values of the tracking error and its rate of change. The resulting aggregated membership degree is mapped to the adaptive gain through a linear interpolation, ensuring that the gain increases monotonically with tracking severity while remaining within predefined bounds.

Stability analysis of the controller was conducted using a Lyapunov function candidate. The analysis demonstrates that the sliding mode stability conditions are preserved even when the adaptive gain varies with time, provided that the gain remains within the prescribed limits. The HFN mechanism enforces these bounds by construction, ensuring that the time-varying term does not compromise the system’s stability. The controller parameters were selected through structured manual tuning, guided by root-mean-square error values, switching-surface behavior, and control-input smoothness. All final parameter values are reported in the study’s tables, and the associated configuration files are included in the reproducibility package.

To verify the effectiveness of the control algorithm, comprehensive simulation studies were conducted using MATLAB and Python programs. The experiments covered two primary cases: straight-line trajectory tracking and circular trajectory tracking. In the linear case, the robot’s actual trajectory quickly aligned with the ideal trajectory, and the tracking errors stabilized within narrow intervals. The angular velocities of the wheels remained constant, satisfying the kinematic constraints of the omnidirectional motion. In the circular case, the robot maintained stable tracking performance, with position errors and heading deviations stabilizing within specified ranges after a short transient period.

A comparative analysis was performed between the proposed AFSMC and a conventional sliding mode controller (SMC) under identical simulation conditions. The results showed that the AFSMC consistently demonstrated superior performance, achieving smoother control inputs, reduced chattering, and more stable convergence behavior. Specifically, the AFSMC exhibited substantially reduced chattering amplitude in the switching-surface evolution compared to the classical SMC. The adaptive gain mechanism helped the controller maintain stability while reducing the abrupt switching behavior characteristic of traditional SMC, leading to more reliable and robust control performance under both linear and nonlinear trajectory conditions.

Robustness was evaluated by introducing three distinct perturbation scenarios: Gaussian white noise in the sensors, a 30% increase in payload mass, and an external sinusoidal force disturbance. Under these conditions, the AFSMC achieved lower root-mean-square error values across all dimensions compared to the SMC. For instance, under sensor noise, the AFSMC maintained position errors below 0.03 meters, whereas the SMC exhibited errors exceeding 0.07 meters. Under payload variation, the AFSMC retained its convergence properties with settling times of less than 3.1 seconds and overshoots of under 4%, compared to 5.5 seconds and over 10% for the SMC. When subjected to external disturbances, the AFSMC showed a 35% lower chattering index and a 20% reduction in control energy, indicating greater efficiency and smoother actuator engagement.

The study highlights the motivation for using hexagonal fuzzy numbers, noting that they extend the structure of standard fuzzy numbers by incorporating six independently adjustable parameters. This geometric structure facilitates smooth yet highly descriptive modeling of uncertainty, which is valuable when the controller must respond differently to small, moderate, and large tracking deviations. The additional degrees of freedom offered by HFNs enable multi-objective gain tuning, allowing the adaptive law to balance responsiveness, robustness, and smoothness. This flexibility supports smoother adaptation and reduces conservativeness in gain tuning, which is critical for nonlinear trajectory tracking under disturbances and model uncertainties.

Despite the encouraging findings, the authors acknowledge that the conclusions are presently limited to a simulation-based evaluation. The analysis assumes ideal actuator characteristics, continuous and noise-free measurements, and perfect state availability. The wheel-motor dynamics are modeled using a second-order approximation rather than full electrical dynamics, and disturbances are introduced in controlled forms rather than arising naturally from unpredictable physical interactions. These simplifications enable a clear comparison between SMC and AFSMC but restrict the generalizability of the results to real-world conditions. The simulations rely on idealized system dynamics and precise timing, which do not fully capture the stochastic and nonlinear complexities present in physical environments.

Future work will focus on implementing the controller on physical robotic hardware to conduct empirical experiments and assess real-time performance, stability, and robustness in the presence of unmodeled dynamics and practical constraints such as actuator delays, sensor quantization, and communication latency. The researchers emphasize that the presented results should be interpreted as preliminary validation of the control strategy under modeled uncertainties. By extending the evaluation to more complex uncertainty conditions and unstructured environments, future experiments will provide deeper insight into the practical applicability and robustness of the proposed approach beyond the controlled conditions considered in this paper.

All simulations reported in the study are fully reproducible using the Python scripts supplied as supplementary material in the authors’ GitHub repository. The implementation uses a fixed-step forward Euler solver with a total simulation duration of 20 seconds. Random seeds are fixed for all disturbance scenarios to ensure numerical repeatability. The code package includes the AFSMC controller with HFN-based adaptation, the baseline SMC implementation, scenario configuration files, and the raw data used to generate each figure and table in the manuscript. This commitment to reproducibility ensures the numerical transparency and validity of the performance metrics presented in the research.

Subject of Research: Adaptive fuzzy sliding mode trajectory tracking control using hexagonal fuzzy numbers

Article Title: Adaptive fuzzy sliding mode trajectory tracking control using hexagonal fuzzy numbers

Article References: Mokhtarzadeh, A. A., Mbala, I. N., Wang, D., Abdalla, A. N., Li, Q., Niabanga, D. M. O., Kifouri, G. O., & Shahzad, N. M. (2026). Adaptive fuzzy sliding mode trajectory tracking control using hexagonal fuzzy numbers. Discover Robotics, 2(1), Article 6. https://doi.org/10.1007/s44430-025-00015-z

Image Credits: AI Generated

DOI: 10.1007/s44430-025-00015-z

Keywords: Adaptive, fuzzy, sliding, mode, trajectory, tracking, control, hexagonal, numbers, scientific research