Adaptive Neuro-Fuzzy Control Improves Trajectory Tracking Precision in Robotic Arms

In a newly published investigation focusing on industrial automation and multi-axis manipulator dynamics, researchers have demonstrated that an Adaptive Neuro-Fuzzy Inference System controller significantly enhances trajectory tracking precision and stability over classical proportional-integral-derivative control in a five-degree-of-freedom robotic arm. Robotic manipulators deployed across manufacturing, medical automation, agriculture, and hazardous material management routinely face performance bottlenecks stemming from nonlinear dynamics, actuator friction, and unmodeled external disturbances. Conventional control architectures frequently struggle to maintain high-accuracy path tracking when executed over complex three-dimensional curves. To confront these fundamental challenges, the research team established an exact mathematical framework encompassing both kinematic structures and Euler-Lagrange dynamic equations tailored to the Mitsubishi RV-2AJ articulated manipulator, validating the theoretical framework directly against dynamic computer-aided simulations before evaluating intelligent adaptive regulation methods.

The physical system examined in the study is the Mitsubishi RV-2AJ robotic arm, an articulated industrial manipulator comprising five serial links interconnected through motorized rotational joints. Each link in the kinematic chain pivots across independent angles denoted from the base to the terminal tooling, actuated by individual torque motor servos designed to execute precise spatial maneuvers. To establish the spatial relationship governing the manipulator, the authors implemented the classical Denavit-Hartenberg convention, attaching individual coordinate reference frames to each joint axis to systematically determine link lengths, offsets, and twist parameters. Through sequential matrix multiplication of individual homogeneous transformation matrices spanning from the fixed base frame to the terminal end effector, the forward kinematic model was explicitly formulated. This analytical formulation enables exact spatial mapping of the end effector’s position and orientation vectors in Cartesian space as direct functions of instantaneous joint rotations.

Complementing the forward geometric derivation, the researchers developed an analytical formulation for the inverse kinematics of the five-degree-of-freedom arm. While inverse kinematics in multi-joint serial robots is frequently resolved through iterative numerical approximation schemes that can introduce computational latency and convergence ambiguities, deriving closed-form analytical solutions provides an essential foundation for deriving inverse dynamic equations and real-time control algorithms. By equating known aggregate transformation matrices with sequential link matrices and systematically premultiplying inverse transformations, the team isolated explicit trigonometric equations for each joint angle. Using two-argument arctangent functions, the analytical approach systematically extracts the precise rotational angles necessary to guide the end effector along predefined three-dimensional continuous coordinates without relying on computationally demanding numerical iterations.

To capture differential motion between the joint actuators and the terminal tool point, the investigation incorporated comprehensive Jacobian analysis. Calculating the analytical Jacobian matrix entailed differentiating the forward kinematic equations with respect to each joint coordinate, yielding linear and angular velocity relationships across the entire robotic structure. By separating the Jacobian operator into distinct linear velocity and angular velocity submatrices, the researchers derived closed-form expressions for the directional velocity components of the terminal link.

Beyond geometric and kinematic representations, capturing real-world physical forces required formulating a comprehensive dynamic model using Euler-Lagrange formulations. The resulting nonlinear matrix equation governs the relationship between joint accelerations, velocities, positions, and driving torques, incorporating an inertia mass matrix, Coriolis and centrifugal force vectors, gravitational torque terms, and non-conservative joint friction components. Each term of the mass matrix was computed through link inertia tensors defined at link centroids and translational-rotational Jacobian transformations across all five segments. To verify the fidelity of the theoretical equations, the team constructed an identical virtual three-dimensional mechanical assembly in MATLAB and Simulink using the Simscape Multibody environment. Simulating dynamic motions across both models revealed that the joint angular positions predicted by the Euler-Lagrange equations matched the Simscape simulation outputs with remarkable consistency.

Having validated the underlying dynamic model, the authors designed and tuned two distinct control strategies to govern trajectory tracking: a baseline classical proportional-integral-derivative system and an intelligent Adaptive Neuro-Fuzzy Inference System. For the classical benchmark, five independent PID controllers were deployed across each corresponding joint servo. The initial controller coefficients were identified through automated tuning utilities in MATLAB and Simulink, after which the proportional, integral, and derivative gains were further refined using Particle Swarm Optimization to suppress steady-state errors.

To overcome the operational limitations of linear control, the researchers engineered an Adaptive Neuro-Fuzzy Inference System based on the Takagi-Sugeno fuzzy model. The ANFIS architecture integrates the transparent rule-based reasoning of fuzzy logic with the data-driven optimization and pattern learning of multi-layer neural networks. Structured across five computational layers—comprising input nodes, fuzzy membership functions, rule evaluation nodes, defuzzification operators, and summary output nodes—the ANFIS controller was configured to process both joint positions and instantaneous joint velocities simultaneously as dual input vectors. The system was trained using a hybrid learning scheme combining least-squares estimation for linear consequence parameters with backpropagation gradient descent for nonlinear premise parameters, utilizing training data sampled at one kilohertz over a sixty-second operating window to effectively capture and counteract dynamic system nonlinearities.

The tracking performance and structural robustness of both control systems were evaluated across three rigorous simulation benchmarks: an N-shaped path containing sharp directional transitions, a smooth circular path, and an intricate three-dimensional complex trajectory. For the N-shaped trajectory, which demands rapid dynamic transitions along multi-axis coordinated lines, both controllers succeeded in completing the profile, but ANFIS exhibited marked performance superiority. Quantitative time-domain analysis revealed that while settling times improved moderately by 1.07 percent to 1.74 percent across joints, ANFIS dramatically suppressed peak overshoot, achieving a 26.70 percent reduction in Joint 5. Steady-state error was curtailed by more than 21 percent in Joints 1 and 3, reaching a 21.74 percent improvement in Joint 3, while cumulative root mean square error decreased by 4.13 percent to 9.60 percent across all links.

Testing along the circular trajectory further corroborated the superior tracking precision of the adaptive neuro-fuzzy structure under continuous rotational motion. Under ANFIS regulation, settling times decreased by up to 1.43 percent in Joint 4, while peak overshoot dropped by 15.28 percent in Joint 2 and by 26.70 percent in Joint 5 relative to the PID baseline. Steady-state tracking errors demonstrated an exact 20.00 percent reduction in both Joint 1 and Joint 3, accompanied by root mean square error reductions ranging between 4.00 percent and 6.88 percent.

When challenged with the complex three-dimensional trajectory, the performance advantages of the adaptive neuro-fuzzy methodology became even more pronounced. The ANFIS controller maintained tighter adherence to the reference coordinates, delivering settling time reductions between 1.36 percent and 1.61 percent and lowering peak overshoot by 16.05 percent in Joint 2 and 24.00 percent in Joint 5. Steady-state tracking errors dropped by 20.00 percent in both Joint 1 and Joint 5, while Joint 1 achieved an 8.00 percent reduction in trajectory root mean square error, demonstrating that the dual-input neuro-fuzzy system successfully compensated for high-frequency dynamic variations.

The study also conducted an extensive comparative evaluation of actuator effort by analyzing the torque signals generated by both control architectures across each trajectory scenario. Evaluating root mean square error between controller torque outputs revealed how each system regulated driving effort under varying geometric complexities. In the abrupt N-shaped path, torque signal RMSE values between PID and ANFIS ranged from 0.4299 in Joint 3 to 0.5086 in Joint 4. In the smoother circular pattern, torque discrepancies decreased, yielding values between 0.3824 in Joint 3 and 0.4740 in Joint 4.

In their concluding assessment, the authors highlighted that combining rigorous Euler-Lagrange dynamic modeling with an adaptive neuro-fuzzy control architecture establishes a highly robust foundation for next-generation industrial manipulators operating in unstructured environments. To translate these simulation findings into operational industrial hardware, the researchers proposed expanding future investigations toward experimental deployment via the Robot Operating System framework.

Subject of Research: Adaptive neuro-fuzzy control and mathematical dynamic modeling for robotic manipulator trajectory tracking

Article Title: A study of advanced mathematical modeling and adaptive control strategies for trajectory tracking in the Mitsubishi RV-2AJ 5-DOF Robotic Arm

Article References: Ben Hazem, Z., Guler, N., & Altaif, A. H. (2025). A study of advanced mathematical modeling and adaptive control strategies for trajectory tracking in the Mitsubishi RV-2AJ 5-DOF Robotic Arm. Discover Robotics, 1(1), Article 2. https://doi.org/10.1007/s44430-025-00001-5

Image Credits: AI Generated

DOI: 10.1007/s44430-025-00001-5

Keywords: Mitsubishi RV-2AJ, Robotic Manipulator, Trajectory Tracking, Euler-Lagrange Equations, Adaptive Neuro-Fuzzy Inference System, ANFIS, PID Controller, Particle Swarm Optimization, Denavit-Hartenberg Convention, Simscape Multibody, MATLAB Simulink, Jacobian Analysis, Forward Kinematics, Inverse Kinematics, Dynamic Modeling, Torque Regulation, Root Mean Square Error, Robot Operating System