In the field of advanced robotics, the integration of robot technology has revolutionized automation, enabling mobile robots to perform complex tasks in dynamic environments. As a researcher focused on enhancing robotic systems, I have observed that precise control of mobile robots remains a critical challenge, particularly when dealing with joint dynamics. Joint angular acceleration is a fundamental parameter that influences the stability, safety, and efficiency of robot operations. In this study, I propose a fuzzy control method that incorporates constraints on joint angular acceleration to improve the performance of mobile robots. By leveraging fuzzy logic and adaptive gain adjustments, this approach aims to mitigate issues like overshoot and instability, which are common in traditional control methods. The growing importance of robot technology in applications such as logistics, surveillance, and industrial automation underscores the need for robust control strategies. Through mathematical modeling, simulations, and experimental validation, I demonstrate how this method can achieve accurate trajectory tracking and rapid stabilization. This work contributes to the broader advancement of robot technology by addressing key limitations in existing control systems, paving the way for more reliable and intelligent robotic solutions.
To begin, I calculate the joint angular acceleration of a mobile robot using its kinematic equations. Let \( p = (\alpha_1, \alpha_2, \cdots, \alpha_n) \) represent the joint drive variables, where \( \alpha_i \) denotes the angle of the \( i \)-th joint, and \( n \) is the total number of joints. The Jacobian matrix \( J \) and the velocity vector \( V = (v_1, v_2, \cdots, v_n) \) are related by the equation \( p = V J \). The joint angle \( \alpha_i \) can be expressed as \( \alpha_i = J_i V \), where \( J_i \) is the Jacobian matrix for the \( i \)-th joint. The kinematic equation of the mobile robot is given by \( V = (\alpha_1, \alpha_2, \cdots, \alpha_n) J \). From this, the joint angular velocity \( \dot{\alpha} \) is derived as \( \dot{\alpha} = V J \). Subsequently, the joint angular acceleration \( \ddot{\alpha} \) is computed using the formula \( \ddot{\alpha} = (r – \dot{J} \dot{\alpha}) J \), where \( \dot{J} \) is the time derivative of \( J \), and \( r \) represents the acceleration of the mobile robot. This calculation is essential for understanding the dynamic behavior of the robot and forms the basis for implementing constraints in the control system. The integration of robot technology in this context allows for real-time monitoring and adjustment of joint parameters, ensuring that the robot operates within safe limits. For instance, in high-speed applications, excessive joint angular acceleration can lead to mechanical failure or trajectory deviations, highlighting the importance of accurate computation.
The fuzzy control method is designed with a constraint on the error between the calculated joint angular acceleration \( \ddot{\alpha} \) and the desired joint angular acceleration \( \ddot{\alpha}_d \), defined as \( e = \ddot{\alpha} – \ddot{\alpha}_d \). This constraint ensures that the control system responds adaptively to dynamic changes, enhancing the stability and precision of the mobile robot. The control law for fuzzy gain adjustment is formulated as \( u = \hat{\beta} \ddot{\alpha}_r + \hat{d} – A (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d)) – Q \), where \( \hat{\beta} \) is the estimated friction coefficient, \( \hat{d} \) is the estimated external disturbance, \( \dot{e} \) is the derivative of the error, \( A \) is a parameter influencing the robot’s dynamic performance, \( \ddot{\alpha}_r \) is a temporary variable for joint angular acceleration, \( \gamma \) is a coefficient matrix for the switching function, and \( Q \) is the switching gain. The term \( \dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) \) serves as the error sliding mode switching function in the fuzzy control system. To evaluate stability, I use the Lyapunov function \( F = 0.5 (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d))^T M (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d)) \), where \( M \) is the inertia matrix of the mobile robot. Differentiating this function and substituting the control law yields \( \dot{F} = (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d))^T (\Delta k – Q) – (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d))^T (\beta + A) (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d)) \), where \( \Delta k = \Delta M \ddot{\alpha}_r + \Delta \beta \dot{\alpha}_r + \Delta d \), with \( \Delta M = \hat{M} – M \), \( \Delta \beta = \hat{\beta} – \beta \), and \( \Delta d = \hat{d} – d \). Here, \( \hat{M} \) is the estimated inertia matrix, and \( \dot{\alpha}_r \) is a temporary variable for joint angular velocity. For \( \dot{F} \) to be negative, indicating stability, the condition \( (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d)) Q \geq 0 \) must hold, ensuring that the sign is consistent. This analysis guides the design of a fuzzy adaptive gain adjustment controller, where \( Q \) is dynamically adjusted based on the input \( \dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) \) using fuzzy logic. The fuzzy system employs a rule base with subsets {NB, NS, ZE, PS, PB} representing {Negative Big, Negative Small, Zero, Positive Small, Positive Big}. The membership function \( \mu(\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d)) \) is defined piecewise: when \( \dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) \leq -\xi \), \( \mu = 0 \); when \( -\xi \leq \dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) \leq B – \xi \), \( \mu = \frac{(\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) + \xi)(1 + C)}{B – (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) – \xi)} \); when \( B – \xi < B’ – \xi \), \( \mu = B + \frac{(\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) + B’ – \xi)(1 + C B)}{1 + C |(\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) + \xi – B|} \); and when \( \dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d) > B’ – \xi \), \( \mu = 1 \). Here, \( \xi \), \( C \), \( B \), and \( B’ \) are control parameters. The fuzzy approximation of \( Q \) is given by \( Q^* = \frac{\sum_{\tau=1}^N \phi_\tau \prod_{\tau=1}^N \mu_\tau(\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d))}{\sum_{\tau=1}^N \prod_{\tau=1}^N \mu_\tau(\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d))} \), where \( N \) is the number of fuzzy rules, \( \phi_\tau \) is the adjustable parameter vector for the \( \tau \)-th rule, and \( \mu_\tau \) is the membership function. Defuzzification using the weighted average method yields the precise approximation \( Q’ = \frac{\sum_{\tau=1}^N \mu_\tau(\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d)) \cdot Q}{\sum_{\tau=1}^N \mu_\tau(\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d))} \). Finally, the adaptive control law is derived as \( u^* = (\dot{e} + \gamma (\ddot{\alpha} – \ddot{\alpha}_d)) Q’ \), which enables precise control of the mobile robot’s motion trajectory, posture, and speed. This fuzzy control approach exemplifies the integration of advanced robot technology to handle uncertainties and disturbances in real-world environments.
In the experimental phase, I applied this method to a mobile robot to evaluate its performance. The robot’s parameters are summarized in Table 1, which includes dimensions, mass, and operational specifications. These parameters are critical for configuring the control system and ensuring that the joint angular acceleration constraints are properly enforced. The use of robot technology in this setup allows for seamless communication between sensors, actuators, and the control unit, facilitating real-time adjustments. For example, the laser positioning sensor with a maximum detection distance of 50 m and an angular resolution of 0.1° provides high-precision data for trajectory tracking. The rigid and flexible arms enable the robot to perform tasks in unstructured environments, such as overcoming obstacles up to 50 mm in height. The computed joint angular acceleration values over time are shown in Figure 2, indicating a gradual increase until stabilizing around ±20 rad/s². This behavior demonstrates the effectiveness of the calculation method in capturing the robot’s dynamic characteristics. Furthermore, the fuzzy control results under no external disturbances are presented in Figure 3, where the set speed was 120 mm/s. The trajectory tracking closely matches the target path, and the speed stabilizes near the set value within approximately 3 seconds, with no overshoot. This rapid and stable response highlights the superiority of the proposed method over traditional approaches, which often suffer from oscillations or delays. The incorporation of robot technology in the control loop ensures that the system can adapt to varying conditions, such as changes in load or terrain, thereby enhancing overall reliability. The experimental validation confirms that the fuzzy control method with joint angular acceleration constraints significantly improves the robot’s performance, making it suitable for applications in fields like autonomous navigation and industrial automation.
| Parameter Name | Value |
|---|---|
| Length (mm) | 300 |
| Width (mm) | 200 |
| Height (mm) | 500 |
| Total Mass (kg) | 3.5 |
| Wireless Control Distance (m) | 10 |
| Mobile Positioning Accuracy (mm) | ±5 |
| Obstacle Height in Unstructured Environment (mm) | 50 |
| Moving Speed on Flat Terrain (mm/s) | 120 |
| Laser Positioning Sensor Max Detection Distance (m) | 50 |
| Laser Positioning Sensor Angular Resolution (°) | 0.1 |
| Rigid Manipulator Length (mm) | 550 |
| Flexible Probe Arm Length (mm) | 300 |
The results from the experiments underscore the efficacy of the proposed fuzzy control method in managing joint angular acceleration constraints. By continuously monitoring and adjusting the control gains based on fuzzy logic, the system maintains stability even in the presence of internal and external perturbations. This adaptability is a hallmark of modern robot technology, which emphasizes intelligent decision-making and autonomous operation. In comparison to conventional methods like sliding mode control or phase control, which may exhibit sensitivity to noise or model inaccuracies, the fuzzy approach reduces overshoot and ensures smoother trajectories. For instance, the error \( e = \ddot{\alpha} – \ddot{\alpha}_d \) is minimized through the adaptive control law, leading to precise motion control. The mathematical framework presented here, including the Lyapunov stability analysis, provides a solid foundation for further developments in robot technology. As robotics continues to evolve, incorporating elements like machine learning and IoT, methods like this will play a pivotal role in creating more resilient and efficient systems. Future work could explore the integration of additional sensors or multi-robot coordination to enhance scalability. Overall, this study demonstrates that constraining joint angular acceleration through fuzzy control not only addresses immediate challenges in mobile robotics but also contributes to the long-term advancement of robot technology.
In conclusion, the fuzzy control method considering joint angular acceleration constraints offers a robust solution for enhancing the performance of mobile robots. Through detailed mathematical modeling and experimental validation, I have shown that this approach enables accurate trajectory tracking, rapid speed stabilization, and improved stability. The use of robot technology in implementing adaptive gain adjustments and fuzzy logic rules ensures that the system can handle real-world complexities effectively. As the demand for autonomous systems grows, such innovations will be crucial in pushing the boundaries of what robot technology can achieve. This work not only addresses specific control challenges but also opens avenues for future research in intelligent robotic systems.