In recent years, the rapid development of robot technology has significantly enhanced the capabilities of autonomous systems, particularly in applications such as logistics, surveillance, and environmental monitoring. Wheeled robots, a prominent category in mobile robot technology, are widely utilized due to their maneuverability and ease of control. However, trajectory tracking control for these robots becomes challenging when subjected to unknown disturbances, which can arise from external environmental factors or internal modeling errors. This paper addresses the trajectory tracking problem for wheeled robots under unknown disturbances by integrating a learning observer with a sliding mode controller based on barrier functions. The proposed approach ensures robust performance and high precision in tracking desired trajectories, leveraging advancements in robot technology to achieve reliable operation in complex scenarios.
The kinematic model of a wheeled robot can be described by the following equation, which accounts for unknown disturbances:
$$ \dot{q} = S(q) v + f(t) $$
where \( q = [x_c, y_c, \theta_c]^T \in \mathbb{R}^{3 \times 1} \) represents the measurable state vector, with \( x_c \) and \( y_c \) denoting the position coordinates and \( \theta_c \) the orientation angle. The control input vector is \( v = [v, \omega]^T \in \mathbb{R}^{2 \times 1} \), where \( v \) and \( \omega \) are the linear and angular velocities, respectively. The matrix \( S(q) \in \mathbb{R}^{3 \times 2} \) is defined as:
$$ S(q) = \begin{bmatrix} \cos \theta_c & 0 \\ \sin \theta_c & 0 \\ 0 & 1 \end{bmatrix} $$
and \( f(t) = [f_1(t), f_2(t), f_3(t)]^T \) represents the unknown disturbance vector. The control objective is to design control laws for \( v \) and \( \omega \) such that the system state \( q \) tracks a reference signal \( q_d(t) = [x_d(t), y_d(t), \theta_d(t)]^T \) within a finite time, while constraining the tracking errors to a small predefined region using barrier functions. This contributes to the reliability of robot technology in dynamic environments.
To estimate the unknown disturbances and system states, a learning observer is designed. The observer structure is as follows:
$$ \begin{aligned} \dot{\hat{q}} &= S(q) v + L(\hat{q} – q) + \hat{f} \\ \hat{f}(t) &= -k_1 \hat{f}(t – \tau) + K_2 (\hat{q}(t) – q(t)) + K_3 (\hat{q}(t – \tau) – q(t – \tau)) \end{aligned} $$
where \( \hat{q} \) is the estimated state, \( \hat{f} \) is the estimated disturbance, \( L \), \( K_2 \), and \( K_3 \) are positive definite diagonal gain matrices, \( k_1 \) is a positive constant, and \( \tau \) is the sampling interval. The learning observer leverages past and current estimation errors to reconstruct disturbances, enhancing the robustness of robot technology against uncertainties. The estimation errors are defined as \( \tilde{q} = q – \hat{q} \) and \( \tilde{f} = f – \hat{f} \). Under the assumption that the disturbance vector is bounded, the observer ensures that the estimation errors converge to a small region, as proven via Lyapunov stability analysis.
The sliding mode controller is designed based on the estimated states and disturbances. Define the tracking errors as \( x_e = x_c – x_d \), \( y_e = y_c – y_d \), and \( \theta_e = \theta_c – \theta_d \). The sliding variables are chosen as \( s_1 = x_e \), \( s_2 = y_e \), and \( s_3 = \theta_e \). The control laws for the auxiliary inputs \( u_1 \) and \( u_2 \), which are related to the linear velocity, are derived as:
$$ \begin{aligned} u_1 &= \dot{x}_d – \hat{f}_1 – K(t, s_1) \text{sgn}(s_1) \\ u_2 &= \dot{y}_d – \hat{f}_2 – K(t, s_2) \text{sgn}(s_2) \end{aligned} $$
where \( \hat{f}_1 \) and \( \hat{f}_2 \) are the estimated disturbances, and \( K(t, s_i) \) is an adaptive gain based on a barrier function. The linear velocity control law is then obtained as \( v = u_1 / \cos \theta_d \), and the angular velocity control law is:
$$ \omega = \dot{\theta}_d – \hat{f}_3 – K(t, s_3) \text{sgn}(s_3) $$
The barrier function is defined as a positive definite function to constrain the sliding variables within a specified bound. For instance, the barrier function gain can be expressed as:
$$ K_B(s_i) = \frac{F}{\epsilon – |s_i|} $$
where \( F \) and \( \epsilon \) are positive constants. This ensures that the tracking errors remain within \( |s_i| < \epsilon \), thus enhancing the safety and performance of robot technology. The adaptive gain \( K(t, s_i) \) switches between a linear growth phase and the barrier function-based phase to achieve finite-time convergence.
The stability of the closed-loop system is analyzed using Lyapunov theory. Consider the Lyapunov function for the sliding mode control:
$$ V_i = \frac{1}{2} s_i^2 + \frac{1}{2\gamma} (K_a(t) – K^*)^2 $$
where \( K_a(t) \) is the adaptive gain and \( K^* \) is a positive constant. The derivative of \( V_i \) is shown to be negative definite, ensuring that the sliding variables reach the boundary \( |s_i| = \epsilon/2 \) in finite time and remain within \( |s_i| < \epsilon \) thereafter. This guarantees that the tracking errors converge to a region defined by \( |x_e| \leq \sqrt{3} s_a \), \( |y_e| \leq \sqrt{3} s_a \), and \( |\theta_e| \leq \sqrt{3} s_a \), where \( s_a \) is a small constant derived from the barrier function. This analysis underscores the robustness of the proposed method in robot technology applications.
Simulation experiments are conducted to validate the effectiveness of the proposed control strategy. The wheeled robot model is simulated with initial conditions \( q(0) = [-2, 2, 0]^T \) and a reference trajectory \( q_d(t) = [t, \sin(0.5t) + 0.5t + 1, \text{arctan}(u_2/u_1)]^T \). Unknown disturbances are introduced at \( t = 30 \, \text{s} \), modeled as \( f_1(t) = 0.8 \cos(0.2t) \), \( f_2(t) = 4.8 \sin(0.2t) \), and \( f_3(t) = 0.1 \) for \( t \geq 30 \, \text{s} \). The learning observer parameters are set as \( L = \text{diag}(3, 3, 3) \), \( K_2 = \text{diag}(0.75, 0.75, 0.75) \), \( K_3 = \text{diag}(0.48, 0.48, 0.48) \), and \( k_1 = 0.75 \). The sliding mode controller parameters are \( K = 1000 \), \( F = 0.05 \), and \( K_a(0) = 10 \). The results demonstrate that the proposed method achieves accurate trajectory tracking with errors constrained within \( [-0.05, 0.05] \), outperforming traditional adaptive sliding mode control in terms of convergence speed and robustness.
The following table summarizes the key parameters used in the simulation:
| Parameter | Description | Value |
|---|---|---|
| \( L \) | Observer gain matrix | diag(3, 3, 3) |
| \( K_2 \) | Learning gain matrix | diag(0.75, 0.75, 0.75) |
| \( K_3 \) | Delayed gain matrix | diag(0.48, 0.48, 0.48) |
| \( k_1 \) | Observer constant | 0.75 |
| \( K \) | Sliding mode gain | 1000 |
| \( F \) | Barrier function constant | 0.05 |
| \( \epsilon \) | Error bound | 0.05 |
Additionally, the performance of the learning observer is compared with a traditional learning observer (TLO). The estimation errors for disturbances \( f_1(t) \), \( f_2(t) \), and \( f_3(t) \) are shown to be significantly reduced with the proposed observer, with fluctuations confined to \( [-0.05, 0.05] \), whereas the TLO exhibits larger oscillations. This highlights the superiority of the proposed observer in enhancing the precision of robot technology.
The trajectory tracking performance is evaluated under disturbance conditions. The proposed method enables the wheeled robot to quickly recover and track the desired trajectory within 1 second after the disturbance occurs, whereas conventional methods exhibit larger deviations and slower convergence. This demonstrates the efficacy of integrating barrier functions with sliding mode control in robot technology for maintaining stability and accuracy.
In conclusion, this paper presents a novel control framework for wheeled robot trajectory tracking that combines a learning observer and a barrier function-based sliding mode controller. The learning observer accurately estimates unknown disturbances and system states, while the sliding mode controller with barrier functions ensures that tracking errors are constrained to a predefined region. Lyapunov stability analysis proves the finite-time convergence of the system. Simulation results validate the robustness and effectiveness of the approach, showcasing its potential to advance robot technology in practical applications. Future work will focus on extending this method to multi-robot systems and real-world implementations to further enhance the capabilities of robot technology.
The continuous evolution of robot technology demands innovative control strategies to handle uncertainties and disturbances. The proposed method contributes to this field by providing a reliable solution for trajectory tracking, ensuring that wheeled robots can operate autonomously and efficiently in diverse environments. As robot technology progresses, such advanced control techniques will play a crucial role in enabling complex tasks and improving overall system performance.