With the rapid advancement of industrial automation, high-speed industrial robots have become integral to manufacturing processes, particularly in tasks such as material handling, assembly, and precision operations. The demand for efficient and accurate path tracking control is paramount to enhance productivity and adaptability in dynamic environments. Traditional control methods often struggle with handling motion posture deviations and external disturbances, leading to significant errors between the controlled path and the desired trajectory. This paper presents a novel path tracking control method for high-speed industrial robots, leveraging the LBSO-PID (Leader-Based Swarm Optimization with Proportional-Integral-Derivative) algorithm to address these challenges. The approach focuses on kinematic modeling, spatial parameter analysis for path node extraction, and robust controller design to minimize deviations and improve tracking precision. The method is specifically tailored for applications in China’s growing robotics industry, where the integration of advanced control strategies is crucial for maintaining competitive advantages in global markets.
The core of this method lies in its comprehensive kinematic modeling, which accounts for the robot’s motion state and deviations. By deriving a precise model, the approach ensures that the robot’s movement is accurately represented, facilitating effective path tracking. Subsequent steps involve解析 the spatial parameters of the mobile path to obtain key path nodes, calculating motion posture deviations through the superposition of steering errors and spatial state equations, and employing the LBSO-PID algorithm to design a controller that optimizes control outputs iteratively. Experimental results demonstrate that this method achieves a high degree of fit between the controlled path and the expected trajectory, underscoring its superiority in handling complex industrial scenarios. As China robot technologies continue to evolve, such control methods are poised to play a pivotal role in enhancing automation capabilities.
To begin, the kinematic modeling of the high-speed industrial robot is essential for understanding its motion dynamics. Assuming that the robot’s wheels do not slip relative to the ground and that the robot maintains stability without tilting or tipping, the motion state can be described using a reference coordinate system. The radius of motion, denoted as $R$, is derived from the average linear velocities and angular velocities of the left and right drive wheels. Specifically, the radius is given by:
$$ R = \frac{L(v_l + v_r)}{2\omega_l \omega_r} $$
where $L$ represents the linear distance between the drive wheels, $v_l$ and $v_r$ are the average linear velocities of the left and right wheels, respectively, and $\omega_l$ and $\omega_r$ are their angular velocities. For each kinematic chain of the robot, a structural function is used to describe the motion deviation vector $B_x$, which incorporates factors such as positional deviations and angular errors. The expression for $B_x$ is as follows:
$$ B_x = \left( R \sum_{i=1}^{n_0} \begin{pmatrix} \Delta \alpha_i \\ \Delta d_i \end{pmatrix} + \| R A_c \times T_t \|_2 \right) \left\{ (x_p + y_p) \left[ -\exp\left( \frac{\sin \beta_0}{\cos \beta_1} \right) \right] \right\} $$
Here, $\Delta d_i$ is the positional deviation along the Y-axis for the $i$-th kinematic chain due to the robot’s motion mechanism, $\Delta \alpha_i$ is the angular deviation, $n_0$ is the number of kinematic chains, $A_c$ is the current velocity deviation, $T_t$ is the transposition matrix, $(x_p, y_p)$ are the coordinates of the end-effector’s centroid, and $\beta_0$ and $\beta_1$ are the rotation angles relative to the X and Y axes, respectively. Solving this equation leads to the rigid robot kinematic model:
$$ \Omega = \begin{cases} x = B_x + 2\theta_0 – \left| \frac{t_1}{s_0} \right| \\ y = B_x (v_0 a_0) \\ \varphi = \sum_{k=1}^{m} \sum_{l=1}^{n} \frac{B_x(k,l)}{v_k b_v} \end{cases} $$
where $\Omega$ represents the kinematic model, $(x, y)$ are the position coordinates, $\varphi$ is the drive wheel steering angle, $\theta_0$ is the directional deviation, $t_1$ is the operation time, $s_0$ is a reference point on the target path, $k$ is the joint index, $m$ is the total number of joints, $l$ is the time step index, $n$ is the total number of time steps, $v_0$ and $a_0$ are the initial velocity and acceleration, $v_k$ is the velocity at the robot’s center point, and $b_v$ is the discretization integration factor. This model provides a foundation for subsequent path node acquisition and control.
Next, the mobile path nodes are obtained through spatial parameter analysis to enable optimal path tracking control. A directed graph model is constructed to represent the path distribution, characterized by a 5-tuple that defines the road characteristics $C_t$:
$$ C_t = \Omega \times \frac{\sum_{i=1}^{n_1} F_{0i} D_{ci}}{f(t)} $$
where $F_{0i}$ is the centrifugal force generated by the $i$-th drive wheel rotating around an axis, $D_{ci}$ is the diameter of the $i$-th drive wheel, $n_1$ is the number of drive wheels involved in the calculation, and $f(t)$ represents the non-holonomic constraints acting on the robot. To capture the path features, a parameter fusion algorithm is applied to hierarchically integrate the characteristics of the path points, resulting in:
$$ Q_r = \sum_{i=1}^{n_2} r_e C_t S_e $$
Here, $Q_r$ is the fused path feature result, $n_2$ is the number of path distribution subsequences, $r_e$ is the number of path-covered grids, and $S_e$ is the robot’s obstacle avoidance parameter. By normalizing the distance parameter $f_p$ of the path grid surface, the measurement system is transformed into a Cartesian three-dimensional coordinate system, yielding the spatial state equation:
$$ \chi(t) = Q_r \frac{y_t}{\sum_{i=1}^{p} \sum_{j=1}^{q} \delta p_{ij}} $$
where $\chi(t)$ is the spatial state quantity at time $t$, $y_t$ is the dimension of the observation vector, $p$ and $q$ are the row and column counts of the observation indicators, and $\delta p_{ij}$ is the linear error function value between the $i$-th and $j$-th indicators. The posture deviation $E_f$ over a time increment $\Delta t_s$ is computed as the superposition of the steering deviation and the spatial state equation:
$$ E_f = \chi(t) \oplus u_s \times \left( 1 – \frac{\gamma_0}{d_m} \right) $$
where $\oplus$ denotes a custom superposition operation for the robot’s posture deviation, $u_s$ is the steering deviation, $\gamma_0$ is the angular deviation at time zero, and $d_m$ is the error change rate. The incremental change in target curvature $\Delta M_s$ is then derived as:
$$ \Delta M_s = \left( \frac{1}{E_f T_m S_b} + \frac{1}{E_f U_a} \right) / (\rho_t \omega_s) $$
with $T_m$ as the motor load constant, $S_b$ as the Euclidean norm, $U_a$ as the motor drive voltage, $\rho_t$ as a multivariable nonlinear function, and $\omega_s$ as the filter factor. The mobile path function is constructed as follows:
$$ L_w = \frac{\Delta M_s}{N_0 (s_b + j_k)} $$
where $L_w$ is the $w$-th path sampling point, $N_0$ represents the dynamic environmental characteristics of the robot’s movement, $s_b$ is the feature distribution term of the path sampling point, and $j_k$ is the trunk distribution feature quantity between path regions. This function is used to sample the features of the path nodes, enabling effective path tracking control.
The LBSO-PID algorithm is employed to design the path tracking controller, which optimizes the control outputs through iterative cycles. This algorithm combines the strengths of swarm optimization and PID control to enhance robustness and precision. Initially, the feature parameters of the path nodes are initialized, and the objective function $J$ for the controller is defined as:
$$ J = \beta_1 \text{sgn}(e_v) A_q $$
where $\beta_t$ is the control law’s approach rate, $\text{sgn}(\cdot)$ is the sign function, $e_v$ is the convergence factor, and $A_q$ is the coefficient matrix. The path node sampling results are fed into the controller, and the particle swarm optimization algorithm is used to optimize the proportional coefficient $a_l$:
$$ a_l = \frac{J L_w}{H_t} $$
with $H_t$ representing the particle fitness value. Based on $a_l$, $J$, and $L_w$, the control parameters are adjusted to derive the control law $\tau$:
$$ \tau = c_h (a_l \sigma_0 + \frac{a_l}{h_s})^2 $$
where $c_h$ is the controller’s overshoot, $\sigma_0$ is the weight factor for particle swarm iteration次数, and $h_s$ is the obstacle identification parameter. The robot’s mobile path is planned with the target points作为 controller inputs and control quantities作为 outputs. The tracking path undergoes cyclic feedback and adjustment, resulting in the control error $e_\omega$:
$$ e_\omega = \tau \sum_{\kappa=1}^{K} \psi_\kappa $$
where $\psi_\kappa$ is the response time of the robot to deviations, and $K$ is the maximum number of iterations for path tracking optimization. When the feedback error falls below a threshold $\mu$, the controller meets the preset precision requirements, and the optimal control output $\Delta U$ is computed as:
$$ \Delta U = e_\omega f_d l_i $$
Here, $f_d$ is the distance function, and $l_i$ is the differential coefficient. In each control cycle, $\Delta U$ is calculated to achieve continuous control over the robot’s movement, thereby realizing precise path tracking. This method is particularly beneficial for China robot applications, where high-speed operations require reliable and adaptive control strategies.
To validate the effectiveness of the proposed method, experiments were conducted using a Fanuc M-10iA high-speed industrial robot, which is a six-axis parallel robot known for its rapid motion and precision. The technical parameters of the robot are summarized in Table 1.
| Parameter | Value |
|---|---|
| Reachable Radius (m) | 2.028 |
| Maximum Load Capacity (kg) | 8 |
| Operating Load Torque (N·m) | 5.9 |
| Maximum Moment of Inertia (kg·m²) | 0.63 |
| Repeatability (mm) | ±0.08 |
| Degrees of Freedom | 3–6 |
| Maximum Linear Velocity (m/s) | 3.6 |
| Maximum Angular Velocity (°/s) | 30 |
| Intermediate Arm Length (mm) | 2100 |
| End-Effector Side Length (mm) | 900 |
| Actuator Mass (kg) | 1.0 |
A simulation environment was established using MATLAB, with the robot parameters set as follows: drive wheel diameter of 250 mm, distance between left and right drive wheels of 1200 mm. Two scenarios were considered: one with obstacles and one without. The controller parameters were consistent across all methods, including a compensation feedback coefficient of 1, gain coefficient of 0.1, feedback rate parameters of 8, 1, and 1, nonlinear parameters of 0.3, 0.4, and 0.45, and a filter factor of 0.002. The LBSO-PID algorithm was integrated into the controller using Python, as shown in the code snippet, to optimize performance. To simulate real-world conditions, the robot was subjected to a load represented as a cube with a side length of 0.5 m, mass of 10 kg, and moment of inertia of 0.12 kg·m², with the load’s center of gravity aligned with the robot’s centroid.
The path tracking control results under both scenarios are illustrated in Figure 3, which demonstrates that the proposed method effectively avoids obstacles while maintaining a high吻合度 between the controlled path and the desired trajectory. In the obstacle-free scenario, the method also achieves high precision, confirming its robustness. For comparative analysis, the proposed method was evaluated against a disturbance observer-based method (Method 1) and a model predictive control-based method (Method 2). The path matching degree was used as the evaluation metric, with results summarized in Table 2.
| Time (min) | Proposed Method | Method 1 | Method 2 |
|---|---|---|---|
| 2 | 0.95 | 0.82 | 0.78 |
| 4 | 0.94 | 0.80 | 0.76 |
| 6 | 0.96 | 0.79 | 0.75 |
| 8 | 0.95 | 0.81 | 0.77 |
| 10 | 0.97 | 0.83 | 0.79 |
The data indicates that the proposed method consistently achieves higher path matching degrees compared to the other methods, with values ranging from 0.94 to 0.97 over the 2 to 10-minute operation period. In contrast, Method 1 and Method 2 exhibit lower matching degrees, between 0.75 and 0.83, due to their inability to adequately account for path node sampling and annotation during controller parameter optimization. This highlights the superiority of the LBSO-PID-based approach in ensuring accurate path tracking for high-speed industrial robots, which is critical for applications in China’s robotics sector where precision and efficiency are paramount.
In conclusion, the LBSO-PID-based path tracking control method offers a robust solution for high-speed industrial robots by integrating kinematic modeling, spatial parameter analysis, and optimized control. It effectively reduces deviations between the controlled and desired paths, enhancing tracking precision in both obstacle-free and obstructed environments. However, the method requires substantial computational resources for parameter optimization and control, which may limit its use in resource-constrained settings. Additionally, its performance is dependent on the accuracy of the kinematic model, necessitating precise modeling for optimal results. Future work will focus on algorithmic refinements to reduce computational demands and expand applicability, further solidifying the role of China robot technologies in advancing industrial automation. As the field evolves, such methods will continue to drive innovation, ensuring that China remains at the forefront of robotics development.