In the era of “Made in China 2025,” industrial robots have emerged as pivotal components in intelligent manufacturing, experiencing rapid advancement in recent years. Among these, Delta parallel robots are extensively utilized in high-speed sorting automation lines within the food industry due to their high velocity and low inertia. Compared to traditional food sorting methods, automated sorting offers advantages such as enhanced efficiency and reduced error rates. However, during the sorting process, parallel food sorting robots often face challenges like poor positioning accuracy and low efficiency of the end effector. These issues can significantly impact sorting performance, leading to inefficiencies in production lines. Therefore, developing an intelligent control method that ensures precise, efficient, and stable position tracking for the end effector is crucial for optimizing food sorting applications.
To address these limitations, I propose an intelligent control approach for the end effector of a parallel food sorting robot that integrates a fuzzy system (FS), fuzzy neural network (FNN), and backstepping control algorithm (BCA). In this method, modeling information is approximated by the FS, unmodeled information is approximated and predicted by the FNN, and the BCA generates the control output. This combined strategy aims to enhance the tracking accuracy and control efficiency of the end effector, ultimately improving the overall sorting performance. The core innovation lies in leveraging the strengths of FS for handling known dynamics and FNN for managing uncertainties, all within a robust BCA framework. This paper details the methodology, experimental validation, and results, demonstrating the effectiveness of the proposed control method.
The parallel food sorting robot consists of servo motors, a static platform, driving arms, intermediate shafts, a moving platform, and driven arms. The intermediate shafts transmit power to the moving platform to control the end effector, while the driven arms, structured as parallelograms, constrain the degrees of freedom of the moving platform, allowing only three-dimensional transformations in space. This configuration is ideal for high-speed operations but requires precise control to maintain end effector accuracy. The dynamics of the robot can be modeled to facilitate control design. The kinetic model is expressed as:
$$ M\ddot{U} + C\dot{U} + KU + G + \delta_d = \tau_i – \tau_f $$
where \( M \) is the generalized mass matrix, \( C \) is the damping matrix, \( K \) is the stiffness matrix, \( G \) is the gravity matrix, \( \ddot{U} \) is acceleration, \( \dot{U} \) is velocity, \( U \) is position coordinates, \( \tau_i \) is the generalized input matrix, \( \tau_f \) is the friction model, and \( \delta_d \) is the disturbance term. This model exhibits key properties essential for control design:
Property 1: \( M \) is bounded, with \( 0 < \lambda_{\min}(M) \leq \|M\| \leq \lambda_{\max}(M) \).
Property 2: \( C\dot{U} \) is bounded, satisfying \( 0 < C_{\min}\|\dot{U}\|^2 \leq \|C\dot{U}\| \leq C_{\max}\|\dot{U}\|^2 \).
Property 3: \( M – 2C \) is a skew-symmetric matrix, so \( (M – 2C) + (M – 2C)^T = 0 \).
Property 4: \( G \) is bounded, with \( \|G\| \leq G_g \), where \( G_g \) is a positive constant.
The proposed control method builds upon the backstepping control algorithm (BCA), which decomposes a complex nonlinear system into subsystems no larger than the system order. It designs Lyapunov functions and virtual control variables for each subsystem, “backstepping” through the system until the control law is derived. For the end effector control, the goal is to ensure the output \( y_o \) tracks the desired angle \( y_d \). Defining the error \( z_1 = y_o – y_d \), and letting \( \alpha_1 \) be an estimate of \( x_2 \), with error \( z_2 = x_2 – \alpha_1 \), the virtual control is chosen as \( \alpha_1 = -\lambda_1 z_1 + \dot{y}_d \), where \( \lambda_1 > 0 \). The Lyapunov function for the first subsystem is \( V_{11} = \frac{1}{2} z_1^T z_1 \), leading to \( \dot{V}_{11} = z_1^T \dot{z}_1 = z_1^T (z_2 + \alpha_1 – \dot{y}_d) = -\lambda_1 z_1^T z_1 + z_1^T z_2 \). Stability is achieved if \( z_2 = 0 \). The control law is then derived as:
$$ \tau = \begin{bmatrix} \tau_1 \\ \tau_2 \end{bmatrix} $$
with \( \tau_1 = -\phi – \lambda_2 z_2 – z_1 \), where \( \phi \) approximates nonlinear functions, and \( \tau_2 = \chi_1 n_{em} + \chi_2 n_{im} \), representing unmodeled internal and external information. For practical control, \( \tau_2 \) is replaced by a control effect vector \( U_a \in \mathbb{R}^{n \times 1} \).
To approximate modeling information, a fuzzy system (FS) is employed. The FS captures the fuzzy characteristics of human brain thinking and excels in describing high-level knowledge. For the nonlinear function \( f = -\dot{\alpha}_1 (C + M) \), which contains modeling information, the FS output is defined as \( y_o = \frac{\sum_{i=1}^N \theta_i \prod_{j=1}^n \mu_j^i(x_j)}{\sum_{i=1}^N \prod_{j=1}^n \mu_j^i(x_j)} = \xi^T(x) \theta \), where \( \xi = [\xi_1(x), \xi_2(x), \dots, \xi_N(x)]^T \), \( \theta = [\theta_1, \theta_2, \dots, \theta_N]^T \), and \( \theta_i \) is the response value when the fuzzy membership function reaches its maximum. Defining \( \Phi = [\phi_1, \phi_2]^T = \begin{bmatrix} \xi_1^T & 0 \\ 0 & \xi_2^T \end{bmatrix} \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix} = \xi^T(x) \theta \), the optimal approximation constant \( \theta^* \) ensures \( \|f – \Phi^*\| \geq \varepsilon \) for an infinitesimal constant \( \varepsilon > 0 \). The adaptive control law is \( \dot{\theta} = \gamma [z_2^T \xi^T(x)]^T – 2\kappa \theta^* \), where \( \gamma > 0 \) and \( \kappa \) are control coefficients.
For approximating unmodeled information, a fuzzy neural network (FNN) is utilized to predict and handle uncertainties that significantly affect stability. The FNN structure comprises four layers: input layer, membership layer, rule layer, and output layer. The input is the tracking error vector, and the output is the control effect vector for the end effector. The membership function uses Gaussian functions: \( \mu_{ji}(z_i) = \exp[-(z_1^i – m_{ji})^2 / (\sigma_{ji})^2] \), where \( m_{ji} \) and \( \sigma_{ji} \) are the mean and standard deviation, respectively. The rule layer output is \( l_k = \prod_{i=1}^n \omega_{kji} \mu_{ji}(z_1^i) \), and the output layer gives \( y_o = \sum_{k=1}^{N_y} \omega_{ok} l_k \). In vector form, \( y_o = [y_1, y_2, \dots, y_{N_o}]^T = W L = U_{IBFN}(z_1, W, m, \sigma) \). The actual control quantity is \( U_{BCA} = U_{IBFN}^*(z_1, W^*, m^*, \sigma^*) = W^* L^* + \varepsilon \), where \( \varepsilon \) is the reconstruction error vector. The FNN-BCA output control is \( U_a = U_{IBFN}'(z_1, W’, m’, \sigma’) = W’ L’ \), with approximation error \( \tilde{U} = U_{BCA} – U_a = \tilde{W} L^* + W’ \tilde{L} + \varepsilon \). Using Taylor series expansion, \( \tilde{L} = [\tilde{l}_1, \tilde{l}_2, \dots, \tilde{l}_{N_y}]^T = L_m \tilde{m} + L_\sigma \tilde{\sigma} + o_{nv} \), where \( \tilde{m} = m^* – m’ \), \( \tilde{\sigma} = \sigma^* – \sigma’ \), and \( o_{nv} \) is a higher-order term vector.
The overall control framework integrates FS, FNN, and BCA for intelligent control of the end effector. Based on FS-BCA and FNN-BCA designs, the FS approximates modeling information, the FNN handles unmodeled information through approximation and prediction, and the BCA serves as the main controller for the parallel robot system to generate output, with \( \tau’ = [\tau_1, U_{IBFN}’]^T \). The control system block diagram illustrates this integration, where \( q_d \) and \( q \) are desired and actual drive angles, \( \chi_1 \) and \( \chi_2 \) are approximation coefficients, \( \lambda_1 \) and \( \lambda_2 \) are Lagrange multipliers, and \( \eta_1, \eta_2, \eta_3, \eta_4 \) are learning rates.
To validate the feasibility and practicality of the proposed control method, experimental comparisons were made with traditional methods such as adaptive active disturbance rejection control (ADRC) and self-learning interval type-2 fuzzy neural network adaptive fuzzy sliding mode control (ST2FNN). The control objective was to enable the end effector of the parallel food sorting robot to track desired trajectories accurately, efficiently, and stably. The Delta robot employed a GTS-400-PV series motion controller, Lexium 23M AC servo drives, BCH1303M11F1C servo motors, and PLX60-5 reducers. A Basler acA2500-14gc camera was used for vision, and a PC with an Intel i5 processor, 8GB RAM, Windows 10, and MATLAB 2018a served as the simulation platform. The FNN network structure was set as 2-20-25-2, with control parameters \( \gamma = 2 \), \( \kappa = 1.5 \), and learning rates \( \eta_1 = 0.4 \), \( \eta_2 = 1 \), \( \eta_3 = 1 \), \( \eta_4 = 0.01 \). The three drive joints are denoted as J1, J2, and J3. Experimental parameters are summarized in Table 1.
| Parameter | Unit | Value |
|---|---|---|
| Length of driving/driven arm | m | 0.2 / 0.5 |
| Maximum acceleration of end effector | m/s² | 50.13 |
| Maximum velocity of end effector | m/s | 2.3 |
| Radius of end effector | m | 0.085 |
| Radius of static platform | m | 0.15 |
| Test object (canned food) | kg | 0.2 |
The experimental results demonstrate the superiority of the proposed control method in terms of driving torque, position error, and sorting efficiency for the end effector. Table 2 compares the maximum transient and steady-state driving torques for different control methods. The proposed method shows lower torque ranges across all drive joints, reducing the impact on driving components and improving dynamic driving characteristics. For instance, compared to ADRC, the average reductions in maximum transient and steady-state driving torques are approximately \( 2.8 \times 10^4 \) N·mm and \( 4.6 \times 10^4 \) N·mm, respectively. Compared to ST2FNN, the average reductions are about \( 3.0 \times 10^4 \) N·mm and \( 1.5 \times 10^4 \) N·mm. These reductions highlight the efficiency of the proposed method in minimizing energy consumption and mechanical stress on the end effector.
| Drive Joint | ADRC Transient | ADRC Steady-State | ST2FNN Transient | ST2FNN Steady-State | Proposed Method Transient | Proposed Method Steady-State |
|---|---|---|---|---|---|---|
| J1 | 0.70 | 0.38 | 0.70 | 0.39 | 0.40 | 0.23 |
| J2 | 0.66 | 0.34 | 0.69 | 0.36 | 0.38 | 0.21 |
| J3 | 0.64 | 0.32 | 0.66 | 0.34 | 0.39 | 0.19 |
Table 3 presents the maximum position errors at the center point of the end effector in the X, Y, and Z directions. The proposed method significantly reduces these errors compared to ADRC and ST2FNN. For example, the average reduction in maximum position error is 0.44 mm relative to ADRC and 0.43 mm relative to ST2FNN. This indicates enhanced tracking accuracy for the end effector, which is critical for precise sorting operations. The errors are maintained below 0.3 mm, demonstrating the method’s capability to achieve high precision in end effector positioning.
| End Effector Position | ADRC | ST2FNN | Proposed Method |
|---|---|---|---|
| X | 0.65 | 0.66 | 0.24 |
| Y | 0.67 | 0.64 | 0.27 |
| Z | 0.82 | 0.80 | 0.30 |
To simulate real-world conditions, the camera capture frequency was set to 100 fps, with a conveyor speed of 200 mm/s. A total of 500 canned food items were used as targets to ensure consistency in food type, distribution density, and grasping speed across trials. Table 4 compares the grasping success rates and efficiencies for different control methods. At a conveyor speed of 100 mm/s, the proposed method achieves a grasping success rate of 100% and an efficiency of 1.99 items/s, outperforming ADRC (94.00% and 1.68 items/s) and ST2FNN (99.40% and 1.93 items/s). As the conveyor speed increases to 200 mm/s, the grasping success rate for the proposed method slightly decreases to 99.60%, with an efficiency of 1.94 items/s, still higher than the other methods. These results underscore the robustness and high performance of the proposed control method in dynamic sorting environments for the end effector.
| Algorithm | Conveyor Speed (mm/s) | Food Items Transported | Food Items Grasped | Grasping Success Rate (%) | Grasping Efficiency (items/s) |
|---|---|---|---|---|---|
| ADRC | 100 | 500 | 470 | 94.0 | 1.68 |
| ADRC | 200 | 500 | 460 | 92.0 | 1.62 |
| ST2FNN | 100 | 500 | 497 | 99.4 | 1.93 |
| ST2FNN | 200 | 500 | 495 | 99.0 | 1.88 |
| Proposed Method | 100 | 500 | 500 | 100.0 | 1.99 |
| Proposed Method | 200 | 500 | 498 | 99.6 | 1.94 |
The effectiveness of the proposed control method can be further analyzed through mathematical formulations. The Lyapunov stability analysis ensures that the system remains stable under the proposed control. For the overall system, consider the combined Lyapunov function \( V = V_{11} + \frac{1}{2} z_2^T M z_2 \). Taking the derivative and substituting the control laws, we obtain:
$$ \dot{V} = -\lambda_1 z_1^T z_1 – \lambda_2 z_2^T z_2 + z_2^T (f – \phi) – z_2^T d $$
Given that \( \phi \) approximates \( f \) and \( d \) is bounded, proper tuning of \( \lambda_1 \) and \( \lambda_2 \) ensures \( \dot{V} \leq 0 \), guaranteeing asymptotic stability for the end effector tracking. Additionally, the adaptive laws for FS and FNN parameters contribute to robustness against uncertainties. For instance, the update rule for FS parameters \( \theta \) is derived from gradient descent minimization of the approximation error, expressed as:
$$ \Delta \theta = -\eta \frac{\partial E}{\partial \theta} $$
where \( E = \frac{1}{2} \| f – \Phi \|^2 \) is the error function, and \( \eta \) is the learning rate. Similarly, for FNN weights \( W \), the update rule follows:
$$ \Delta W = -\eta \frac{\partial \tilde{U}^T \tilde{U}}{\partial W} $$
These adaptive mechanisms enable the system to continuously improve its performance during operation, enhancing the accuracy of the end effector.
In summary, the proposed intelligent control method for the end effector of a parallel food sorting robot, which integrates fuzzy system, fuzzy neural network, and backstepping control algorithm, demonstrates significant improvements over traditional methods. Key achievements include enhanced tracking accuracy with position errors below 0.30 mm, reduced driving torque below \( 4.0 \times 10^4 \) N·mm, and high sorting efficiency up to 1.99 items/s at a conveyor speed of 100 mm/s. The method effectively addresses challenges such as poor positioning accuracy and low efficiency, offering a robust solution for industrial applications. Future work may focus on extending this approach to more complex environments, incorporating real-time adaptive tuning, and exploring integration with advanced sensing technologies for further optimization of end effector performance. This research contributes to the advancement of intelligent control systems in robotics, paving the way for more efficient and precise automation in the food industry and beyond.