In the field of industrial automation, robot technology has revolutionized various manufacturing processes, including glazing applications. The optimization of robot trajectories is crucial for enhancing efficiency and quality in glazing operations. This article presents a multi-objective optimization approach based on the Non-dominated Sorting Differential Evolution (NSDE) algorithm to address the challenges in glazing robot trajectory planning. The integration of advanced robot technology enables precise control over spray parameters, leading to improved uniformity and reduced operational time.
The glazing process involves depositing a uniform layer of glaze on ceramic surfaces, which requires careful trajectory planning to avoid defects such as uneven thickness. Traditional methods, such as manual teaching or offline programming, often result in suboptimal trajectories that necessitate extensive trial-and-error adjustments. By leveraging robot technology, we can automate and optimize these trajectories through computational models. The proposed NSDE algorithm optimizes key parameters, including spray gun velocity and height, to achieve a balance between glaze thickness uniformity and glazing time.
To model the glaze deposition, we consider both planar and curved surfaces. For a planar surface, the glaze deposition rate follows a double-beta distribution model. Let $a$ and $b$ represent the semi-major and semi-minor axes of the elliptical spray pattern, respectively, $h$ denote the gun height, $v$ the gun velocity, $q$ the deposition rate, and $Q$ the glaze thickness. The deposition function for any point $(x, y)$ within the elliptical region is given by:
$$q(x, y) = q_{\text{max}} \left(1 – \frac{y^2}{b^2}\right)^{\beta_2 – 1} \left(1 – \frac{x^2}{a^2(1 – y^2/b^2)}\right)^{\beta_1 – 1}$$
for $-b \leq y \leq b$ and $-a(1 – y^2/b^2)^{1/2} \leq x \leq a(1 – y^2/b^2)^{1/2}$, and similarly for the alternate form. Parameters such as $a$, $b$, $\beta_1$, $\beta_2$, and $q_{\text{max}}$ are determined through experimental fitting. For curved surfaces, the deposition model accounts for the surface geometry and orientation. Let $S$ be a point on the surface, $l$ the distance from the gun to $S$, $\theta$ the angle from the vertical, and $\phi$ the angle between the surface normal and the line $OS$. The glaze accumulation rate at $S$ is:
$$q_s = q_{s’} \cdot \left(\frac{h}{l}\right)^2 \cdot \frac{\cos \phi}{\cos^3 \theta}, \quad \text{for } \phi < 90^\circ$$
This model ensures that the glaze distribution is accurately represented for complex surfaces, which is essential in robot technology applications.
The key factors influencing glaze deposition include gun velocity and height. Analysis shows that gun velocity has a more significant impact on thickness uniformity compared to height. To illustrate, simulations were conducted with fixed height and varying velocity, and vice versa. The results are summarized in Table 1, which highlights the slope of thickness curves as a measure of uniformity. Lower slopes indicate better uniformity.
| Factor | Range | Min Slope | Max Slope |
|---|---|---|---|
| Velocity (mm/s) | 300-500 | 0.00101 | 0.06326 |
| Height (mm) | 400-500 | 0.00183 | 0.01006 |
Based on this, the multi-objective optimization model aims to minimize both the glaze thickness error and the total glazing time. Let $S = \{s_j, 1 \leq j \leq m\}$ be the discretized surface points, $q = [q_1, q_2, \ldots, q_m]$ the glaze thickness vector, $T$ the total glazing time, and $n$ the number of trajectories. The total time is computed as:
$$T = n \sum_{i=1}^k \frac{\Delta l_i}{v_i} = \text{ceil}\left(\frac{W}{d}\right) \sum_{i=1}^k \frac{\Delta l_i}{v_i}$$
where $W$ is the surface width, $d$ the trajectory spacing, $\Delta l_i$ the segment length, and $v_i$ the velocity segment. The glaze thickness at point $s_j$ is:
$$q_j = \sum_{i=1}^{nk} \dot{q}(r_i) \left(\frac{h}{l}\right)_i^2 \frac{\cos \gamma_i}{\cos^3 \theta_i} \frac{\Delta l_i}{v_i}$$
The objective function is defined as:
$$\min L = (E, T)$$
subject to:
$$|q_j – q_d| \leq q_e, \quad h_{\text{min}} \leq h \leq h_{\text{max}}, \quad v_{\text{min}} \leq v_i \leq v_{\text{max}}, \quad d_{\text{min}} \leq d \leq d_{\text{max}}$$
where $E$ represents the thickness error, calculated as:
$$E = |q_{\text{max}} – q_d| + |q_{\text{min}} – q_d|$$
and $q_d$ is the desired glaze thickness.
The NSDE algorithm is employed to solve this multi-objective problem. The steps include population initialization, mutation, crossover, selection, fast non-dominated sorting, and crowding distance calculation. The population is initialized as:
$$x_{j,i}(0) = x_{j,i}^L + \text{rand}(0,1) \times (x_{j,i}^U – x_{j,i}^L)$$
for $i = 1, 2, \ldots, NP$ and $j = 1, 2, \ldots, D$, where $NP$ is the population size. Mutation is performed using the differential strategy:
$$\nu_i(g+1) = x_{r1}(g) + F \cdot (x_{r2}(g) – x_{r3}(g))$$
with $i \neq r1 \neq r2 \neq r3$. Crossover generates new individuals:
$$u_{j,i}(g+1) = \begin{cases}
v_{j,i}(g+1), & \text{if rand}(0,1) \leq CR \text{ or } j = j_{\text{rand}} \\
x_{j,i}(g), & \text{otherwise}
\end{cases}$$
Selection uses a greedy approach:
$$x_i(g+1) = \begin{cases}
u_i(g+1), & \text{if } f(u_i(g+1)) \leq f(x_i(g)) \\
x_i(g), & \text{otherwise}
\end{cases}$$
Fast non-dominated sorting assigns ranks based on dominance, and crowding distance ensures diversity. The traditional crowding distance is computed as:
$$I[i]_{\text{distance}} = I[i]_{\text{distance}} + \frac{I[i+1].m – I[i-1].m}{f_m^{\text{max}} – f_m^{\text{min}}}$$
To improve this, an entropy-based crowding distance (ICEBE) is introduced:
$$ICEBE = -\frac{I[i+1]_{\text{dis}} + I[i-1]_{\text{dis}}}{f_m^{\text{max}} – f_m^{\text{min}}} \times \left[ \frac{I[i+1]_{\text{dis}} \cdot \log_2\left(\frac{I[i+1]_{\text{dis}}}{I[i]_{\text{dis}}}\right)}{I[i]_{\text{dis}}} + \frac{I[i-1]_{\text{dis}} \cdot \log_2\left(\frac{I[i-1]_{\text{dis}}}{I[i]_{\text{dis}}}\right)}{I[i]_{\text{dis}}} \right]$$
This enhancement reduces the polarization of solutions in the Pareto front.
Simulation experiments were conducted to validate the approach. The NSDE algorithm generated a Pareto solution set, from which an optimal point was selected. The parameters were: spacing $d = 70.56$ mm, velocity $v = 406$ mm/s, and height $h = 410$ mm. A comparison of glaze thickness uniformity before and after optimization is shown in Table 2.
| Condition | Max Thickness (μm) | Min Thickness (μm) | Uniformity Error (%) |
|---|---|---|---|
| Initial Trajectory | 950 | 700 | 15.8 |
| Optimized Trajectory | 890 | 810 | 9.21 |
The optimized trajectory demonstrates improved uniformity, with a reduction in error from 15.8% to 9.21%. This highlights the effectiveness of the NSDE algorithm in robot technology for glazing applications.
Further, robotic glazing experiments were performed on ceramic specimens. The glaze thickness was measured at multiple points using electronic microscopy. The results, presented in Table 3, show the thickness deviations from the desired value of 850 μm.
| Section | Thickness (μm) | Deviation (μm) |
|---|---|---|
| Section 1 | 764.95 | -85.05 |
| Section 1 | 724.15 | -125.85 |
| Section 1 | 713.95 | -136.05 |
| Section 2 | 826.14 | -23.86 |
| Section 2 | 846.54 | -3.46 |
| Section 2 | 856.74 | 6.74 |
| Section 3 | 764.95 | -85.05 |
| Section 3 | 734.42 | -115.58 |
| Section 4 | 815.94 | -34.06 |
| Section 4 | 805.75 | -44.25 |
The root mean square error of thickness is 78.35 μm, and the uniformity error is 9.21%, confirming the feasibility of the optimized trajectory. The integration of robot technology ensures consistent and efficient glazing operations.
In conclusion, the multi-objective NSDE algorithm effectively optimizes glazing robot trajectories by balancing thickness uniformity and glazing time. The use of robot technology enables precise parameter control, leading to significant improvements in quality and efficiency. Future work could explore real-time adaptation and broader applications in industrial robot technology.