In the realm of advanced manufacturing, particularly for aero-engine components, the pursuit of high efficiency and superior surface quality consistency in blade polishing has been a persistent challenge. Traditional single-sided polishing methods often induce unstable grinding due to unilateral forces, leading to compromised surface integrity and shape accuracy. Leveraging advancements in robot technology, this study introduces a novel double-sided symmetric polishing trajectory planning method. By integrating a seventh axis into the robotic system, we aim to achieve simultaneous polishing on both sides of the blade, thereby enhancing processing stability, reducing deformation errors, and improving overall efficiency. The core of this approach lies in the development of a path planning strategy based on the blade’s mean line, which ensures symmetrical contact between the polishing wheels and the blade surfaces. This method not only addresses the limitations of conventional techniques but also capitalizes on the inherent geometric correlations between the concave and convex surfaces of the blade. Through rigorous theoretical analysis and experimental validation, we demonstrate the feasibility and effectiveness of this approach in achieving high surface quality consistency and contour accuracy, all while halving the polishing time compared to single-sided methods. The integration of robot technology here is pivotal, as it enables precise control over complex trajectories and adaptive adjustments, making it ideal for handling the intricate geometries of aero-engine blades.
The foundation of our trajectory planning method rests on the concept of the mean line, which serves as a reference for generating symmetrical polishing paths on both the concave and convex surfaces of the blade. The mean line is defined as the locus of points equidistant from the blade’s concave and convex curves, essentially representing the midline of the blade’s cross-section. To derive this, we begin by extracting cross-sectional lines from the blade model using parallel, equally spaced planes. Each cross-sectional line is discretized using a curvature-arc length sampling method, which ensures a higher density of points in regions of high curvature, thereby preserving critical geometric information. The discrete points on the concave and convex curves are represented as sets $A = \{(x_i, y_i)\}$ and $B = \{(x_j, y_j)\}$, respectively. These points are then fitted into NURBS curves to facilitate smooth and accurate representations. The arc length factor $l(t)$ and curvature factor $k_p(t)$ are computed as follows:
$$ l(t) = \int_{t_1}^{t_2} \| r'(u) \| \, du $$
$$ k_p(t) = \frac{\| r'(t) \times r”(t) \|}{\| r'(t) \|^3} $$
where $r(t)$ is the parametric representation of the curve. The hybrid quality parameter $p(t)$ is then determined by balancing the arc length and curvature factors:
$$ p(t) = (1 – \lambda) \frac{l(t)}{l_{\text{total}}} + \lambda \frac{k_p(t)}{k_{p,\text{max}}} $$
with $\lambda = 0.5$ typically used to weigh both factors equally. This parameter guides the resampling of points to ensure uniformity in the hybrid quality metric. Next, the minimum and maximum offset distances between the concave and convex curves are calculated. The offset distance set is defined as:
$$ \text{offset} = \{ \text{offset}_{\text{min}} + s \cdot \Delta r \mid s = 1, 2, \ldots, q \} $$
where $\Delta r = (\text{offset}_{\text{max}} – \text{offset}_{\text{min}}) / q$. For each offset distance, the points on the concave and convex curves are shifted along their normal vectors, and the resulting curves are intersected to find points on the mean line. The endpoints are extended tangentially to connect with the leading and trailing edges, completing the mean line construction.
Based on the mean line, we generate symmetrical polishing paths by projecting normals from the mean line onto the blade surfaces. For each discrete point on the mean line, the normal vector $K$ is computed and extended to intersect both the concave and convex surfaces, defining the contact points for the upper and lower polishing wheels, denoted as $WA$ and $WB$, respectively. The deviation errors $D_A$ and $D_B$ between the actual contact points and the ideal positions are minimized to ensure precise contact. The condition for acceptable error is $D_A + D_B \leq \epsilon$, where $\epsilon$ is a predefined threshold. If this condition is not met, the sampling density is adjusted by modifying $\lambda$ in the hybrid quality parameter. To further refine the path, a chord height error check is performed. The chord height error $\epsilon_i$ between successive points on the polishing path is calculated as:
$$ \epsilon_i = R_i – \sqrt{R_i^2 – \left( \frac{L}{2} \right)^2} $$
where $R_i$ is the radius of curvature at point $i$, and $L$ is the distance between points. If $\epsilon_i$ exceeds a maximum allowable error $\Delta \epsilon$, additional points are interpolated along the mean line to reduce the error. This iterative process continues until all segments meet the chord height error requirement, resulting in smooth and accurate polishing paths.
The robot technology plays a crucial role in executing the planned trajectories. We define multiple coordinate systems to describe the relative positions and orientations of the robot, tool, and workpiece. The base coordinate system $\{O\}$, tool coordinate system $\{T\}$, and workpiece coordinate system $\{N\}$ are established. The transformation between these systems is represented by homogeneous transformation matrices. For instance, the transformation from $\{N\}$ to $\{T\}$ is given by:
$$ ^T_N\mathbf{T} = \begin{bmatrix} ^T_N\mathbf{R} & ^T_N\mathbf{P} \\ \mathbf{0} & 1 \end{bmatrix} $$
where $^T_N\mathbf{R}$ is the rotation matrix and $^T_N\mathbf{P}$ is the translation vector. At each contact point $WA_i$ on the concave surface, a local coordinate system $\{C\}$ is defined with its Z-axis aligned with the normal vector $K_i$. The transformation from $\{T\}$ to $\{C\}$ is computed to ensure the polishing wheel maintains normal contact with the blade surface. The constraint equations for the robot’s orientation are derived to minimize the deviation between the tool axis and the surface normal. Specifically, the objective is to minimize the function:
$$ f(u_1, u_2) = |v_3| + |u_3| $$
where $u_1, u_2, u_3$ and $v_1, v_2, v_3$ are components of the orthonormal vectors defining the orientation of $\{C\}$. This optimization ensures that the robot’s end-effector orientation aligns with the desired contact normal, thereby enhancing polishing consistency and quality.
The control of the robot’s seventh axis is essential for adjusting the position of the upper polishing wheel relative to the blade. The distance $z_i$ between contact points on the concave and convex surfaces is calculated as:
$$ z_i = \| WA_i – WB_i \| = \sqrt{(b_1 – a_1)^2 + (b_2 – a_2)^2 + (b_3 – a_3)^2} $$
Given the radii of the upper and lower polishing wheels, $r_1$ and $r_0$, respectively, the position $Z_i$ of the upper wheel’s center is determined by:
$$ Z_i = Z_0 – (r_0 + r_1 + z_i) $$
where $Z_0$ is the reference position of the lower wheel. This model allows precise control of the seventh axis to maintain optimal contact conditions during polishing. Additionally, to account for potential misalignment during blade clamping, an improved TCP (Tool Center Point) calibration method is employed. Four calibration points on the blade are used to compute the transformation between the flange coordinate system $\{E\}$ and the workpiece coordinate system $\{N\}$. The position deviations of these points relative to $\{E\}$ are used to solve for the transformation matrix $^E_N\mathbf{T}$, ensuring accurate registration of the blade within the robot’s workspace.
In the experimental validation, two aero-engine blades made of GH4169 and TC4 alloys are selected. The minimum radii of curvature for the concave and convex surfaces are analyzed to determine appropriate polishing wheel dimensions. For Blade 1, the concave surface has a minimum curvature radius ranging from 3.1 mm to 60.7 mm, and the convex surface from 0.1 mm to 353.5 mm. Thus, an upper wheel diameter of 20 mm and a lower wheel diameter of 80 mm are chosen. For Blade 2, both surfaces have minimum curvature radii between 0.2 mm and 200.6 mm, so wheels with 40 mm diameter are used for both sides. The polishing parameters are summarized in the table below:
| Parameter | Blade 1 (GH4169) | Blade 2 (TC4) |
|---|---|---|
| Abrasive Material | Brown Fused Alumina | Brown Fused Alumina |
| Abrasive Size (#) | 400 | 400 |
| Wheel Speed (r/min) | 2500 | 2500 |
| Feed Rate (mm/min) | 200 | 200 |
| Upper Wheel Diameter (mm) | 20 | 40 |
| Lower Wheel Diameter (mm) | 80 | 40 |
| Wheel Width (mm) | 20 | 20 |
The polishing paths generated using the mean line method show significantly lower normal contact errors compared to traditional methods based on cross-sectional normals. For Blade 1, the average surface roughness is reduced from 0.5 μm to 0.29 μm, and for Blade 2, from 0.8 μm to 0.36 μm. The contour errors are also improved, with Blade 1’s maximum error decreasing from 1.301 mm to an average of 0.449 mm across sections, and Blade 2’s from 0.384 mm to 0.137 mm. The double-sided polishing approach reduces the total processing time by approximately 50%, showcasing the efficiency gains enabled by advanced robot technology. The consistency in surface quality is further evidenced by the reduction in roughness variance, as detailed in the following table:
| Surface Area | Mean Ra (μm) | Variance of Ra (μm²) |
|---|---|---|
| Blade 1 Concave (Before) | 0.5061 | 0.0149 |
| Blade 1 Concave (After) | 0.2490 | 0.0025 |
| Blade 1 Convex (Before) | 0.4771 | 0.0099 |
| Blade 1 Convex (After) | 0.2976 | 0.0033 |
| Blade 2 Concave (Before) | 0.7894 | 0.0233 |
| Blade 2 Concave (After) | 0.3689 | 0.0020 |
| Blade 2 Convex (Before) | 0.7758 | 0.0095 |
| Blade 2 Convex (After) | 0.3527 | 0.0013 |
The experimental results confirm that the proposed double-sided symmetric polishing trajectory planning method, underpinned by sophisticated robot technology, effectively enhances surface quality consistency and contour accuracy while significantly reducing processing time. By leveraging the geometric properties of the blade’s mean line and integrating precise control of the robot’s seventh axis, this approach mitigates issues such as blade deformation and uneven material removal. The method’s robustness is demonstrated through its application to different blade materials and geometries, highlighting its potential for widespread adoption in aero-engine manufacturing. Future work could focus on further optimizing the path planning algorithm for real-time adaptability and expanding the system to handle more complex blade designs. Ultimately, this research underscores the transformative impact of robot technology in advancing precision manufacturing processes.