Trajectory planning for shoe sole glue spraying based on cutting plane and multi-scale neighborhood normal vector estimation

Abstract

This paper focuses on the trajectory planning of glue spraying on shoe sole sidewalls, addressing issues such as low spraying efficiency, high health risks from adhesives, low spraying precision. A method based on cutting plane slicing and multi-scale neighborhood normal vector estimation (TP-MNVE) is proposed. Curvature features and tensor voting are introduced to remove point cloud noise. Boundary search calipers are used to search heel and toe, cutting plane slicing from heel to toe provides an extended line description (XLD) and identifies the highest points on both sides of the contour curves. These highest points, combined with the angle threshold of the spraying process, determine the glue spraying trajectory on the sidewalls. The surface variation of the shoe sole sidewall is incorporated to develop a multi-scale neighborhood weighted normal vector estimation function, yielding the normal vectors of trajectory points. Experimental results on sample shoe soles show that the number of denoised point clouds is reduced by 16,990, with a signal-to-noise ratio (SNR) of 42.156 and entropy of 3.19 × 10^5, outperforming Laplacian, bilateral, and median filtering methods. Boundary segmentation of the toe and heel is accurate, and the cutting plane extraction for the glue spraying trajectory is stable. The trajectory accuracy is 0.931 mm, the root mean square (RMS) of the deviation angle between the estimated and theoretical normal vectors is 0.66°, processing time is less than 1.6 s. This method solves the challenges of glue spraying for various shoe sole styles and promotes automation in the textile industry.

Keywords

cutting plane multi-scale neighborhood normal vector estimation shoe sole glue spraying trajectory planning

Introduction

In 2022, China’s annual production of sports footwear reached 1.439 billion pairs. The market size is expected to grow by 13.2% year-on-year in 2023. Projections indicate that by 2025, the proportion of individuals regularly engaging in physical exercise will reach 38.5%, driving a significant increase in the demand for sports footwear. The cold bonding process employs specialized chemical adhesives to bond the shoe sole to the upper under ambient temperature conditions. This technique integrates elements of other footwear manufacturing processes, ensuring the structural integrity of the shoes while simultaneously enhancing production efficiency.

Footwear manufacturing is an important sector of the textile. In the production of shoe soles, spraying adhesive is a key process that bonds the sole to the textile upper. The textile upper is typically made from various types of mesh fabric, which are required to have certain breathability and water permeability. As a result, there are challenges in bonding the textile upper to the sole. Cold adhesive shoe manufacturing is a typical labor-intensive industry, with most workstations relying on manual or semi-automated operations, resulting in low production efficiency and poor working conditions. The bonding of shoe soles and shoe uppers using water-based and vulcanized adhesives releases toxic fumes, posing serious health risks to workers and hindering the development of the footwear industry. The use of robotic arms for glue spraying can effectively address issues such as adhesive waste, unstable spraying quality, and low efficiency.¹ The key to robotic arm glue spraying is the trajectory planning for the shoe sole sidewall. Existing studies have used robotic teaching,² CAD models,³ and shoe sole point cloud data^4–6 to obtain spraying trajectories. However, teaching and CAD models cannot address issues such as shoe sole deformation and size changes. By using 3D vision, the spatial position and normal vector of the glue spraying trajectory can be obtained online, bringing innovative changes to shoe sole glue spraying. However, current 3D vision-based glue spraying trajectory planning only considers a single characteristic of the shoe sole, without incorporating the specific features of the spraying area and the glue spraying industry. This leads to issues such as material wastage, unstable spraying quality, and low efficiency.

To meet the manufacturing requirements of sports shoes, the deviation in glue spraying trajectories should not exceed 1.0 mm. The glue spraying trajectories for different sole styles depend on the boundary geometric features of the sole contours, with the spraying area primarily originating from the sidewall regions. As an online detection method for shoe soles, 3D vision effectively addresses issues such as multiple styles and shoe sole deformation.^7,8 The glue spraying unit for sports shoe soles is shown in Figure 1. The glue spraying area on the shoe sole sidewall has characteristics such as small spraying area, large curvature fluctuations, and high precision requirements for boundary spraying, making 3D vision-based trajectory planning for glue spraying more complex. Semantic segmentation methods have been used to extract glue spraying trajectories for multiple shoe sole styles,⁹ but shoe sole styles change rapidly, and the constructed training sets cannot keep up with the timeliness of shoe sole updates. The Canny edge detection method has been used to obtain the outermost boundary contour of the shoe sole and generate spraying trajectories by evenly offsetting the contour lines.^10,11 However, Canny edge detection can only capture the outermost boundary contour of the shoe sole, which is not always the top edge, and the evenly offset trajectories cannot accommodate the changes in the glue spraying surface of the shoe sole.^12,13 In summary, current 3D vision-based glue spraying trajectory planning for shoe soles does not meet the production demands for style updates.

Figure 1.

The glue spraying unit for sports shoe sole.

The acquisition of shoe sole point cloud data is influenced by external environmental factors, measurement equipment defects, the reflective characteristics of the shoe sole surface, and point cloud 3D reconstruction algorithms, resulting in noise such as scattering, blockiness, and spikes during the data acquisition process.^14,15 Mainstream point cloud filtering methods cannot effectively handle both large-scale and small-scale noise,¹⁶ and they are sensitive to input point cloud data, often failing to preserve the geometric features of the shoe sole while removing noise.¹⁷ To solve the problem of not being able to remove shoe sole noise as a whole, the noise in the shoe sole point cloud is divided into large-scale noise outside the sole’s boundary and small-scale noise within the sole. Tensor voting is used to differentiate between boundary noise and external noise, and the shoe sole’s internal noise is removed by combining the curvature features of the shoe sole sidewall.^18,19 The top edge of the shoe sole sidewall directly reflects the contour features of the glue spraying area. Boundary perception and deep learning segmentation methods are used to extract the boundary of the shoe sole sidewall point cloud data,^20,21 but these methods require the preparation of a training set in advance and cannot adapt to the frequent changes in shoe styles. To address the efficiency problem of shoe style switching, a cutting plane method is used to extract the boundary contour.²² Obtain the XLD contour from the intersection of the cutting plane and the point cloud. Based on curvature variations, divide the XLD into multiple segments, and determine the spray trajectory points at the locations where the angle difference between adjacent segments is maximal. In addition to the spatial position in the robotic arm guidance task, the normal vector estimation of the point cloud is also used as essential data. The spraying trajectory posture is constructed using a rotation matrix based on the normal vector of the shoe sole sidewall spraying area.^2–25 The shoe sole spraying area has a relatively large curvature, and when the neighborhood search radius is small, the number of neighboring point clouds is limited, making it difficult to accurately estimate the normal vector in areas with missing point clouds. When the neighborhood search radius is large, the number of neighboring point clouds increases, offering some resistance to point cloud loss, noise, and outliers. However, an excessively large search radius cannot accurately reflect the true shape of the original spraying area.²⁶

Based on the above, this paper proposes a glue spraying trajectory planning method TP-MNVE. The main research contents of this paper are as follows:

(1) Use tensor voting to remove large-scale point cloud noise and introduce the shoe sole sidewall curvature factor to eliminate small-scale noise. A quantitative comparison is made with Laplacian filtering, bilateral filtering, and median filtering in terms of SNR, entropy, and efficiency.

(2) Define a grayscale contrast threshold to develop a search caliper for segmenting the boundaries of the heel and toe. Create a cutting plane for vertical cutting of the shoe sole to obtain the XLD of the shoe sole. From the edge regions on both sides of the contour curve, extract the highest points as reference markers. Using the highest points as references, define an angle threshold for the glue spraying process to determine the spraying trajectory.

(3) Introduce the surface variation of the shoe sole sidewall and establish a multi-scale neighborhood weighted normal vector estimation function to estimate the normal vector of the shoe sole sidewall glue spraying area. Compare the results with normal vector estimation using principal components analysis (PCA) with fixed and adaptive neighborhoods, and perform glue spraying simulations and experiments on different shoe sole styles.

Trajectory planning of shoe sole glue spraying

Preprocessing of shoe sole point cloud

Large scale denoising based on tensor voting

The three-dimensional point cloud of the shoe sole is represented by a symmetric positive semi-definite matrix $3 \times 3$ . Let the shoe sole point cloud be $P = {p_{1}, p_{2}, \dots, p_{n}}$ , $p_{a} (a \in [1, n])$ be a sampling point on the shoe sole. $N_{a} = {p_{a 1}, p_{a 2}, p_{a 3}, \dots, p_{a k}}$ , $k \leq n$ denotes the k-nearest neighborhood point set of $p_{a}$ in the point cloud model $P$ , obtained through fast nearest neighbor search techniques in high-dimensional space. The tensor voting matrix of the neighborhood point set $N_{a}$ is²⁷

E = \sum_{k \in N_{a}} ε_{k} (E_{3} - \frac{u_{k} {u_{k}}^{T}}{‖ u_{k} {u_{k}}^{T} ‖})

(1)

where

E_{3}

is the unit matrix of

3 \times 3

;

u_{k}

is the direction vector from the sampling point

p_{a}

to the neighborhood point

p_{a k}

;

ε_{k} = \exp (- \frac{{‖ u_{k} ‖}^{2}}{v^{2}})

;

v = \sum_{p_{a k} \in N_{a}} \frac{p_{a} - p_{k}}{k}

λ_{1}

λ_{2}

λ_{3}

(

λ_{1} \geq λ_{2} \geq λ_{3} \geq 0

)is the eigenvalue of the tensor voting matrix

E

The geometric features of the sampled points can be classified into three categories as follows: when $λ_{1} \geq λ_{2} \approx λ_{3} \approx 0$ , the point $p_{a}$ is a point in the shoe sole point cloud; when $λ_{1} \approx λ_{2} ≫ λ_{3} \approx 0$ , the point $p_{a}$ is a point on the boundary of the shoe sole point cloud; when $λ_{1} \approx λ_{2} \approx λ_{3} \geq 0$ , the point $p_{a}$ is a corner or isolated point of the shoe sole point cloud. Summary, when the eigenvalues of the voting matrix $λ_{1} \approx λ_{2} \approx λ_{3} \geq 0$ , remove large scale noise outside the boundary of the shoe sole point cloud.

Small scale denoising based on curvature features

The point cloud of the shoe sole after removing large scale noise is denoted as $P^{'} = {p_{1}, p_{2}, \dots, p_{m}}$ , $p_{b} (b \in [1, m], m \leq n)$ represents the tensor voting denoised shoe sole point cloud, $k \leq m$ refers to the k-nearest neighbor point set obtained by using $k d - t r e e$ search for any point $p_{b}$ in the shoe sole point cloud $P^{'}$ . The normal vector of point $p_{b}$ is calculated using the moving least squares method. Assuming the equation of the cutting plane at point $p_{b}$ is $A x + B y + C z + D = 0$ , the normal vector at point $p_{b}$ is denoted as $n_{b} = {[A, B, C]}^{T}$ . By using the distance from point $p_{b}$ to the targent plane, the observation equation for the k-nearest neighbor point set $M_{b}$ is constructed as follows

[\begin{array}{l} x_{b} & y_{b} & z_{b} \\ x_{b, 1} & y_{b, 1} & z_{b, 1} \\ \dots & \dots & \dots \\ x_{b, k} & y_{b, k} & z_{b, k} \end{array}] [\begin{array}{l} A \\ B \\ C \\ D \end{array}] = [\begin{array}{l} D_{b, 0} \\ D_{b, 1} \\ \dots \\ D_{b, k} \end{array}]

(2)

where

(x_{b}, y_{b}, z_{b})

is the coordinate of the point

p_{b}

;

(x_{b, j}, y_{b, j}, z_{b, j})

is the coordinates of the

k

neighborhood of the point

p_{b}

j = 1, 2, \dots, k

;

D_{b, 0}

is the distance of the point

p_{b}

to the targent plane;

D_{b, 1}, D_{b, 2}, \dots, D_{b, k}

is the distance of each of the

k

neighboring points of the point

p_{b}

to the cutting plane.

Suppose $p_{b k}$ is any point among the $k$ neighboring of the point $p_{b}$ and the distance of the point $p_{b k}$ to the targent plane is calculated as $‖ p_{b k} - p_{b} ‖$ using Gaussian function, then the weight of the point $p_{b k}$ is

ω_{b, k} = e \frac{- ‖ p_{b k} - p_{b} ‖}{h^{2}}

(3)

where h is the distance constant between neighboring points. The larger the value of h, the wider the range of approximation. The distance constant h is set as the maximum distance between the k-nearest neighbor points.

Establish the minimum constraint criterion for the distance of each the $k$ neighboring points to the targent plane is

Define the minimum-distance constraint function for the $k$ neighboring points with respect to the targent plane is

\min \sum_{k = 0}^{k} D_{b, k} ω_{b, k}

(4)

The constraint gives the normal vector $n_{b} = {[A, B, C]}^{T}$ of point $p_{b}$ , and the curvature $C_{b}$ of the point $p_{b}$ is

C_{b} = \frac{1}{k} \sum_{k = 0}^{k} ‖ n_{b} - n_{b, k} ‖

(5)

where

n_{b, k}

is the normal vector of the

k^{'} th

neighboring of the point

p_{b}

By formula (6), the curvature $C_{b}$ of the shoe sole point cloud $p_{b}$ after tensor voting denoising is calculated. If $C_{b} > ε$ , where $ε$ is the curvature threshold, the shoe sole point cloud $p_{b}$ is determined to be small scale noise. The curvature threshold $ε$ is set based on factors such as the number of shoe sole points and external interference, in order to achieve optimal denoising performance.

Evaluation for shoe sole point cloud denoising

Accuracy, simplicity, and efficiency are the evaluation metrics for shoe sole point cloud denoising. Existing evaluation functions cannot balance all three. To objectively assess the denoising effect, SNR and entropy are introduced to evaluate the denoising effect of the shoe sole point cloud. Let the shoe sole point cloud after small scale denoising based on curvature features be denoted as $P^{″} = {p_{1}, p_{2}, \dots, p_{o}}$ , $p_{c} (c \in [1, o], o \leq m)$ . Then the $S N R_{s}$ is²⁸

S N R_{s} = 10 \lg (\frac{\sum_{P} {| P |}^{2}}{\sum_{P} {| P - P^{″} |}^{2}}) = 10 \lg (\frac{\sum_{P} {x_{a}}^{2} + {y_{a}}^{2} + {z_{a}}^{2}}{| {P} | \times \frac{\sum_{P} {(x_{a} - x_{c})}^{2} + {(y_{a} - y_{c})}^{2} + {(z_{a} - z_{c})}^{2}}{| {P} |}})

(6)

where

(x_{a}, y_{a}, z_{a})

is the coordinate of the sampled shoe sole point before denoising;

(x_{c}, y_{c}, z_{c})

is the coordinate of the shoe sole point after denoising;

| {P} |

represents the number of shoe sole point clouds.

Entropy theory describes the characteristic information of the denoised shoe sole point cloud. The greater the entropy in the shoe sole point cloud, the larger of information and the more detailed features. The entropy theory $H_{c}$ of the denoised shoe sole point cloud $P^{″}$ is

H_{c} = - p_{c}^{'} lo {g_{2}}^{{p^{'}}_{c}} - \sum_{k = 1}^{k} p_{c k}^{'} lo {g_{2}}^{{p^{'}}_{c k}}

(7)

where

p_{c}^{'}

is the probability distribution of the shoe sole point

p_{c}

;

p_{c} = C_{c} / C_{c} + \sum_{k = 0}^{k} C_{k}

;

p_{c k}^{'}

is the probability distribution of the

k

neighbors of the shoe sole point

p_{c}

p_{c k}^{'} = C_{k} / C_{c} + \sum_{k = 1}^{k} C_{k}

;

C_{c}

is the curvature of the shoe sole point

p_{c}

;

C_{k}

is the curvature of the

k

neighbors of the shoe sole point

p_{c}

The entropy theory $H_{c}$ of the shoe sole point $P^{″}$ as the entropy to evaluate the denoising

The sum of entropy $H = \sum_{c = 1}^{o} H_{c}$ as an evaluation of denoising effect of shoe sole point cloud.

Trajectory extraction of shoe sole glue spraying

Search for heel and toe

A search caliper, adapted to the heel and toe search length and width, was established in the image coordinate system to define the start and endpoint of the segmentation. The search principle for the heel and toe boundary is illustrated in Figure 2. The search caliper primarily consists of the search start point, the search area width, the search length, and the search width of a single caliper. Since the shoe sole does not have a fixed placement area, binary large object (Blob) analysis was employed to determine the center point of the shoe sole contour and the shoe sole length. Based on these measurements, the location of the search caliper was determined. Due to the curvature of the heel and toe, the minimum point of the heel was selected as the starting point, and the maximum point of the toe was identified as the terminal point.

Figure 2.

Search principle of heel and toe boundary.

In the search length direction, the search region width is projected from a two-dimensional image to a one-dimensional image. The segmentation principle of projection threshold is shown in Figure 3. The average pixel value of search caliper region image in the projected image is obtained, and the edge lines that do not meet the contrast threshold are eliminated using a contrast threshold. Points with higher contrast are selected as the search points for the caliper. A line passing through the minimum search point of the heel caliper, parallel to the search region width direction is established as the heel boundary; a line passing through the maximum search point of the toe caliper, parallel to the search region width direction is established as the toe boundary. The contrast of the projection data is

C = \sum δ {(i, j)}^{2} P_{δ} (i, j)

(8)

where

δ (i, j) = | i - j |

is the grayscale difference between adjacent pixels;

i

and

j

are the grayscale values of adjacent pixels;

P_{δ} (i, j)

is the probability of pixel distribution where the grayscale difference between adjacent pixels

δ (i, j)

Figure 3.

Segmentation principle of projection threshold.

For the boundary segmentation of shoe heels and toes in various styles, design experiments and configure the parameters, including the caliper’s search area width, number of calipers, search length, single caliper search width, and other relevant parameters, as detailed in Table 1. Since the heel width varies slightly across different shoe styles, the search area width for heel is set to 70.00 mm, 80.00 mm, and 90.00 mm, the RMSE error of the search points is calculated for different condition combinations. When the heel search area width is 80.00 mm, the number of search calipers is 20, the search length is 25.00 mm, the single caliper search width is 3.00 mm, and the contrast threshold is 0.20, the RMSE error of the search points is minimized to 0.12. Since the width of the toe area is smaller than that of the heel area, the search area width for the toe is set to 60.00 mm, 70.00 mm, and 80.00 mm, and the RMSE error of the search points is calculated for different condition combinations. When the search area width is 60.00 mm, the number of search calipers is 20, the search length is 35.00 mm, the single caliper search width is 5.00 mm, and the contrast threshold is 0.30, the RMSE error of the search points is minimized to 0.14.

Table 1.

Setting for caliper parameter.

Search area	Search area width (mm)	Number of calipers (pcs)	Search length (mm)	Single caliper search width (mm)	Contrast threshold parameters	RMSE error
Heel caliper parameter	70.00	20	25.00	3.00	0.20	0.15
			40.00	5.00	0.30	0.88
			50.00	8.00	0.40	1.14
	80.00	20	25.00	3.00	0.20	0.12
			40.00	5.00	0.30	0.62
			50.00	8.00	0.40	0.91
	90.00	20	25.00	3.00	0.20	0.26
			40.00	5.00	0.30	1.21
			50.00	8.00	0.40	1.56
Toe caliper parameter	60.00	20	25.00	2.00	0.2	0.21
			35.00	5.00	0.3	0.14
			45.00	10.00	0.4	0.84
	70.00	20	25.00	2.00	0.2	0.29
			35.00	5.00	0.3	0.61
			45.00	10.00	0.4	1.23
	80.00	20	25.00	2.00	0.2	0.51
			35.00	5.00	0.3	0.34
			45.00	10.00	0.4	1.64

Note: The bold indicates the optimal selected parameters.

In conclusion, the search parameters are set as follows: for the heel, the search area width is 80.00 mm, the number of search calipers is 20, the search length is 25.00 mm, the single caliper search width is 3.00 mm, and the contrast threshold is 0.20; for the toe, the search area width is 60.00 mm, the number of search calipers is 20, the search length is 35.00 mm, the single caliper search width is 5.00 mm and the contrast threshold is 0.30.

Trajectory planning of shoe sole sidewall

Since the consistency of the orientation of the shoe sole can not be guaranteed, establish the smallest external rectangle of the shoe sole, establish the coordinate $o (x, y, z)$ based on the center of the shoe sole boundary as the origin. The rectangle box length direction is $x$ , the rectangle box width direction is $y$ , the rectangle box height direction is $z$ . Establish the cutting plane in the $y z$ plane, make the cutting plane along the $x$ direction from the starting point of the heel to the end of the toe is shown in Figure 4(a). The length of the cutting plane is greater than the width of the shoe sole, and the width of the cutting plane is greater than the height of the shoe sole. Each time the specified length is moved along the $x$ axis, the cutting plane forms a point cloud intersection with the point cloud of the shoe sole, and the point cloud intersection is mapped to the 2D plane to form an XLD 2D contour intersection line is shown in Figure 4(b). In the area of heel and toe curvature changes greatly, adopt the boundary line of the heel and toe as the reference, establish the heel and toe area of width 30 mm respectively, and increase the number of cuts to the heel and toe area.

Figure 4.

Principle of acquisition glue spraying trajectory points. (a) Tanget plane cutting (b) point cloud intersection of tenget plane (c) tanget plane projected of yz plane (d) curvature segmentation.

The projection of the XLD contour curve obtained by cutting the shoe sole with the cutting plane in the YZ plane is shown in Figure 4(c). The highest point $A$ on the sidewall of the shoe sole is determined using the nearest neighbor search method. Combined with the shoe making craft, the actual boundary of the shoe sole and upper spraying is not the highest point $A$ , but the point $B$ near the highest point where the slope change is larger. The x and y coordinates of point B are obtained from the projection of point B on the x and y axes, the cutting plane moves on the x axis to obtain the x coordinate of point B. The highest point $A$ is used as the boundary to establish the 30 × 30 mm search area of the XLD contour line, and the XLD contour line is divided into $ξ - 1$ segments is shown in Figure 4(d). The angular difference is introduced to search for the spraying trajectory point, Using $p_{ξ - 2}$ as the reference and two edge points on each side for straight line fitting to obtain the slope of the line $k_{p_{ξ}}$ , invert the cutting angle $t a$ , the $p_{ξ}$ point with the largest angle difference between the two adjacent straight lines is used as the spraying trajectory point of the shoe sole sidewall. The angle difference between two adjacent straight lines is

\max Δ θ = | \arctan k_{p_{ξ}} - \arctan k_{p_{ξ - 1}} |

(9)

where

p_{ξ}

is the

ξ

point on the XLD profile;

k_{p_{ξ}}

represents the slope of the

ξ

segment ;

k_{p_{ξ - 1}}

represents the slope of the

ξ - 1

segment.

In summary, the XLD contour curve of the shoe sole point cloud is initially obtained using a cutting plane. The highest point along the contour curve is then identified. Using this highest point as a reference, an angle threshold is applied to search for and determine the glue spraying trajectory of the shoe sole.

Normal vector estimation of glue spraying

The normal vector of the shoe sole adhesive spraying points is estimated by fitting the plane of the trajectory point neighborhood point cloud using PCA, incorporating the surface variation degree. However, the accuracy of the normal vector estimation is affected by different neighborhood search radii. When the neighborhood search radius is small, the normal vector cannot be accurately estimated in regions with missing point clouds. When the neighborhood search radius is large, it has some resistance to point cloud missing and noise, but an excessively large search radius cannot accurately reflect the true shape of the adhesive spraying surface. Therefore, a multi-scale normal vector estimation approach is used, where the normal vector estimation results with different search radii are weighted to increase robustness against the influence of point cloud missing and noise. The principle of multi-scale neighborhood search is shown in Figure 5.

Figure 5.

Principle of multi-scale neighborhood search.

The boundary between the midsole and sidewall regions of the shoe sole is determined using an angle threshold. The sidewall of the shoe sole is then segmented based on this boundary line. Obtain the shoe sole sidewall region based on the signed surface variation (SSV).²⁹ The trajectory point $p_{φ}$ is obtained based on the cutting plane, and the projection point $p_{φ}^{'}$ is obtained by inwardly shrinking the trajectory point $p_{φ}$ by 5.0 mm. The multi-scale neighborhood search area with the radius of $r_{1}$ $r_{2}$ and $r_{3}$ is established with the center of $p_{φ}^{'}$ . The multi-scale normal vector estimation of $r_{1}$ $r_{2}$ and $r_{3}$ is carried out by the PCA method. The normal vector estimation results from different scales are then weighted and fused to serve as the normal vector for the adhesive spraying trajectory. The normal vector weighting function is

\hat{n} = N_{1} \times ω_{1} + N_{2} \times ω_{2} + N_{3} \times ω_{3}

(10)

where

N_{i} (i = 1, 2, 3)

is the estimate of the normal vector at different scales;

ω_{i}

is the weighted value.

{\begin{cases} ω_{1} = {e^{-}}^{δ} \\ ω_{2} = 1 - {e^{-}}^{δ} \\ ω_{3} = 2 (1 - {e^{-}}^{δ}) \end{cases}

(11)

where:

δ

is the degree of variation of the shoe sole sidewall surface;

ω_{1}

ω_{2}

and

ω_{3}

correspond to the weight parameters at scales with radii

r_{1}

r_{2}

and

r_{3}

, respectively.

Trajectory evaluation

Obtain the point cloud of the sample shoe sole by 3D scanner, and the shoe sole and upper bonding trajectory line is obtained by point cloud inversion. Calculate the Euclidean distance between the glue spraying trajectory and the bonding trajectory, take the average error as the evaluation criterion for trajectory accuracy. The average error is

\bar{e r r} = \frac{\sum_{φ = 1}^{n} d i s t (p_{φ}, p_{ψ})}{n}

(12)

where

p_{φ}

is the spraying trajectory point;

p_{ψ}

is the nearest point on the shoe sole and upper bonding trajectory line to

p_{φ}

, dist is the euclidean distance; n is the number of glue trajectory points.

The angle between the estimated normal vector and the theoretical normal vector is the normal deviation angle. The RMS of the estimated normal vector deviation angle is used to evaluate the accuracy of the estimated normal vector. The larger the RMS, the greater the error in the estimated normal vector. The RMS of normal deviation angle is

R M S = \sqrt{\frac{1}{n} \sum_{i = 1}^{N} \arccos {(\frac{\hat{n} \cdot n^{*}}{| \hat{n} | | n^{*} |})}^{2}}

(13)

where

\hat{n}

is the estimation normal vector of multi-scale;

n^{*}

is the theoretical normal vector.

Deviations in the shoe sole glue spraying trajectory are permissible provided that the average error towards the inner side of the sole, based on the shoe upper edge of the shoe sole sidewall as the reference, is controlled within 1.0 mm and the RMS of the normal vector deviation angle is less than 1.5. If these conditions are met, the 3D camera will recapture the sole’s point cloud to compute the glue spraying trajectory. Otherwise, automated glue spraying will not be performed on the sole, and it will be handled manually.

Experiment and analysis of shoe sole spraying

The core hardware parameters of the shoe sole glue spraying experiment platform are shown in Table 2. The PCL point cloud library and OpenCV function library were used to design the algorithm in this paper. The TP-MNVE method is validated through trajectory planning simulations, adhesive spraying experiments, and shoe sole adhesive spraying experiments. To mitigate interference from the external environment during the 3D camera alignment process, image acquisition was conducted within an enclosed space. The experiment platform of shoe sole glue spraying is shown in Figure 6.

Table 2.

The core hardware parameters of the shoe sole glue spraying experimental platform.

Model and name	Category	Parameters
3D camera SR6260	Wavelength	405 nm
	Line width	210 mm
	Depth of field	290 mm
	Repeated accuracy	0.08 mm
	Scanning speed	2500 Hz/s
ESTUN ER10-900	Number of axes	6
	Arm span	900 mm
	Repeatability accuracy	0.03 mm
Lenovo 9000k	CPU	i7-10875H CPU
	RAM	16 GB
	GPU	GeForce RTX2060 8 GB

Figure 6.

The experiment platform of shoe sole glue spraying.

Denoising experiment

The large scale and small scale noise is removed using a step-by-step method. Experimental process of shoe sole denoising is shown in Figure 7, the color codes in Figure 7 are consistent with those in Figure 5. The 3D camera captures the shoe sole point cloud is shown in Figure 7(a). The large scale noise of the shoe sole is extracted is shown in Figure 7(b). The noise is subsequently removed is shown in Figure 7(c). The shoe sole data after removing the large scale noise still contains small scale noise mixed within the point cloud is shown in Figure 7(d). In regions (1) and (2), there are scattered noise points with irregular distribution in the laser lines, which affect the normal vector estimation of the shoe sole sidewall glue spraying trajectory points. Small scale noise is removed based on curvature characteristics is shown in Figure 7(e). Through point cloud denoising, the large scale noise is completely removed, and the scattered small noise in the laser lines is mostly eliminated, resulting in valid shoe sole point cloud data, which lays the foundation for extracting the glue spraying trajectory.

Figure 7.

Experimental process of shoe sole denoising. (a) Shoe sole point cloud (b) large scale noise (c) remove large scale noise (d) remove small scale noise (e) denoising of shoe sole point cloud.

The shoe sole point cloud in Figure 7(a) is processed using tensor voting with curvature feature filtering, Laplacian filtering, bilateral filtering, and median filtering. A comparison of the denoising results in terms of point cloud count, SNR, entropy and processing time, as shown in Table 3. Using tensor voting and curvature feature denoising, the number of point clouds is reduced by 16,990, which is slightly fewer than the point cloud removal achieved by laplacian filtering, bilateral filtering, and median filtering. The SNR is 42.156, the entropy is 3.19 × 10⁵, and the processing time is 896 ms, all of which outperform laplacian filtering, bilateral filtering and median filtering.

Table 3.

Comparison of denoising results.

Serial number	Method	Number of points (pcs)	Signal noise ratio	Information entropy value	Processing time (ms)
1	Original data	317882	20.164	3.65 × 10⁵	—
2	Methodology of this paper	300892	42.156	3.19 × 10⁵	896
3	Laplace’s algorithm	285681	29.245	2.83 × 10⁵	1205
4	Bilateral filtering	299458	34.168	2.89 × 10⁵	1198
5	Median filtering	286455	25.346	2.54 × 10⁵	982

Trajectory planning of glue spraying

Segmentation boundary of heel and toe

The 3D camera is set to capture images from the toe to the heel direction. In the camera coordinate system, the heel data is calibrated to be smaller than the toe data. The denoised shoe sole point cloud data is converted into a 16-bit depth map, the shoe sole point cloud is rotated to ensure that the heel and toe axes of the shoe sole are aligned with the direction of the 3D camera. According to the caliper parameters in Table 1, the heel search area width is set to 80.00 mm, search length to 25.00 mm, individual caliper search width to 3.00 mm, and the contrast threshold to 0.2. The single caliper projection search for the heel is shown in Figure 8(a). The toe search area width is set to 60.00 mm, search length to 35.00 mm, individual caliper search width to 5.00 mm, and the contrast threshold to 0.3. The single caliper projection search for the toe is shown in Figure 8(b), the color codes in Figure 8 are consistent with those in Figure 5.

Figure 8.

Projection data of single caliper. (a) Projection data of single heel caliper (b) projection data of single toe caliper.

Through the projection data of single caliper in Figure 8. Both the heel and toe boundaries are extracted using 20 search calipers. The results of heel and toe search boundary are shown in Figure 9. Based on the shoe sole depth map of Figure 9(a), Blob analysis is used to determine the center of the shoe sole contour and the sole length. Taking the shoe sole contour center as a reference, search calipers for the heel and toe are established by moving half the length of the shoe sole in both the heel and toe directions is shown in Figure 9(b). 20 search calipers are evenly distributed along the search width direction for both the heel and toe. 20 caliper search points are obtained for both the heel and toe is shown in Figure 9(c). A line passing through the minimum search point of the heel caliper, parallel to the search area width direction, is established as the heel boundary. Similarly, a line passing through the maximum search point of the toe caliper, parallel to the search area width direction, is established as the toe boundary. The heel and toe boundary segmentation is shown in Figure 9(d). The heel and toe boundary search experiment indicates that the caliper search method is stable and accurate, it is suitable for heel and toe positioning and segmentation.

Figure 9.

The results of heel and toe search boundary. (a) Depth map of shoe sole (b) establishment of search caliper (c) result of search caliper (d) boundary segmentation.

Trajectory of glue spraying

The cutting range of the shoe sole is limited based on the boundary of the heel and toe. The cutting plane is then established from the direction of the heel to the toe, and the glue spraying trajectory is extracted, as shown in Figure 10. Given that the contour curves of the toe and heel of different shoe sole styles do not vary significantly, and the design width of both the heel and toe is within 30 mm. So the boundary line of heel and toe is used as the reference to set the cutting area of heel and toe as 30 mm, the interval is 3 mm for cutting, and the spacing of the cutting plane in the middle of the shoe sole is 5 mm, and the cutting plane is established, as shown in Figure 10(a), extract the XLD cutting curve of the cutting plane and the shoe sole is shown in Figure 10(b). Cyclically cutting the shoe sole based on the set cutting area and cutting plane spacing. Extracted the highest points of shoe sole sidewall is shown in Figure 10(c). The trajectory point spacing is relatively dense at the heel and toe is shown in Figure 10(d) and (e). In Figure 10(b), the XLD contour curve is divided into region A and region B by the midpoint of the X-axis of the XLD contour curve. Point P₁ is the highest point of region A, and point P₄ is the highest point of region B. Using formula (10), the maximum angle threshold search is performed for points P₁ and P₄, resulting in points P₂ and P₃ as the actual glue spraying trajectory points. Using formula (10), the maximum angle threshold search is performed on the highest points of the shoe sole sidewall in Figure 10(c), yielding the shoe sole sidewall glue spraying trajectory points is shown in Figure 10(f). In Figure 10(f), the cutting in the middle of the shoe sole is relatively uniform, while the cutting spacing at the heel and toe is smaller than in the middle of the shoe sole, resulting in a denser distribution of trajectory points at the heel and toe. Due to the larger curvature variations in the heel and toe regions, the spacing between the glue spraying trajectory points slightly increases, but it still meets the application requirements.

Figure 10.

Extraction of glue spraying trajectory. (a) Target plane cutting (b) angle threshold extraction (c) the highest point of the shoe sole sidewall (d) heel area (e) toe area (f) glue spraying trajectory.

Based on the method of extracting glue spraying trajectory from Figure 10. Since the average projection width of the shoe sole sidewall on the sole plane is 10 mm, to ensure that the normal vector of the glue spraying points lies at the center of the spraying surface, the glue spraying trajectory points are offset inward by 5 mm to obtain the projection points of the shoe sole sidewall. Considering the curvature of the shoe sole sidewall and the density of the captured point cloud, multi-scale search regions with radii of 0.5 mm, 1.0 mm, and 1.25 mm are established centered on the projection points, as shown in Figure 11. PCA is used to solve for the normal vectors in different scale regions, and the normal vector of the glue spraying trajectory point is calculated using formula (10). To verify the stability under multi-scale neighborhood weighted normal vector estimation, uniform sampling, gradient sampling, and fringe density sampling are used in the regional density of the spraying trajectory points on the shoe- sole sidewall, and Gaussian noise with standard deviations of $σ = 0.002$ , $σ = 0.012$ and $σ = 0.024$ is added to verify the stability of the multi-scale neighborhood weighting compared with fixed neighborhood and adaptive neighborhood normal vector estimation. The RMS of normal vector deviation angles is shown in Table 4.

Figure 11.

Establishing multi-scale search area.

Table 4.

RMS of normal vector deviation angle.

Sampling method	Noise	Fixed neighborhood	Adaptive neighborhood	Multi-scale neighborhood weighting
Uniform sampling	None	1.76	1.69	0.81
	$σ = 0.002$	2.79	1.87	1.83
	$σ = 0.012$	9.16	9.54	3.69
	$σ = 0.024$	15.19	35.87	9.53
Gradient sampling	None	1.26	1.64	0.76
	$σ = 0.002$	2.70	1.69	1.36
	$σ = 0.012$	9.13	12.66	3.53
	$σ = 0.024$	15.24	39.63	12.69
Fringe density sampling	None	5.21	2.83	0.66
	$σ = 0.002$	2.96	3.28	5.29
	$σ = 0.012$	10.17	13.64	7.35
	$σ = 0.024$	16.91	38.4	13.64

From Table 4, under different sampling methods and noise levels, the RMS of the multi-scale domain weighted normal vector deviation angle is the smallest, which is better than the fixed neighborhood and adaptive neighborhood, outperforming both fixed neighborhood and adaptive neighborhood methods. The RMS of the normal vector deviation angle for multi-scale neighborhood weighting, adaptive neighborhood, and fixed neighborhood increases with the increase in noise standard deviation. Under the noise condition of $σ = 0.024$ , the RMS of the maximum deviation angle for multi-scale neighborhood weighting, adaptive neighborhood, and fixed neighborhood are 13.64°, 39.63°, and 16.91°, indicating that noise has a significant impact on normal vector estimation in multi-scale neighborhood weighting, adaptive neighborhood, and fixed neighborhood methods. Under noise-free conditions with uniform sampling, gradient sampling, and stripe sampling, the RMS of the estimated normal vector deviation angle with multi-scale neighborhood weighting are 0.81°, 0.76°, and 0.66°. Indicating that point cloud density has a minimal effect on normal vector estimation with multi-scale neighborhood weighting.

Experiment and analysis of glue spraying trajectory for different shoe sole styles

Trajectory planning is conducted for different styles of shoe soles, and experiments are performed on 273 soles of various styles to verify the robustness of the proposed method. The experimental results demonstrated that 273 shoe soles were correctly identified in terms of their adhesive spraying trajectories. Both the accuracy and efficiency of the trajectory planning met the application requirements of the production line. The glue spraying trajectory recognition results for different shoe sole styles shown in Figure 12, the color codes in Figure 12 are consistent with those in Figure 5. Figure 12(a) and (b) show different shoe sole styles, while Figure 12(c) and (d) display the left and right shoe soles of the same pair of shoes, Figure 12(e) and (f) show shoe soles with special shapes featuring curved glue spraying areas. The glue spraying trajectory points and normal vectors of the shoe soles can be identified. The cutting planes move with equal intervals in the direction, due to the curvature changes of the shoe sole sidewall, the spacing between the trajectory points is inconsistent. The curvature is larger at the heel and toe, so if the same cutting interval as the middle part of the sole is used, the trajectory point spacing at the heel and toe becomes larger, which affects the interpolation accuracy of the trajectory. For the heel and toe regions, the cutting area is divided by the heel and toe boundaries with a 30 mm width and a cutting plane spacing of 3 mm. The cutting plane spacing for the middle part of the sole is set to 5 mm. In Figure 12, the cutting density at the heel and toe is greater than at the middle of the sole. However, the significant curvature changes in the heel and toe regions cause the spacing between the glue spraying trajectory points to slightly increase, but this does not affect the interpolation accuracy of the robotic arm for the glue spraying trajectory points.

Figure 12.

Glue spraying trajectory recognition results for different shoe sole styles. (a) Style one (b) style two (c) style three (d) style four (e) style five (f) style six.

To quantify the accuracy of the glue spraying trajectory, the Euclidean distance between the glue spraying trajectory points and the closest point on the bonding curve is calculated using formula (12), and the average is taken. The planning of the glue spraying trajectory is related to the spraying area; therefore, the accuracy standards for the spraying process are not directly influenced by the shoe sole style. The accuracy and efficiency of the glue spraying trajectory for different shoe sole styles are shown in Table 5. The comparison results between TP-MNVE and references 8 and 9 demonstrate that TP-MNVE outperforms the other methods in terms of both glue spraying average error and variance. Specifically, the maximum error for different shoe sole styles using TP-MDNVE is 0.931 mm, whereas reference 8 exhibits a maximum error of 0.984 mm, and reference 9 shows the lowest accuracy with a maximum error of 1.352 mm. However, when considering trajectory planning efficiency, the TP-MNVE method does not yield the best performance, as both references 8 and 9 show fluctuations in their efficiency. The proposed method achieves an efficiency of 1.568 s, which, when combined with the shoe sole production cycle, satisfies the accuracy and efficiency requirements for automated production.

Table 5.

Accuracy and efficiency of the algorithm.

Figure number	Method	Average error (mm)	Variance	Efficiency (ms)
12(a)	TP-MNVE	0.931	0.679	1457
	Li⁸	0.984	0.986	1325
	Wang⁹	1.025	0.864	1438
12(b)	TP-MNVE	0.436	0.825	1328
	Li⁸	0.654	0.968	1254
	Wang⁹	0.526	1.035	1368
12(c)	TP-MNVE	0.889	0.947	1426
	Li⁸	0.972	1.365	1689
	Wang⁹	1.168	1.547	1254
12(d)	TP-MNVE	0.759	0.771	1568
	Li⁸	0.864	0.965	1502
	Wang⁹	0.826	1.341	1697
12(e)	TP-MNVE	0.825	0.824	1354
	Li⁸	0.981	0.934	1483
	Wang⁹	0.879	0.917	1587
12(f)	TP-MNVE	0.819	0.724	1407
	Li⁸	0.934	0.862	1508
	Wang⁹	1.352	0.964	1634

Glue spraying simulation and experiment

Glue spraying simulation

According to the spraying trajectory, the simulation experiment of shoe sole glue spraying is shown in Figure 13. The left and right views confirm full coverage of the glue film in the spraying area without overflow on the outer side of the sidewall. The top-down views of shoe sole A and shoe sole B reveal neat outer boundaries with no overflow, though minor irregularities appear on the inner side in areas with larger curvature. For shoe sole B, the glue film is slightly thicker in the middle and thinner at the edges, but the simulation results still comply with the production process requirements.

Figure 13.

Simulation experiment of shoe sole glue spraying.

Glue spraying experiment

By calibrating the robotic arm and camera, the glue spraying trajectory in the visual coordinate system is converted to the robotic arm coordinate system, and the spraying experiment is conducted on the experimental platform is shown in Figure 6. A fluorescent agent is added to the glue to highlight the spraying effect, the glue spraying effect of difference shoe sole styles is shown in Figure 14. There is no glue overflow at the edge of the sidewall, and the glue coverage in the spraying area of the sidewall is uniform. The curvature of the glue spraying areas at the toe and heel shows significant fluctuations, but there is no overflow beyond the boundaries or areas that were missed. The spraying experiment demonstrates that the glue spraying trajectory extracted using the cutting plane and multi-scale neighborhood point cloud normal vector estimation method is stable, and the spraying effect meets the requirements of the footwear production process.

Figure 14.

Glue spraying effect of difference shoe sole styles.

Conclusion

The cutting plane and multi-scale normal vector estimation method effectively identifies the glue spraying trajectory on the sidewalls of shoe sole. The feasibility of TP-MNVE method has been verified through simulation and experiments, solving the bottleneck of multi-style shoe sole glue spraying, realize automation of glue spraying for shoe soles significantly mitigates the labor-intensive challenges prevalent in the footwear industry and promoting the automation transformation of textile industry. At the same time, it provides theoretical support for the future adhesive spraying of shoe soles. The main research conclusions are as follows:

(1) The tensor voting and curvature feature denoising method was employed to remove both large-scale and small-scale noise from the sole point cloud, resulting in a reduction of 16,990 points, slightly fewer than the reductions achieved by Laplacian filtering, bilateral filtering, and median filtering. The proposed method demonstrated superior performance, the SNR is 42.156, the entropy is 3.19 × 10⁵, and processing time is 896 ms, outperforming Laplacian filtering, bilateral filtering, and median filtering. These results indicate that the tensor voting and curvature feature method effectively eliminates noise while maximizing the preservation of local details in shoe sole point cloud.

(2) To address the diversity of shoe styles, boundary search calipers for the heel and toe are established, cutting regions are created based on these boundary search calipers. Cutting planes are used to obtain XLD contour curves, glue spraying trajectory points are obtained using angle thresholding. The simulation of the shoe sole glue spraying trajectory shows that the method identifies the spraying trajectory stably, with no missed or misjudged trajectory points.

(3) Biasing the glue spraying trajectory point as the reference, establishing the multi-scale normal vector estimation region. Under the conditions of uniform sampling, gradient sampling and fringe sampling, the normal vector estimation is carried out by using fixed neighborhood, adaptive domain and multi-scale neighborhood. The simulation of normal vector estimation shows that under different sampling conditions, the RMS of the normal vector deviation angles using multi-scale neighborhood weighting are 0.81°, 0.76°, and 0.66°, which are better than fixed and adaptive neighborhoods. Glue spraying simulation and experiments on different shoe styles show that the trajectory accuracy is 0.931 mm, processing time is less than 1.6 s, and the glue spraying effect meets the required standards.

Future work

(1) Further improvements are needed to enhance the accuracy and efficiency of glue spraying trajectory planning.

(2) The glue gun significantly influences the spraying effect. Therefore, introducing a glue gun model is essential to optimize the glue spraying trajectory.

ORCID iD

Hao Zhang https://orcid.org/0000-0001-9210-6811

Footnotes

Statements and declarations

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