Inverse Kinematics Proposal to Automatize the 3D Scanning of Handball Shoes by Using a Robotic Arm with 3 Actuated Joints

Abstract

The purpose of this paper is to present a procedure for automating the scanning process based on a mathematical model for a handheld 3D scanner for shoes used in some indoor sports. The study aims to use inverse kinematics to automate 3d footwear scanning used for indoor sport (handball) using a robotic arm with three joints. A modeling of the robotic arm and final effector was performed, to simulate the minimum and maximum trajectory of the robot arm according to the angles shown based on the mathematical model and inverse kinematics. With an easy-to-use interface and ergonomic design, this 3D scanning solution offers the versatility to scan various objects (such as scanning two shoe models used in indoor sports) and complex surface types. This scanning manner represents the state of the art of 3D scanning solutions as well as a benchmark in the 3D measurement equipment industry. The data obtained as a result of this research provide new directions and solutions for sports shoe scanning for indoor sports based on scanning trajectories preset by inverse kinematics in order to automate the scanning process using a handheld 3D scanner. Based on the mathematical model presented in the paper, automation of the scanning process can be achieved by maintaining the proposed trajectory using an automated arm operating through a control program that can be run on a simple controller.

1. Introduction

The development of robust, compact, and low-cost optical, mechanical, and electronic components makes it possible to build accurate and reliable 3D scanning systems. In particular, in the textile industry, surface digitization allows for the assessment of textile surface quality, such as changes in appearance or the detection of fabric defects [1,2]. Internationally, the automation and optimization of the scanning process has been considered through different variants using diverse methods such as:

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Mounting a handheld 3D scanner on a machine tool for which inverse kinematics was performed in order to perform automated scanning [3,4,5];

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Mounting a handheld 3D scanner on a robot [6];

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Mounting a handheld 3D scanner on a tripod, the object to be scanned being positioned on a rotating table [7].

When performing 3D scanning with structured light while using a robotic arm, it is not necessary to pulverize the surface with special spray for antireflection (as in the case of laser scanning). Three-dimensional scanning of sneakers using a robotic arm is faster compared to other methods such as photogrammetry or laser scanning along a line and point (which takes more time). The robotic arm together with a 3D handle scanner can be quickly guided around the object, which allows flexibility in scanning complex surfaces or objects with uneven geometry. Compared to the above-mentioned methods, the authors of the paper propose a technical and economical solution that is much more advantageous compared to the existing variants on the market.

The research in this paper also aims to optimize the scanning process based on a mathematical model for generating a 3D scanning trajectory of sports shoes used by handball players. However, few works have dealt with the generation of scanning trajectories on shoes used for indoor sports. Manual scanning performed by several people is different even if the procedures are standardized, and in applications requiring a high degree of accuracy and repeatability, this leads to measurement variations. In order to scan more sports shoes, automation minimizes a lot of repetitive work performed by the human factor (i.e., manually performing the scanning procedures, which can become tiring for the human factor) [8,9].

Consistent results and reduced errors compared to the manual scanning process can be achieved by automating the 3D scanning process by using a robotic arm, leading to long-term cost savings, and thus minimizing the need for additional human resources for important tasks and system control [10].

Kinematic and reverse analysis has been performed internationally for prismatic and other shaped parts made of solid materials, but has not been performed for products made of flexible materials such as the sneakers used for indoor sport analyzed in this paper.

This paper uses inverse kinematics to automate 3d scanning for footwear used for indoor sport with the help of a robotic arm for which the minimum and maximum trajectory was simulated. The robotic arm proposed with a low investment and maintenance cost for automated 3D scanning is composed of three actuated joints with three degrees of mobility which can be controlled by a simple and non-expansive controller based on Arduino or Raspberry Pi. Sustainability is increasingly important in the footwear industry as consumers look for environmentally-friendly and sustainable options [11].

The use of a handheld 3D hand scanner contributes to increased sustainability by creating scanned models (databases) that are processed using specialized software to generate a 3D digital model of the feet for custom footwear design purposes, resulting in the production of only the exact amount of material required for each pair of shoes.

By 3D scanning the existing shoes used for indoor sports before a new product is made, this design can be checked and adjusted to reduce material loss and minimize energy consumption. Sustainability is important in the 3D scanning of indoor sports shoes for additive manufacturing using environmentally friendly materials and low energy technologies. Three-dimensional scanning of indoor sports shoes can be beneficial considering the sustainability perspective as it can help reduce waste and resource consumption by optimizing the production process. Three-dimensional scanning can enable the production of customized shoes without the need for standard sizes or large stocks of products [12].

This can reduce the amounts of materials and energy used in the manufacturing process, and helps reduce the environmental impact. Three-dimensional scanning can also help to achieve greater efficiency in the production of indoor sports footwear by identifying and eliminating potential problems and inconsistencies in product design and manufacture more quickly. Three-dimensional scanning can be a sustainable and efficient way to reproduce high-quality, customized shoes, reducing the impact on the environment and natural resources.

Sustainability is an important topic in any industry, including footwear production. By 3D scanning shoes, it is possible to gain an accurate picture of their shape and dimensions. This information can then be used to create virtual models of the shoes, which can be tested and adjusted to ensure a perfect fit. This process reduces the amount of material needed to produce sports footwear for indoor games, as there is no need to make multiple prototypes to find the right size. It also helps to reduce waste by producing shoes that can be worn and do not have to be discarded due to an inadequate fit.

Furthermore, the implementation of 3D footwear scanning can enhance the effectiveness of the manufacturing process by automating the cutting and pre-processing stages of materials, leading to a diminution in energy and resource consumption. To ensure sustainability in 3D shoe scanning, it is crucial to employ environmentally-friendly technologies and materials. By applying 3D scanning, designers can create precise and detailed models of sports shoes, enabling them to make accurate modifications for a perfect fit. Additionally, 3D scanning can be employed for prototyping and testing various materials and technologies, facilitating the identification of optimal design and production choices. Consequently, this aids in reducing the time and expenses associated with the development and production of sports shoes [13,14,15,16,17,18,19,20].

2. Materials and Methods

The GoSCAN SPARK, developed by Creaform, is a compact and portable 3D scanner that offers precise measurement and scanning capabilities. It is specifically designed for a wide range of applications, including creating 3D models for the textile industry, product development, quality control, reverse engineering, and heritage preservation. With its advanced features and versatility, it provides an efficient solution for object reconstruction and measurement necessities. The GoSCAN SPARK is a handheld 3D scanner that utilizes structured light technology to capture precise and detailed 3D scans of objects. It offers measurement accuracy as low as 0.05 mm and scanning resolution as low as 0.1 mm. Being user-friendly, the scanner is easy to handle, making it suitable for scanning complex objects. Additionally, it uses advanced technology to rapidly generate high-quality 3D scans of objects and surfaces [21,22,23,24,25,26,27].

The GoSCAN SPARK scanner does not require the surface of the indoor sports shoes to be pulverized with special spray for antireflection, compared to other types of laser scanners that require this preliminary operation. This scanner utilizes 3D hand scanning and reverse engineering technology, along with prototyping technology, to quickly measure and analyze the geometric data of sports shoes designed for indoor use/matches. This process involves analyzing the shape and appearance of real-world objects or environments to create three-dimensional digital models. The 3D hand scanning of sports shoes presented in this paper was performed on the outer surface of the shoe because a different approach is required for inner scanning, which will be presented in a future paper. Considering manual scanning with a 3D scanner with structured light for scanning several objects, as in the case of the present work (shoes used in indoor sports), scanning errors may occur. The GoSCAN 3D SPARK shown in Figure 1 is an optical device that projects white light strips onto scanned objects (in this case shoes used in indoor sports) and then registers the distortion of the lines by means of three cameras on the entire light strip, creating a surface formed by a cloud of points [28].

The necessary steps to build a 3D model of scanned sports footwear using a handheld 3D scanner are scanning the point cloud and real-time processing as a mesh in the VXelements software [29]. VXelements is a software application developed by Creaform, specifically designed for the acquisition and processing of 3D scan data from various scanning devices [30,31].

The processing of the point cloud resulting from the scanning of the sports shoes analyzed in this paper was performed using VXelements software by reducing the noise and the number of points using filtering algorithms from different assemblies and subassemblies in the case of products made from multiple 3D scans [32].

Structured light scanners use trigonometric triangulation, but instead of using laser light, these systems use a projector to display a series of linear patterns on an object such as the footwear intended for indoor sports [33]. Then, by examining each line in the pattern, it calculates the distance from the scanner to the surface of the sports shoe analyzed in this paper [34].

Figure 2 shows the principle of scanning indoor sports footwear using a 3D hand scanner with structured light [35].

Figure 3 presents model A and B of the footwear used for indoor sport (handball) which went through the 3D scanning process.

In the case of the GoSCAN SPARK scanner that was used for the scanning of the two models of shoes used for indoor sports, it is not necessary to spray the contact surface of the shoes analyzed, compared to other types of laser scanners which require this preliminary operation.

Figure 4a shows the steps of the 3D scanning process of footwear A and the scanning software, VX scan, which was an integral part of the scanning phase, meaning that the software parameters would have a real impact on the final rendering of the scans. The first scanning phase consisted of fixing shoe A on a table with predefined targets for 3D scanning, using a portable 3D scanner. The scanning was performed on the X and Y coordinate measurement directions relative to the Z coordinate system of the scanner.

In Figure 4b, the scanning process for footwear A’s sole is depicted. The sole of shoe A was fixed on a table with predetermined targets, allowing for 3D scanning. The scanning was performed in the X and Y coordinate measurement directions relative to the Z coordinate system of the scanner. Moving on to Figure 4c, it illustrates the software processing steps for shoe A after the 3D scanning process. Lastly, Figure 4d displays the 3D mesh shoe sole type A, which was processed using the VX scan software, at the end of the 3D scanning process.

In the initial scanning phase, shoe B was securely placed on a table with predefined targets for 3D scanning. A portable 3D scanner was used to measure the X and Y coordinates in relation to the Z coordinate system of the scanner. Figure 5a illustrates the scanning process for the sole of footwear B, where the sole was fixed to a table with predetermined targets for 3D scanning. The scanning was performed in real time, measuring the X and Y coordinates relative to the Z coordinate system of the scanner, as depicted in Figure 5b. Figure 5c showcases the shoe sole type B fixed on a table with preset targets for 3D scanning. Finally, Figure 5d displays the 3D mesh of shoe sole type B, processed using VX scan software.

Figure 6 shows the outer surface of the shoe type B in 3D mesh, processed using VX scan software, at the end of the 3D scanning process.

Given the weight of the handheld 3D scanner, which is about 2 kg, and the fact that the human factor must constantly maintain the same scanning distance and trajectory (and the software warns us in real time by distinct colors when the scanner is moving away from or towards the object), the human factor cannot constantly maintain the same distance between the scanner and the object.

Over the entire surface of the object to be scanned, the authors propose a solution for automating the scanning process that eliminates the above-mentioned situations for mass scanning of objects. The present work comes with a solution for automating the scanning process that eliminates the circumstances mentioned above in view of mass scanning of objects. Direct kinematics is a mathematical technique used to determine the position and orientation of an end effector (such as a 3D hand scanner), given the joints or displacements of an automated arm. Meanwhile, inverse kinematics is the mathematical technique used to determine the joints or displacements of an automated arm, given the desired position and orientation of a 3D hand scanner.

In the context of automated scanning, inverse kinematics can be used to determine the joint configurations required for the 3D hand scanner to reach a particular target point in space; in order to automate the scanning process by making use of a 3D hand scanner using inverse kinematics, the coordinate system for each joint of the arm is defined, as well as the scanner.

3. Results and Discussion

Automation of the Scanning Procedure

If the process of scanning has to be repeated multiple times and/or on multiple samples, a good time-saving solution would be to design a mechanical arm which can follow a programed path with a structured light scanner attached to the end point of the mechanism. A schematic of the scanning path is given in Figure 7.

The program is designed in such a way that the path composed of straight lines and arcs of circles is approximated by a set of points along this path linked with segments of a line of length given by an allowed approximation error.

Based on automated scanning with a 3D handheld scanner for the mechanical properties of the footwear scanned for finite element and thermal analysis in order to identify the areas that provide the best foot ventilation for comfort in movement, are going to be presented in a future paper.

Figure 8 shows the forward kinematic schematics used to automate the scanning process using a 3D handheld scanner.

In order to obtain automated scanning of a product we could use a robotic arm but this is not economically feasible. In our opinion, it is better to have a robot-like arm with three degrees of mobility (three actuated joints) which can be controlled by a simple and non-expensive controller.

The mechanical arm would have a planar motion to follow the path (like the one described in Figure 7) and in order to scan different sides of the object we can rotate the object (for example to scan the lateral side of the shoe).

A diagram of such a device is given in Figure 8. In order to control the movement of the scanner on a prescribed path, the program will need to be written by developing the forward and reverted kinematics of the device.

We assume that point C of the device has to reach consecutive points of the path—in Figure 8, the second quadrant of circle M, which is part of the scanning path. There is also a constraint that the l₃ segment of the arm has to be perpendicular on the tangent to the circle, so having the same orientation as the radius from point M to point C, defined by angle α. The perpendicularity constraint is given by the requirement that the scanner is attached to the l₃ segment with the scanning of the structured light the same orientation as the l₃ segment. If the path is a straight line, angle α is or the line’s slope. Then we can define angle α in accordance with angle δ which is the usual trigonometric orientation of the radius defined by points M and C, describe in Equation (1) as follows:

α = π − δ

(1)

Next is described the development of basic relations of the forward and inverse kinematics of the device. In forward kinematics the goal is to obtain the coordinates y_C and z_C when θ₁, θ₂, and θ₃ are known.

Using the notations in Figure 8, we can infer the values of the coordinates of point C as follows:

We can write angle α as a function of the known angles describe in Equations (2) and (3) as follows:

θ₁ − θ₂ = θ₃ − α

(2)

$${\mathsf{\theta }}_3+{ \mathsf{\theta }}_2-{\mathsf{\theta }}_1=\alpha$$

(3)

Equations (4) and (5) describe the coordinates of point B in terms of the length l₁ and angle θ₁.

$${\mathrm{y}}_{\mathrm{B}}={ \mathrm{l}}_1\cos {\mathsf{\theta }}_1$$

(4)

$${\mathrm{z}}_{\mathrm{B}}={\mathrm{l}}_1\sin {\mathsf{\theta }}_1$$

(5)

Equations (6) and (7) represent the distances DB and AD in terms of the length l₂ and the difference between angles θ₁ and θ₂.

$$\overline{\mathrm{DB}} ={\mathrm{l}}_2 \ast \cos \ast \left({\mathsf{\theta }}_1{-\mathsf{\theta }}_2\right)$$

(6)

$$\overline{\mathrm{AD}} ={\mathrm{l}}_2\sin \left({\mathsf{\theta }}_1{-\mathsf{\theta }}_2\right)$$

(7)

Equations (8) and (9) correspond to the distances FC and HC in terms of the length l₃ and angle α.

$$\overline{\mathrm{FC}} = {\mathrm{l}}_3\cos \left(\alpha \right)$$

(8)

$$\overline{\mathrm{HC}} ={\mathrm{l}}_3\sin \left(\alpha \right)$$

(9)

Equations (10) and (11) express the coordinates of point A in terms of the coordinates of point B and the distances DB and AD.

$${\mathrm{y}}_{\mathrm{A}}={ \mathrm{y}}_{\mathrm{B}}+\overline{\mathrm{DB}}$$

(10)

$${\mathrm{z}}_{\mathrm{A}}={\mathrm{z}}_{\mathrm{B}}+\overline{\mathrm{AD}}$$

(11)

Equations (12) and (13) provide the coordinates of point A in terms of the length l₁, the angles θ₁ and θ₂, and the length l₂.

$${\mathrm{y}}_{\mathrm{A}}={\mathrm{l}}_1\cos {\mathsf{\theta }}_1+{\mathrm{l}}_2\cos \left({\mathsf{\theta }}_1{ - \mathsf{\theta }}_2\right)$$

(12)

$${\mathrm{z}}_{\mathrm{A}}={ \mathrm{l}}_1\sin {\mathsf{\theta }}_1+{\mathrm{l}}_2\sin \left({\mathsf{\theta }}_1{ - \mathsf{\theta }}_2\right)$$

(13)

Equations (14) and (15) give the coordinates of point C in terms of the coordinates of point B and the distances DB, FC, AD, and HC.

$${\mathrm{y}}_{\mathrm{C}}={\mathrm{x}}_{\mathrm{B}}+\overline{\mathrm{DB}} +\overline{\mathrm{FC}}$$

(14)

$${\mathrm{z}}_{\mathrm{C}}={ \mathrm{z}}_{\mathrm{B}}+\overline{\mathrm{AD}} +\overline{\mathrm{HC}}$$

(15)

Equations (16) and (17) express the coordinates of point C in terms of the length l₁, the angles θ₁ and θ₂, the lengths l₂ and l₃, and angle α.

$${\mathrm{y}}_{\mathrm{C}}={ \mathrm{l}}_1\cos {\mathsf{\theta }}_1+{\mathrm{l}}_2\cos \left({\mathsf{\theta }}_1{-\mathsf{\theta }}_2\right)+{\mathrm{l}}_3\cos \left(\alpha \right)$$

(16)

$${\mathrm{z}}_{\mathrm{C}}={\mathrm{l}}_1\sin {\mathsf{\theta }}_1+{\mathrm{l}}_2\sin \left({\mathsf{\theta }}_1{-\mathsf{\theta }}_2\right)+{\mathrm{l}}_3\sin \left(\alpha \right)$$

(17)

Equations (1) to (17) represent the direct kinematic calculation of the positions for the end effector in which the 3D handle scanner is positioned to move on the established trajectory provided that the orientation of the structured light is perpendicular to the scanned surface.

The forward kinematic model is useful for modeling and simulation purposes in order to check the operation of the device.

Figure 9 shows the inverse kinematic scheme used to automate the scanning process using a handheld 3D scanner.

In the inverse kinematic model, the parameters y_C, z_C, l₁, l₂, l₃, and α are known, and the parameters of the controlled joints (represented by points O, B, and A in Figure 9), θ₁, θ₂, and θ₃ must be obtained. In the following stage we present the deduction of the joint control parameters θ₁, θ₂, and θ₃ for point C with orientation α and as a result of all the points along the scanning path. Equations (18) to (32) represent the inverse kinematic calculation of the positions for the end effector in which the 3D handle scanner is positioned to move on the established trajectory provided that the orientation of the structured light is perpendicular to the scanned surface.

Equation (18) relates the y-coordinate of point A to the y-coordinate of point C (after considering the length FC).

$${\mathrm{y}}_{\mathrm{A}}={\mathrm{y}}_{\mathrm{C}} - \overline{\mathrm{FC}}$$

(18)

Equation (19) relates the z-coordinate of point A to the z-coordinate of point C (after considering the length AF).

$${\mathrm{z}}_{\mathrm{A}}={ \mathrm{z}}_{\mathrm{C}}+\overline{\mathrm{AF}}$$

(19)

Equation (20) gives the y-coordinate of point A in terms of the y-coordinate of point C and the length l₃ and the cosine of angle α.

$${\mathrm{y}}_{\mathrm{A}}={\mathrm{y}}_{\mathrm{C}} - {\mathrm{l}}_3\cos \left(\alpha \right)$$

(20)

Equation (21) gives the z-coordinate of point A in terms of the z-coordinate of point C and the length l₃ and the sine of angle α.

$${\mathrm{z}}_{\mathrm{A}}={\mathrm{z}}_{\mathrm{C}}+{\mathrm{l}}_3\sin \left(\alpha \right)$$

(21)

Equation (22) establishes the relationship between the length l and the coordinates of point A using the Pythagorean theorem.

$${\mathrm{l}}^2={\mathrm{y}}_{\mathrm{A}}^2+{\mathrm{z}}_{\mathrm{A}}^2$$

(22)

Equation (23) determines the length of OK (one of the sides of the triangle) in terms of length l₁, length l₂, and the cosine of angle θ₂.

$$\overline{\mathrm{OK}} ={ \mathrm{l}}_1+{\mathrm{l}}_2\cos {\mathsf{\theta }}_2$$

(23)

Equation (24) determines the length of AK (another side of the triangle) in terms of length l₂ and the sine of angle θ₂.

$$\overline{\mathrm{AK}} ={ \mathrm{l}}_2\sin {\mathsf{\theta }}_2$$

(24)

Equation (25) relates the length l to the lengths of OK and AK using the Pythagorean theorem.

$${\mathrm{l}}^2={\overline{\mathrm{OK}} }^2+{\overline{\mathrm{AK}} }^2$$

(25)

Equation (26) expands Equation (25) to express the length l in terms of l₁, l₂, and the trigonometric components of angle θ₂.

$${\mathrm{l}}^2={\left({\mathrm{l}}_1+{\mathrm{l}}_2\cos {\mathsf{\theta }}_2\right)}^2+{\left({\mathrm{l}}_2\sin {\mathsf{\theta }}_2\right)}^2$$

(26)

Equation (27) relates the sum of the squares of the y-coordinate and z-coordinate of point A to the sum of the squares of l₁ and l₂, and the trigonometric components of angle θ₂.

$${\mathrm{y}}_{\mathrm{A}}^2+{\mathrm{z}}_{\mathrm{A}}^2={\mathrm{l}}_1^2 + 2{\mathrm{l}}_1{\mathrm{l}}_2\cos {\mathsf{\theta }}_2+{\mathrm{l}}_2^2{\left(\cos {\mathsf{\theta }}_2\right)}^2+{\mathrm{l}}_2^2{\left(\sin {\mathsf{\theta }}_2\right)}^2$$

(27)

Equation (28) simplifies Equation (27) by combining the trigonometric components.

$${\mathrm{y}}_{\mathrm{A}}^2+{\mathrm{z}}_{\mathrm{A}}^2={\mathrm{l}}_1^2 + 2{\mathrm{l}}_1{\mathrm{l}}_2\cos {\mathsf{\theta }}_2+{\mathrm{l}}_2^2\left[{\left(\cos {\mathsf{\theta }}_2\right)}^2+{\left(\sin {\mathsf{\theta }}_2\right)}^2\right]$$

(28)

Equation (29) solves the equation for angle θ₂ based on the expressions derived in Equation (28).

$$\cos \left({\mathsf{\theta }}_2\right)=\frac{{\mathrm{y}}_{\mathrm{A}}^2+{\mathrm{z}}_{\mathrm{A}}^2 - {\mathrm{l}}_1^2 - {\mathrm{l}}_2^2}{2·{\mathrm{l}}_1·{\mathrm{l}}_2}\to {\mathsf{\theta }}_2=\mathrm{acos}\left(\frac{{\mathrm{y}}_{\mathrm{A}}^2+{\mathrm{z}}_{\mathrm{A}}^2 - {\mathrm{l}}_1^2 - {\mathrm{l}}_2^2}{2·{\mathrm{l}}_1·{\mathrm{l}}_2}\right)$$

(29)

Equation (30) calculates the tangent of angle β using the z-coordinate and y-coordinate of point A.

$$\tan \mathsf{\beta }=\frac{{\mathrm{z}}_{\mathrm{A}}}{{\mathrm{y}}_{\mathrm{A}}}\to \mathsf{\beta }=\mathrm{atan}\left(\frac{{\mathrm{z}}_{\mathrm{A}}}{{\mathrm{y}}_{\mathrm{A}}}\right)$$

(30)

Equation (31) calculates the tangent of angle γ using the lengths AK and OK, and solves the equation for angle γ based on those trigonometric functions.

$$\tan \mathsf{\gamma } =\frac{\overline{\mathrm{AK}} }{\overline{\mathrm{OK}} }=\frac{{\mathrm{l}}_2\sin {\mathsf{\theta }}_2}{{\mathrm{l}}_1+{\mathrm{l}}_2\cos {\mathsf{\theta }}_2}\to \mathsf{\gamma } =\mathrm{atan}\left(\frac{{\mathrm{l}}_2\sin {\mathsf{\theta }}_2}{{\mathrm{l}}_1+{\mathrm{l}}_2\cos {\mathsf{\theta }}_2}\right)$$

(31)

Equation (32) describes the value of θ₁ as a function of various variables. It starts with θ₁ equaling the sum of β and γ expressed by relationships (30 and 31)

$${\mathsf{\theta }}_1=\mathsf{\beta }+\mathsf{\gamma } \to {\mathsf{\theta }}_1=\mathrm{atan}\left(\frac{{\mathrm{y}}_{\mathrm{A}}}{{\mathrm{x}}_{\mathrm{A}}}\right)+\mathrm{atan}\left(\frac{{\mathrm{l}}_2\sin {\mathsf{\theta }}_2}{{\mathrm{l}}_1+{\mathrm{l}}_2\cos {\mathsf{\theta }}_2}\right)$$

(32)

Introducing the coordinates of the set of points y_C and z_C and computing the joint angles with the above-presented relations, we can develop the control program of the device in order to follow the prescribed scanning path.

The programming of low-cost controllers is much easier and the cost of the device is much lower than in the case of a robot.

Figure 10a shows the modeling of the robot arm and the end effector, as we used V-RealmBuider (Matlab-Simulink design program) with the help of which it was possible to simulate the minimum and maximum trajectories of the robot arm as a function of angles θ₁, θ₂, and θ₃ shown in Figure 9.

Based on the diagram shown in Figure 10b we can analyze the feedback from the real environment where the scanner will be used.

Simulations were performed to generate the trajectory of the proposed arm for automated scanning using a Solidworks design program. In order to simulate the trajectories of the arm’s micrometer, the Motion Control function in Solidworks was used following the scanned path represented in Figure 8 and Figure 9.

Figure 11 shows the virtual simulation of the scanning of sneaker type A using a robotic arm. In this case, scanning it took 20 s to follow the proposed trajectory based on the mathematical model and inverse kinematics.

Figure 12 shows the time required for automated 3D scanning using a robotic arm for footwear type A along the point path on scanning trajectory shown in Figure 11.

Figure 13 shows the virtual simulation of the scanning of sneaker type B using a robotic arm. In this case it took 17.8 s to follow the proposed trajectory based on the mathematical model and inverse kinematics.

Figure 14 shows the time required for automated 3D scanning using a robotic arm for footwear type B along the point path on scanning trajectory shown in Figure 13.

Following the simulations performed, the scanning process and the times required for scanning the shoe types A and B are shown. In the simulation shown below, the paths are identical for both sneakers but the path times differ depending on the structure of the scanned sneaker.

4. Conclusions and Future Directions

In this paper, a proposal for automating the 3D scanning of footwear used in indoor sports (handball) was presented, following a predefined trajectory based on a proposed mathematical model for a robotic arm with three actuated joints, which can be controlled by a simple controller.

Digital 3D models of both sides of the shoes used in indoor sports were acquired and processed using VXelements software.

The results demonstrated effectiveness and fast scanning technique compared to other types of scanning in terms of the accuracy of the results, the possibility to obtain a complete picture of the condition of outdoor sports shoes, and the possibility of an automatic scanning procedure based on inverse kinematics using a robotic arm.

In fact, the presented inspection technique offers the possibility to study the condition of sports shoes as a whole and not only in specific points or target surfaces, in order to improve the data on their condition.

Based on the original 3D models of footwear used in indoor sports, it is possible to superimpose the scanned model with the original one, in order to measure the geometrical discrepancies, representing wear and surface deterioration.

Automated 3D scanning is a complex task that involves a combination of hardware and software, which saves time and increases the number of objects that can be scanned with a high degree of accuracy and precision.

The mechanical arm would have a planar motion to follow the trajectory and to scan different parts of the object. The research in this paper is concerned with the optimization of the hand scanning method and the use of a mathematical model to generate 3D scanning strategies on equipment intended for indoor sports with a focus on shoes used in handball matches. However, few papers have dealt with the generation of scanning paths on indoor sports footwear.

Direct kinematics is a mathematical technique used to determine the position and orientation of a 3D hand scanner given the joints and displacements of an automated arm.

In the context of automated scanning, inverse kinematics can be used to determine the joint configurations required for the 3D hand scanner to reach a particular target point in space.

In order to automate the scanning process using a 3D hand scanner using inverse kinematics, the chord system for each joint of the arm is defined, as well as the scanner.

The novelty of this article consists of generating the direct and inverse kinematics of a robotic arm in order to optimize the scanning process for footwear used in indoor sports, based on which a simulation was performed to determine the minimum and maximum trajectory depending on the angles presented having the mathematical model as a source.

The direct arm and the inverse kinematic model were defined taking into account the constraint that the last segment of the arm (l₃, represented in Figure 8 and Figure 9) has to be perpendicular to the circular trajectory. The mathematical model was deduced based on the arm geometry completed with the constraint that the scanner has to face the scanned objects on the whole trajectory.

Future Recommendations

Based on the mathematical model developed in this paper, it is possible to automate the scanning process by maintaining the proposed trajectory using an automated arm.

After obtaining the 3D model of two sneakers used in indoor sports through the 3D scanning process, the authors plan to perform finite element and thermal analyses to identify the areas that provide the best foot ventilation for comfort in movement, which will be presented in a future paper. The authors propose the manufacture of a robotic arm operating on a program assisted by a simple controller based on Arduino or Raspberry Pi in order to scan the self-mathematical model presented in this paper. The described solution represents a device with a low investment maintenance cost.