ABSTRACT This paper presents a novel axial-normal plane mutual interconversion algorithm for ball screws, which effectively bridges the gap between design and actual product. It is pivotal

in streamlining processes across the design, manufacturing, and inspection phases of ball screws. By forming the ball track, the mutual mapping matrix between the axial and normal planes

addresses key challenges and enhances engineering applications has been developed. The method simplifies the complex interaction within ball screws, offering a comprehensive model for better

understanding and application. Empirical evaluations demonstrate the efficiency and precision of our algorithm. In 3D modeling tests, it processes 10,000 points in merely 5 ms, with an

benefits, thereby advancing ball screw technology. SIMILAR CONTENT BEING VIEWED BY OTHERS ENABLING ADDITIVE MANUFACTURING PART INSPECTION OF DIGITAL TWINS VIA COLLABORATIVE VIRTUAL REALITY

Article Open access 30 November 2024 ERROR CONTROL METHOD USING CROSSED AXES STRATEGY IN THE FIVE-AXIS THREAD MILLING PROCESS Article Open access 01 April 2025 OPTIMIZING 2D IRREGULAR

PACKING VIA IMAGE PROCESSING AND COMPUTATIONAL INTELLIGENCE Article Open access 10 April 2025 INTRODUCTION The ball screw is a transmission component that can convert linear motion to rotary

motion and vice versa1, making it indispensable in modern machinery. Due to its high precision, large load capacity, long lifespan, and high efficiency, ball screws are extensively employed

in aerospace, CNC machine tools, automotive, military, and nuclear power industries. Consequently, research on ball screws has also been a significant area of focus in recent years. These

studies can be broadly categorized into two main areas. One category focuses on predicting their service life by establishing non-linear dynamic models2,3,4 to investigate the dynamic

characteristics of ball screws. This approach involves integrating reliability models5,6,7 of ball screws, fault diagnosis techniques8,9,10, and artificial intelligence11,12,13 to develop

comprehensive lifespan prediction systems for these critical components. The other category focuses on investigating methods to enhance the service life of ball screws, such as conducting

research into their friction and lubrication14,15,16,17, as well as their thermodynamic characteristics18,19,20. The focus of these studies is predominantly on finished ball screws. There is

a notable scarcity of research conducted from the perspective of ball screw manufacturers, investigating how to enhance the performance, and consequently the lifespan, of ball screws

through improvements in the process chain. The helical pair of the ball screw comprises balls and ball tracks, with the design and machining quality of these tracks directly influencing ball

screw performance21,22,23. There are two essential planes for the ball track: the axial plane and the normal plane, as illustrated in Fig. 1. The axial plane passes through the central axis

at any given position along the ball track, while the normal plane is perpendicular to the guiding helix at their intersection point on the axial plane. The angle between the axial plane

and the normal plane corresponds to the lead angle (\(\lambda\)) of the guiding helix. Although ball track design occurs in the normal plane, accurately locating this plane during practical

product testing is challenging. Consequently, direct detection from the normal plane suffers from poor repeatability and low efficiency, as only one groove can be tested at a time. In

contrast, the axial plane is easier to locate, allowing most ball track inspections to occur there before converting data to the normal plane via an axial-to-normal transformation algorithm.

Additionally, the assessment of the machining quality and performance of ball screws are typically conducted in the axial plane. Errors in the normal cross-section can lead to contact angle

deviations during machining, which in turn affect load-bearing capacity, wear, and other factors, eventually leading to lead errors. Alexander Denisenko24 has demonstrated computationally

that deviations in the normal cross-section have a significant impact on performance, and the lead error has a prevailing influence on the dispersion of the probabilistic distribution of the

axial force. Therefore, the reciprocal conversion algorithm between the axial and normal planes is crucial, as incorrect methods could adversely affect the ball track inspection,

manufacturing processes, and design, resulting in substandard products entering the market. Experimental results25,26 demonstrate that commercial profilometers equipped with algorithms that

convert axial cross-sections to normal cross-sections (A2N) simplify three-dimensional point motions into two-dimensional projections by projecting points from the normal plane directly onto

the axial plane. While straightforward, this method struggles with accuracy since it overlooks the complex geometric relationships and potential nonlinear variations between the axial and

normal planes. Wang et al.25 attempted to address it by connecting points in the two planes using a helical line and establishing equations through coordinate projection relationships,

devising a new A2N method. However, the resulting cross-section represents the normal plane within the axial plane rather than being an accurate normal plane cross-section, limiting its

applicability to small-lead ball screws. Building upon Wang’s model, Wu et al.26 optimized for computational efficiency, achieving an improvement of at least 70%. However, the fundamental

model remained unchanged. Miao et al.27 developed an algorithm converting the normal cross-section to the axial cross-section (N2A) based on the geometric positional relationship between two

points, focusing on a single circular raceway. This model lacks universality due to its reliance on single circular raceway characteristics, which have become largely obsolete with

advancements in manufacturing technology favoring double circular raceways (gothic grooves). Most of the ball screws on the market now use the double circular raceway. Previous studies have

investigated cross-section transformation algorithms unidirectionally without systematicity or thorough validation of algorithmic accuracy, often testing on small-lead ball screws where

inherent errors are minimal but overshadowed by manufacturing inaccuracies. To address these limitations, Section “Modeling” analyzes the groove machining process and establishes a mutual

mapping model of the two planes. The proposed model has been validated through theoretical analysis and practical experimentation in Section “Experiment and validation”, exploring its

applicability across various scenarios in Section “Analysis of application scenarios”. The application scenarios presented in Section “Analysis of application scenarios” directly correspond

to the three core stages of the process chain, demonstrating that the research in this paper can, to a certain extent, address the existing gap in improving the performance of ball screws

from a process chain perspective. MODELING The ball track of a ball screw can be conceptualized as being machined28 by a forming tool along a guiding helical path, as depicted in Fig. 1.

This guiding helix represents the ideal trajectory of the ball center as it traverses the ball track, conforming to a standard cylindrical helix. The diameter of this helix is termed as the

pitch circle diameter (PCD) \(D_{m}\) of the screw, while its axial advance per one complete turn is known as the lead of the screw \(P_{h}\). The relationship between these parameters gives

rise to the lead angle \(\lambda\), which can be mathematically expressed as follows: $$\lambda = \arctan \left( {\frac{{P_{h} }}{{\pi D_{m} }}} \right)$$ (1) The forming tool operates in

alignment with the normal plane. As it progresses along the guiding helical path, each point on the tool follows a spiral motion with a lead identical to that of the screw. Therefore, the

ball track can be considered as composed of a cluster of helical lines sharing the same pitch as the guiding helix. As shown in Fig. 2, in the axial plane, we define the central axis of the

screw as the \(z_{a}\) axis. Draw a perpendicular line from the intersection of the guiding helix with the axial plane to the \(z_{a}\) axis, intersecting at the point \(O\). The Cartesian

coordinate system established is denoted as \(Ox_{a} y_{a} z_{a}\), where the \(x_{a}\) axis represents the intersection line between the axial plane and the normal plane. Using the

\(x_{a}\) axis as the axis of rotation, rotate the \(z_{a}\) axis from the axial plane to the normal plane, resulting in the new coordinate system \(Ox_{n} y_{n} z_{n}\). For the right-hand

helical groove shown in Fig. 2, \(Ox_{n} y_{n} z_{n}\) can be regarded as obtained by counterclockwise rotation of the \(Ox_{a} y_{a} z_{a}\) around the \(z_{a}\) axis by an angle

\(\lambda\). Therefore, we can establish the transformation relationship29 between the two coordinate systems as follows: $${}^{a}P = rotx\left( \lambda \right){}^{n}P = \left[

{\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {\cos \lambda } & { - \sin \lambda } \\ 0 & {\sin \lambda } & {\cos \lambda } \\ \end{array} } \right]{}^{n}P$$ (2) $${}^{n}P

= rotx\left( { - \lambda } \right){}^{a}P$$ (3) where \({}^{n}P\) and \({}^{a}P\) respectively denote the description of the same point \(P\) in the coordinate systems \(Ox_{n} y_{n} z_{n}\)

and \(Ox_{a} y_{a} z_{a}\). Choose any helix from the cluster of helical lines. Its intersections with the axial plane and the normal plane are denoted as \(P_{i}^{\prime }\) and \(P_{i}\)

respectively. Thus, the problem of transforming the cross-section between the axial and normal planes can be framed as the problem of finding \(P_{i}^{\prime }\) from \(P_{i}\), or vice

versa. The process is referred to as helical mapping. MATRIX FOR MAPPING FROM THE NORMAL PLANE TO THE AXIAL PLANE Given that the coordinates of the point \(P_{i}\) in the normal plane are

represented as \(^{n} P_{i} = \left( {^{n} x_{i} ,0,^{n} z_{i} } \right)\) in the \(Ox_{{\text{n}}} y_{n} z_{n}\). After applying Eq. (2), we obtain its coordinates \(^{a} P_{i} = \left(

{^{a} x_{i} ,^{a} y_{i} ,^{a} z_{i} } \right)\) in the \(Ox_{a} y_{a} z_{a}\). According to the parameterization of the spiral curve30, we have: $$\left\{ {\begin{array}{*{20}l} {^{a} x_{i}

= r_{hi} \cos \theta_{i} } \hfill \\ {^{a} y_{i} = r_{hi} \sin \theta_{i} } \hfill \\ {^{a} z_{i} = \frac{{P_{h} }}{2\pi }\theta_{i} + z_{0i} } \hfill \\ \end{array} } \right.$$ (4) where

\(r_{hi}\) is the radius of the helical curve, \(\left( {0,0,z_{0j} } \right)\) is the coordinates of the helix’s starting point. In Eq. (4) the unknowns are \(r_{hi}\), \(\theta_{i}\) and

\(z_{0i}\). With three unknowns and three equations, there exists a unique solution. Thus, we can derive the parametric equations of the helical curve passing through the point \(P_{i}\) in

the \(Ox_{a} y_{a} z_{a}\): $$\left\{ {\begin{array}{*{20}l} {x = {}^{n}x_{i} \frac{\cos \theta }{{\cos \theta_{i} }}} \hfill \\ {y = - {}^{n}z_{i} \sin \lambda \frac{\sin \theta }{{\sin

\theta_{i} }}} \hfill \\ {z = \frac{{P_{h} }}{2\pi }\left( {\theta - \theta_{i} } \right) + {}^{n}z_{i} \cos \lambda } \hfill \\ \end{array} } \right.$$ (5) Here, \(\theta_{i}\) is referred

to as the helical travel angle, and its calculation formula is as follows: $$\theta_{i} = - \arctan \left( {\frac{{{}^{n}z_{i} \sin \lambda }}{{{}^{n}x_{i} }}} \right)$$ (6) When the

parameter \(\theta\) in Eq. (5) is such that \(\theta = 0\), the helical curve intersects the axial plane, yielding the point \(P_{i}^{\prime }\). Considering \(^{n} y_{i} =

{}^{a}y_{i}^{\prime } = 0\), it is not possible to obtain a non-singular matrix when expressed in matrix form. Thus, by omitting the y-axis coordinate, the following matrix form is obtained:

$$\left[ {\begin{array}{*{20}c} {{}^{a}x_{i}^{\prime } } \\ {{}^{a}z_{i}^{\prime } } \\ 1 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{1}{{\cos \theta_{i} }}} & 0

& 0 \\ 0 & {\cos \lambda } & { - \frac{{P_{h} }}{2\pi }\theta_{i} } \\ 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{}^{n}x_{i} } \\ {{}^{n}z_{i} } \\

1 \\ \end{array} } \right]$$ (7) This represents the mapping relationship between two 2D plane figures in 3D space. It can be abbreviated as follows: $${}^{a}P_{xzi}^{\prime } = {}_{n}^{a}

{\mathbf{T}}_{xzj} {}^{n}P_{xzi}$$ (8) Refer to \({}_{n}^{a} {\mathbf{T}}_{xzj}\) as the normal-to-axis mapping matrix. It’s determinant \(\det \left( {{}_{n}^{a} {\mathbf{T}}_{xzj} }

\right) \ne 0\). It has an inverse matrix: $${}_{n}^{a} {\mathbf{T}}_{xzj}^{ - 1} = \left[ {\begin{array}{*{20}c} {\cos \theta_{i} } & 0 & 0 \\ 0 & {\frac{1}{\cos \lambda }}

& {\frac{1}{\cos \lambda } \cdot \frac{{P_{h} \theta_{i} }}{2\pi }} \\ 0 & 0 & 1 \\ \end{array} } \right]$$ (9) MATRIX FOR MAPPING FROM THE AXIAL PLANE TO THE NORMAL PLANE Given

that the coordinates of a point \(P_{i}^{\prime }\) in the axis plane are \({}^{a}P_{i}^{\prime } = \left( {{}^{a}x_{i}^{\prime } ,{}^{a}y_{i}^{\prime } ,{}^{a}z_{i}^{\prime } } \right)\),

the helical curve passing through \(P_{i}^{\prime }\) in the \(Ox_{a} y_{a} z_{a}\) can be described as: $$\left\{ {\begin{array}{*{20}l} {x = {}^{a}x_{i}^{\prime } \cos \theta } \hfill \\

{y = {}^{a}x_{i}^{\prime } \sin \theta } \hfill \\ {z = {}^{a}z_{i}^{\prime } + \frac{{P_{h} }}{2\pi }\theta } \hfill \\ \end{array} } \right.$$ (10) This is the helical curve cluster,

denoted as \(Q\). Substituting into Eq. (3), we can describe it in the \(Ox_{a} y_{a} z_{a}\): $${}^{n}Q = rotx\left( { - \lambda } \right){}^{a}Q = \left[ \begin{gathered}

{}^{a}x_{i}^{\prime } \cos \theta \hfill \\ {}^{a}x_{i}^{\prime } \sin \theta \cos \lambda + \sin \lambda \left( {{}^{a}z_{i}^{\prime } + \frac{{P_{h} }}{2\pi }\theta } \right) \hfill \\ -

{}^{a}x_{i}^{\prime } \sin \theta \sin \lambda + \cos \lambda \left( {{}^{a}z_{i}^{\prime } + \frac{{P_{h} }}{2\pi }\theta } \right) \hfill \\ \end{gathered} \right]$$ (11) Let the helical

curve cluster \(^{n} Q\) intersect with the normal plane, yielding: $${}^{a}x_{i}^{\prime } \sin \theta \cos \lambda + \sin \lambda \left( {{}^{a}z_{i}^{\prime } + \frac{{P_{h} }}{2\pi

}\theta } \right) = 0$$ (12) It is a transcendental equation involving only one unknown, which can be solved to find the helical travel angle \(\theta_{i}\). Substituting \(\theta_{i}\) into

Eq. (11), it can be simplified into the following matrix form: $${}^{n}P_{xzi} = {}_{a}^{n} {\mathbf{T}}_{xzj} {}^{a}P_{xzi}^{\prime }$$ (13) where $${}_{a}^{n} {\mathbf{T}}_{xzj} = \left[

{\begin{array}{*{20}c} {\cos \theta_{i} } & 0 & 0 \\ 0 & {\frac{1}{\cos \lambda }} & {\frac{1}{\cos \lambda } \cdot \frac{{P_{h} \theta_{i} }}{2\pi }} \\ 0 & 0 & 1 \\

\end{array} } \right]$$ (14) Refer to \({}_{a}^{n} {\mathbf{\rm T}}_{xzj}\) as the axis-to-normal mapping matrix. It is identical to Eq. (9), thus mutually corroborating each other. THE

RESOLUTIONS OF CORE PROBLEMS The two mapping matrices described above are applicable to all helical components. However, the axis-to-normal mapping matrix is derived through the process of

deducing design parameters from the normal cross-section. At this stage, the coordinate system of the design parameters is unknown. It is necessary to deduce the coordinate system of the

normal cross-section based on the raceway data in the axial plane, specifically identifying the intersection between the axial plane and the normal plane. For the double circular raceway

ball screw, specifically referring to the \(x_{a} /x_{n}\) in Fig. 2. For the purely theoretical ball track, this intersection line represents the axis cross-section’s symmetrical line.

However, in practical machining, various errors can exist, causing the left and right circular arcs to be asymmetrical. Furthermore, for the transcendental equation (Eq. 12), approximate

analytical solutions may have lower accuracy, while iterative numerical solutions may not be computationally efficient. In engineering applications, it is essential to balance efficiency and

accuracy. DETERMINING THE INTERSECTION LINE Typically, in the design of the normal cross-section, the profile consists of dual circular arcs. Although transformed through normal-to-axis

mapping, it retains the characteristics of these dual circular arcs. The “ball-dropping method” can be used to establish the coordinates of the ball center. Specifically, this involves

dropping a virtual circle representing the ball and allowing it to naturally contact the ball track’s normal cross-section. By leveraging geometric properties, the coordinates of the ball’s

center can be calculated when it reaches equilibrium with the raceway. The intersection line \(x_{a} /x_{n}\) can be determined by drawing a perpendicular line from the ball center to the

central axis. There are two types of normal cross-section forms as shown in Fig. 3, referred to as the lower raceway and upper raceway. Let \(O_{o}\) denote the center of the virtual circle,

and \(O_{l} = \left( {x_{lc} ,y_{lc} } \right)\), \(O_{r} = \left( {x_{rc} ,y_{rc} } \right)\) represent the coordinates of the centers of the left and right circular-arc raceways, with

radii \(R_{l}\) and \(R_{r}\) respectively. The coordinates of \(O_{o}\) can be calculated based on the positions of the three centers \(O_{o}\), \(O_{l}\) and \(O_{r}\), as well as the

coordinate transformation. The coordinates of the lower raceway are: $$\left\{ {\begin{array}{*{20}c} {x_{o} = O_{r} O_{o} \cos \left( {\beta_{2} - \beta_{1} } \right) + x_{rc} } \\ {y_{o} =

O_{r} O_{o} \sin \left( {\beta_{2} - \beta_{1} } \right) + y_{rc} } \\ \end{array} } \right.$$ (15) The coordinates of the upper raceway are: $$\left\{ {\begin{array}{*{20}c} {x_{o} = O_{r}

O_{o} \cos \left( {\beta_{2} { + }\beta_{1} } \right) + x_{rc} } \\ {y_{o} = O_{r} O_{o} \sin \left( {\beta_{2} { + }\beta_{1} } \right) + y_{rc} } \\ \end{array} } \right.$$ (16) where

$$\left\{ {\begin{array}{*{20}l} {\beta_{1} = \arccos \frac{{\overline{{O_{r} O_{l} }}^{2} + \overline{{O_{r} O_{o} }}^{2} - \overline{{O_{l} O_{o} }}^{2} }}{{2 \cdot \overline{{O_{r} O_{l}

}} \cdot \overline{{O_{r} O_{o} }} }}} \hfill \\ {\beta_{2} = \arctan 2\left( {\frac{{y_{l} - y_{r} }}{{x_{l} - x_{r} }}} \right)} \hfill \\ \end{array} } \right.$$ (17) Here, the \(y_{o}\)

axis we sought is what we refer to as the \(x_{a} /x_{n}\). Let \(x_{o} = 0\), which means \(x_{i} - x_{o}\), indicating that the \(y_{o}\) axis coincides with the \(x_{a} /x_{n}\). However,

the coordinates \(O_{l}\),\(O_{r}\) and radii \(R_{l}\),\(R_{r}\) of the dual circular arcs in the axis plane are currently unknown. These can be obtained using circle fitting algorithms.

HIGH-PRECISION CIRCLE FITTING ALGORITHM Currently, the most used circle fitting algorithm is the least squares method. In the ball screw inspection systems developed by Feng31 and Zhao et

al.32, they employ the widely used Kåsa least squares algorithm33. The advantage of this algorithm is its efficiency and independence from coordinate systems. However, a drawback is that the

fitting result may tend to align towards minimizing the radius of the arc rather than minimizing the deviation. Henceforth, this paper employs the optimized least squares fitting algorithm

proposed by Pratt34. Formulate the problem of fitting a circle as an optimization task aimed at minimizing the following function: $$F_{p} \left( {x_{c} ,y_{c} ,R} \right) =

\frac{{\sum\limits_{i = 1}^{m} {\left[ {r_{i}^{2} - R^{2} } \right]}^{2} }}{{4R^{2} }}$$ (18) where \(\left( {x_{c} ,y_{c} } \right)\) represents the coordinates of the circle’s center,

\(R\) denotes the radius of the circle, \(\left( {x_{i} ,y_{i} } \right),i = 1,2, \ldots m\) are the data points involved in the fitting, and \(r_{i}\) represents the Euclidean distance from

\(\left( {x_{i} ,y_{i} } \right)\) to the fitted circle’s center. For generality, the equation of the circle is described as: $$A\left( {x^{2} + y^{2} } \right) + Bx + Cy + D = 0$$ (19)

When \(A = 0\), we obtain a circle with zero curvature, which corresponds to a straight line. When \(A \ne 0\), the coordinates of the circle’s center and the radius of the arc can be

expressed as: $$\left\{ {\begin{array}{*{20}l} {x_{c} = - \frac{B}{2A}} \hfill \\ {y_{c} = - \frac{C}{2A}} \hfill \\ {R = \frac{{\sqrt {B^{2} + C^{2} - 4AD} }}{2A}} \hfill \\ \end{array} }

\right.$$ (20) Then, Eq. (18) can be transformed into the following problem: $$\left\{ {\begin{array}{*{20}l} {\mathop {{\text{minimize}}}\limits_{A,B,C,D} \, f_{p} \left( {A,B,C,D} \right)

= \sum\limits_{i = 1}^{m} {\left[ {A\left( {x_{i}^{2} + y_{i}^{2} } \right) + Bx_{i} + Cy_{i} + D} \right]^{2} } } \hfill \\ {{\text{subject to }}g\left( {A,B,C,D} \right) = B^{2} + C^{2} -

4AD - 1 = 0} \hfill \\ \end{array} } \right.$$ (21) Expressing \(f_{p}\) and \(g\) as quadratic functions of \({\mathbf{a}}^{T} = [\begin{array}{*{20}c} A & B & C & D \\

\end{array} ]\), their respective Hermitian matrices are denoted as \({\mathbf{H}}_{f}\) and \({\mathbf{H}}_{p}\). Introduce the Lagrange multiplier \(\eta\) and simplify Eq. (21) into the

Lagrangian function: $$L\left( {{\mathbf{a}},\eta } \right) = {\mathbf{a}}^{T} {\mathbf{H}}_{f} {\mathbf{a}} - \eta \left( {{\mathbf{a}}^{T} {\mathbf{H}}_{p} {\mathbf{a}} - 1} \right)$$ (22)

Solving it yields the following equation: $${\mathbf{H}}_{p}^{ - 1} {\mathbf{H}}_{f} {\mathbf{a}} = \eta {\mathbf{a}}$$ (23) Given $${\mathbf{M}} = {\mathbf{H}}_{p}^{ - 1} {\mathbf{H}}_{f}

= \left[ {\begin{array}{*{20}c} { - \frac{1}{2}\sum\limits_{i = 1}^{m} {z_{i} } } & { - \frac{1}{2}\sum\limits_{i = 1}^{m} {x_{i} } } & { - \frac{1}{2}\sum\limits_{i = 1}^{m} {y_{i}

} } & { - \frac{m}{2}} \\ {\sum\limits_{i = 1}^{m} {x_{i} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i}^{2} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} } } & {\sum\limits_{i =

1}^{m} {x_{i} } } \\ {\sum\limits_{i = 1}^{m} {y_{i} z_{i} } } & {\sum\limits_{i = 1}^{m} {x_{i} y_{i} } } & {\sum\limits_{i = 1}^{m} {y_{i}^{2} } } & {\sum\limits_{i = 1}^{m}

{y_{i} } } \\ { - \frac{1}{2}\sum\limits_{i = 1}^{m} {z_{i}^{2} } } & { - \frac{1}{2}\sum\limits_{i = 1}^{m} {x_{i} z_{i} } } & { - \frac{1}{2}\sum\limits_{i = 1}^{m} {y_{i} z_{i} }

} & { - \frac{1}{2}\sum\limits_{i = 1}^{m} {z_{i} } } \\ \end{array} } \right]$$ (24) where \(z_{i} = x_{i}^{2} + y_{i}^{2}\). The eigenvector \({\mathbf{a}}_{{0}}\) corresponding to the

smallest non-negative eigenvalue \(\eta_{{0}}\) of the matrix \({\mathbf{M}}\) is the desired least squares solution35. By substituting into Eq. (20), the center and radius of the fitted

circular arc can be calculated. Compared to Kåsa’s method, this approach not only eliminates the influence of radius, allowing correct operation even in cases of very large curvature radii

and large noise of measured data but also integrates linear equations. This makes it highly effective in situations where it’s unclear whether the data to be fitted represents a line or an

arc. This is particularly suitable for the ball track composed solely of circular arcs and straight lines, as it does not require the selection of arc segments and enables automatic fitting.

EFFICIENT METHOD FOR CALCULATING THE HELICAL TRAVEL ANGLE To improve accuracy, a higher sampling frequency is typically used, and the number of data points for one cross-section can reach

tens of thousands. Therefore, the computational time of this model primarily depends on the speed of calculating the helical travel angle. This leads to the recommendation of preferring

analytical solutions when choosing algorithms. Transcendental equations do not have exact analytical solutions; only approximate analytical solutions are available. When the calculation

accuracy of the helical travel angle is not high, it significantly affects the accuracy of subsequent fitting computations. Therefore, it is preferable to prioritize using numerical

solutions in such cases. This paper combines the two approaches. Equation (12) becomes a transcendental equation because it simultaneously involves \(\sin \theta\) and \(\theta\). By setting

\(\sin \theta = \theta\), two approximate solutions are obtained: $$\left\{ {\begin{array}{*{20}l} {\theta_{ia} = - \frac{{{}^{a}z_{i}^{\prime } \sin \lambda }}{{{}^{a}x_{i}^{\prime } \cos

\lambda + \frac{{P_{h} }}{2\pi }\sin \lambda }}} \hfill \\ {\theta_{ib} = - \arcsin \frac{{{}^{a}z_{i}^{\prime } \sin \lambda }}{{{}^{a}x_{i}^{\prime } \cos \lambda + \frac{{P_{h} }}{2\pi

}\sin \lambda }}} \hfill \\ \end{array} } \right.$$ (25) The true solution \(\theta_{i}\) of Eq. (12) lies between \(\theta_{ia}\) and \(\theta_{ib}\). Using the interval \(\left[ {\min \{

\theta_{ia} ,\theta_{ib} \} ,\max \{ \theta_{ia} ,\theta_{ib} \} } \right]\), and then employing the robust and highly efficient bisection method36, to narrow down and find the solution. The

criterion for convergence can be set as \(eps < 10^{ - 9}\) to ensure high precision. EXPERIMENT AND VALIDATION VALIDATION OF MODELING ACCURACY BASED ON THE 3D MODEL The model proposed

in this paper includes the normal-to-axis mapping matrix, the axis-to-normal mapping matrix, and the core problem-solving algorithm. Previous research has not directly validated their

algorithms due to the challenge of machining purely theoretical raceways. In industry, ball screws are generally designed using 3D modeling software. By using 3D modeling software, it is

possible to generate theoretical ball tracks. VALIDATION OF THE NORMAL-TO-AXIS AND AXIS-TO-NORMAL MAPPING MATRICES This paper uses CATIA V5R20 for 3D modeling. To highlight errors, a large

lead ball screw with a model number of 1616 is employed, and its parameters are detailed in Table 1. The validation process is illustrated in Fig. 4. The validation results for the

normal-to-axis mapping matrix are shown in Table 2, and the validation results for the axis-to-normal mapping matrix are depicted in Fig. 5. From Table 2, it can be observed that the maximum

absolute error among the randomly selected 20 data points is \({2}{\text{.6}} \times {10}^{{ - 6}}\) mm. Upon checking the “Options” settings in CATIA V5R20 software, it is noted that the

software guarantees an accuracy of 0.001 mm. This matches the highest machining accuracy achievable in mechanical processing. Therefore, CATIA V5R20 itself can ensure that floating-point

calculation errors do not exceed \(1 \times 10^{ - 4}\) mm. This validates the accuracy of the proposed normal-to-axis mapping matrix. As shown in Fig. 5, it is evident that the absolute

errors for all parameters are less than \({10}^{{ - 3}}\). The largest absolute error is observed for the contact angle, which is \(2^{\prime \prime }\), with a relative error of only

0.001%. The maximum angular error in machining is \(30^{\prime \prime }\). The absolute error in the groove radius is \(7.7 \times 10^{ - 7}\) mm, while the highest machining precision in

mechanical processing is 0.001 mm. All of these validate the accuracy of the proposed axis-to-normal mapping matrix. VALIDATION OF CIRCLE FITTING ALGORITHMS To verify the superiority of the

Pratt’s algorithm over the Kåsa’s algorithm in the application scenario of this paper, an ideal circular arc was constructed with the coordinate origin (0,0) as the center and half the

diameter of the ball \(D_{b}\) in Table 1 as the radius \(r\). To simulate real-world machine deviations, noise was introduced to the radius by sampling from a normal distribution with mean

\(\mu = r\) and standard deviation \(\sigma\), i.e., \(r_{i} \sim N\left( {r,\sigma^{2} } \right)\). According to the Six Sigma methodology37, the standard deviation of typical raceways

varies depending on machining precision, generally set as \(\sigma = 0.001r\sim 0.005r\). The coordinates of the points were calculated using the parametric equations of a circle. The

parameter range was defined as \(\left( {0,\Psi } \right)\), where \(\Psi\) represents the central angle, set to 30°, 60°, and 90°, respectively. The interval \(\left( {0,\Psi } \right)\)

was equally divided into N segments. Both the Pratt’s and Kåsa’s algorithms were applied to fit the N data points, with evaluation metrics including fitted radius, distance between the

fitted center and the ideal center, and elapsed time. The experimental results are shown in Fig. 6. As shown in Fig. 6a and b, the radius fitted by the Kåsa’s algorithm is smaller than that

of the Pratt’s algorithm, consistent with theoretical analysis. The Kåsa’s algorithm performs adequately only under low noise (\(\sigma = 0.001r\)). When \(\sigma = 0.005r\), its maximum

radius error reaches a significant 19.1%. In contrast, Pratt’s algorithm demonstrates higher robustness. Although its fitting accuracy slightly degrades under high-noise conditions compared

to low-noise scenarios, this can be compensated by increasing the data size N. For N = 10,000, both the radius and center fitted under high-noise conditions approach the ideal values, with a

maximum error of only 0.025%. Under identical noise levels, Pratt’s algorithm exhibits lower accuracy than the Kåsa’s algorithm only when N = 1,000. Regarding the central angle, the

accuracy of both algorithms improves as \(\Psi\) increases. However, the Kåsa’s algorithm is more sensitive to this effect: when \(\Psi\) increases from 30° to 90°, its error decreases from

19.1% to 0.3%. Pratt’s algorithm remains much more robust, with a maximum error below 0.6%. As illustrated in Fig. 6c, both algorithms exhibit comparable elapsed times, averaging around 3

ms. The computational efficiency of Pratt’s algorithm is not significantly inferior. The analysis leads to the following conclusions: The computational efficiencies of the Kåsa’s and Pratt’s

algorithms are similar. However, the Kåsa’s algorithm demonstrates advantages only under low noise, small datasets, and large arcs. In contrast, the Pratt algorithm offers higher robustness

and superior accuracy in most scenarios, except for small datasets. Considering the application context here, where \(\Psi\) typically approximates 60° and the collected data points N

generally exceed 10,000, Pratt’s algorithm is more advantageous. VALIDATION OF INTERSECTION LINE DETERMINATION In engineering practice, the typical constraint is that \(\lambda \le 32.5^{

\circ }\). A larger lead angle can lead to vibrations, compromising the stability of the screw. The maximum lead angle \(\lambda = 32.5^{ \circ }\) is chosen here for experimental design. To

simulate the asymmetry of left and right circular arcs encountered in actual machining, we adjusted the parameters such as the contact angle and conformity for left and right circular arcs.

Twelve sets of normal cross-sections are designed accordingly. The normal-to-axis mapping matrix is utilized to obtain data points in the axial plane. Subsequently, the circle fitting

algorithm is applied to get the centers and radii of the arcs. Finally, the “ball-dropping method” is employed to compute the coordinates of the ball’s center \(\left( {x_{o} ,y_{o} }

\right)\). As \(\Delta x = \left| {x_{o} } \right|\) approaches 0, it indicates that the circle fitting algorithm is more accurate, resulting in more precise intersection determination.

Refer to the calculation results in Table 3. As shown in Table 3, the absolute errors are all less than \(1 \times 10^{ - 9}\) mm. These errors are mainly due to computational precision

limitations. This is sufficient evidence to demonstrate the high accuracy of the algorithm proposed in this paper. VALIDATION OF THE HELICAL TRAVEL ANGLE CALCULATION ALGORITHM In the field

of computer science, algorithm evaluation metrics are typically based on computational complexity. This is due to the characteristics of this domain, namely high concurrency, high

performance, and large-scale data handling. In contrast, the domain relevant to this paper clearly does not exhibit these characteristics. The field of mechanical engineering is primarily

concerned with achieving optimal real-time performance at a sufficient level of accuracy. Consequently, a direct comparison of the time and accuracy required to process the same data volume

on the same hardware platform suffices. To satisfy the requirement for the highest angular machining accuracy, the error must meet at least \(eps \le 10^{ - 4}\). Considering that the

helical travel angle error influences the subsequent fitting error, a value of \(eps = 10^{ - 9}\) is therefore adopted. The helical travel angle calculation algorithm proposed in this

document is referred to as m.dic. There are multiple methods available to solve Eq. (12). Firstly, there are some approximation approaches. In Eq. (25), the two approximate solution formulas

are denoted as m.a and m.b. Taking the average of \(\theta_{ia}\) and \(\theta_{ib}\) as the solution, denoted as \(\theta_{{\text{m}}}\), is referred to as method m.m. Expand \(\sin

\theta\) at \(\theta = 0\) using Taylor expansion to form a cubic polynomial and a quintic polynomial. The eigenvalues of the companion matrix of these polynomials are the solutions. These

two methods are denoted as m.n3 and m.n5, respectively. The above cubic polynomial also has Cardano’s formula solution method, which is denoted as m.car. Subsequently, there are numerical

solution methods. In addition to the previously mentioned bisection method, commonly used methods include the golden section method and Newton–Raphson method. The golden section method,

using the same initial interval as the m.dic, is denoted as m.gol. The Newton–Raphson method requires initial values. It is computed with initial values of 0, \(\theta_{ia}\),

\(\theta_{ib}\), \(\theta_{{\text{m}}}\), denoted as m.new0, m.newa, m.newb, m.newm. Let \(\theta_{i}^{\prime }\) represent the helical travel angle calculated by various algorithms for a

chosen point in the normal plane, and \(\theta_{i}\) denote the true value. The absolute error is given by \(\Delta \theta = \left| {\theta_{i}^{\prime } - \theta_{i} } \right|\). Given

$$L\Delta \theta = \log_{10} \left( {\Delta \theta } \right)$$ (26) \(L\Delta \theta\) indicates the order of magnitude of the error. An error reaching \(1 \times 10^{ - 19}\)(that is,

\(L\Delta \theta = - 19\)) is considered as effectively zero. The algorithms were executed using Octave 8.4 software on a computer equipped with an Intel i9-13980HX processor (2.20 GHz),

running a 64-bit Windows operating system. The elapsed time for each algorithm is shown in Fig. 7, while the errors are depicted in Fig. 8. In both figures, the algorithm proposed in this

paper m.dic is highlighted in red for distinction. As can be seen from Fig. 7, the algorithm proposed in this paper requires only around 5 ms for every 10,000 computations, with only m.a,

m.b, and m.m showing higher efficiency. Meanwhile, from Fig. 8, it is evident that the errors of these three algorithms increase with the rise in lead angle, and none of them essentially

meets the requirements of \(eps < 10^{ - 9}\). Overall, the algorithm proposed in this paper combines high efficiency with high accuracy. VALIDATION OF MODELING ACCURACY BASED ON THE

ACTUAL PRODUCT Validation through the actual product can demonstrate the practical engineering applicability of the model proposed in this paper. The current optimal approach involves

directly measuring the axial plane profiles and normal plane profiles of the same track of an actual machined large lead screw to verify the accuracy of the model. EXPERIMENTAL EQUIPMENT The

experimental equipment is a commercial profilometer, such as the one shown in Fig. 9, specifically the Optacom VC-10 by Dantsin. The mechanical part of the equipment consists of three main

components: the motion unit, the probe, and the rotating swivel vise. The motion unit consists of two linear modules, providing only two degrees of freedom. The probe is mounted on a carbon

fiber rod and makes direct contact with the test specimen. The vise is divided into three layers, enabling rotation and tilting to specific angles by rotating different layers. When the vise

clamps the screw and the first layer is rotated to the lead angle, enabling direct measurement of the screw’s normal profile. EXPERIMENTAL PROCEDURE AND RESULTS The experimental procedure

is depicted in Fig. 10. For the same track of a single screw, data of the cross-sections in both the normal and the axial plane are collected by varying the installation method on the vise.

The normal profile data is processed using the built-in algorithm of the profilometer to calculate its parameters, denoted as N1. For the axial profile data, two different transformation

algorithms are employed. The results obtained from the built-in algorithm of the profilometer are recorded as A1, and the results from the Axis-to-Normal Mapping Algorithm (ANMA, Eqs. 14–25)

proposed in this paper are recorded as A2. Three tracks on the screw are randomly selected to follow the procedure for inspection. The average of these measurements is taken as the final

inspection result, as shown in Fig. 11. As shown in Fig. 11, from the perspective of contact angle, the absolute errors of A1 compared to N1 are \(\left[ {2^{ \circ } 30^{\prime } 17^{\prime

\prime } ,2^{ \circ } 45^{\prime } 39^{\prime \prime } } \right]\), and the absolute errors of A2 compared to N1 are \([7^{\prime } 1^{\prime \prime } ,7^{\prime } 32^{\prime \prime } ]\).

From the perspective of ball track radius, the absolute errors of A1 compared to N1 are \(\left[ { - 0.0021\;{\text{mm}},0.0159\;{\text{mm}}} \right]\), and the absolute errors of A2

compared to N1 are \([ - 0.0023\;{\text{mm}}, - 0.0070\;{\text{mm}}]\). It can be observed that the error between A2 and the direct measurement of the normal profile is small. The maximum

relative error in the contact angle is 0.28%, and the relative error in the ball track radius is no more than 0.46%. Considering the manufacturing and measurement errors, the accuracy of the

results from A2 is quite high. However, the results in the ball track radius from A1 are marginally acceptable. The errors in contact angle from A1 exceed \(2^{ \circ } 30^{\prime }\),

which is unacceptable and could significantly mislead onsite machining processes. ANALYSIS OF APPLICATION SCENARIOS INSPECTION The most direct application of the mutual mapping model

proposed in this paper is in the detection of ball track parameters. Currently, various manufacturers use commercial profilometers for inspection. Due to drawbacks such as low efficiency and

difficulty in locating the rotation axis with direct detection from the normal plane, it is common to perform detection from the axial plane and then transform the data to the normal plane.

However, as indicated by the detection results in Section “Validation of modeling accuracy based on the actual product”, the algorithm in commercial profilometers exhibits significant

errors in the case of large leads. The mutual mapping model can be used to calculate and provide the applicable range for this measurement method. First, analyze the impact of each design

parameter on the error. Using the parameters in Table 1 as a baseline, vary one design parameter at a time to obtain the normal cross-section. Then, use the normal-to-axial mapping matrix

(Eqs. 6 and 7) to obtain the axial cross-section. The axial cross-section is transformed into the normal cross-section using the ANMA as well as the algorithm from commercial profilometers

and the ball track parameters are calculated subsequently. The difference in contact angle and arc radius calculated by the two algorithms serves as the evaluation metric, with specific

results shown in Fig. 12. Figure 12 consists of a series of atypical stem plots. In stem plots, the length of the stems represents the contact angle error of the commercial profilometer,

while the area of the circles terminating each stem is proportional to the error in raceway radius; specifically, larger circles correspond to larger errors in raceway radius. Moreover, when

the diameters of circles are relatively similar, their colors differentiate their relative sizes. The more the color leans towards blue, the relatively smaller the circle; conversely, the

more it leans towards yellow, the relatively larger it is. It can be observed that design parameters such as lead angle, pitch circle diameter, ball diameter, designed contact angle, and

conformity all influence the error of the commercial profilometer. Specifically, the error in contact angle is positively correlated with changes in lead angle and ball diameter, and

negatively correlated with changes in pitch circle diameter, designed contact angle, and conformity. The error in raceway radius is positively correlated with changes in lead angle, ball

diameter, and conformity, and negatively correlated with changes in pitch circle diameter and designed contact angle. Based on this analysis, the applicable range of the axial plane

measurement method using a commercial profilometer can be determined to meet certain error requirements. In engineering, manufacturers prioritize whether the contact angle meets

specifications. Typically, when the contact angle error \(\varepsilon_{\alpha } \le 1^\circ\), the resulting raceway radius error generally falls well below the specified tolerance.

Moreover, the conformity of most ball screw designs is set to 0.55, with a contact angle of 45°. Based on this, the applicable range charts for commonly used specifications (screw diameter

4-100 mm) are provided, shown in Figs. 13 and 14, respectively. Where the contact angle error \(\varepsilon_{\alpha } \le 0.5^\circ\) is suitable for the design of high-precision ball

screws, and \(\varepsilon_{\alpha } \le {1}^\circ\) is suitable for the design of standard-precision ball screws. For these two charts, first, determine your requirements by selecting

\(\varepsilon_{\alpha } \le 0.5^\circ\) or \(\varepsilon_{\alpha } \le 1^\circ\) based on your needs. Then, locate the desired screw diameter on the vertical axis. The length of the row

represents the corresponding lead value, while the numerical value outside the row indicates the diameter of the balls. For instance, if one needs to inspect a high-precision screw with a

diameter of 40mm, they should locate the row corresponding to a diameter of 40mm in the vertical axis in Fig. 13. From the chart, it can be seen that where the designed pitch circle diameter

\(D_{m} \ge 40{\text{mm}}\), lead \(P_{h} \le 15{\text{mm}}\), ball diameter \(D_{b} \le 7.5{\text{mm}}\), contact angle \(\alpha \ge 45^{ \circ }\), and conformity \(f_{v} \ge 0.55\), then

the indirect measurement method using a commercial profilometer is applicable. DESIGN In cases where there are specific requirements for the axial profile, it is necessary to use the mutual

mapping model proposed in this paper for simulation and assist with design work in the normal plane. For example, in the design of a ball screw, when the lead is fixed, increasing the ball

diameter is necessary to enhance load-bearing capacity. At present, the maximum allowable ball diameter is typically determined based on empirical experience. This issue can be addressed

using the normal-to-axial mapping matrix (Eq. 7) or the axial-to-normal mapping matrix (Eq. 14). For commonly used leads, standard steel balls are selected according to ISO 3290-1:200838 to

participate in the simulation. The maximum diameter of the ball that ensures no interference between adjacent axial grooves is considered the maximum steel ball diameter for the

corresponding lead. It is illustrated in Fig. 15. A ball screw manufacturer received a special order requiring the design of a 1604 model ball screw with a rated dynamic load capacity of up

to 7 kN within a specific space. The spatial dimensions were constrained, limiting the number of ball turns to a maximum of three. Based on empirical practice, the maximum usable ball size

for the 1604 model is typically 3.175 mm, a solution universally adopted by leading international manufacturers. However, as indicated in Fig. 15, for a screw shaft with a lead of 4 mm, the

maximum usable ball size can be up to 3.5 mm. With our assistance, the manufacturer successfully designed and produced a 1604 model ball screw utilizing 3.5 mm balls, as illustrated in Fig. 

16. The optimized ball screw exhibited a 13.8% increase in dynamic load rating, reaching 7.4 kN, thereby satisfying the customer’s requirements. Furthermore, the static load rating also

improved by 9.8%, achieving 13.2 kN. The ball screw has successfully passed customer acceptance and is currently being utilized in volume production within the customer’s equipment.

MANUFACTURING In factories, ball tracks are usually created by the grinding wheels, which are dressed by diamond rollers. Diamond rolls shape the profile of the grinding wheel, which then

acts like a forming tool like the one shown in Fig. 1 with an angle relative to the screw, known as the wheel tilt angle. Typically, the wheel tilt angle is equal to the lead angle.

Factories generally develop a series of diamond rolls for grinding wheels tailored to standard series of screws. When customers have specific requirements, the use of existing diamond rolls

and how to apply them effectively can often be a challenge. Most enterprises rely on workers to adjust the wheel tilt angle based on experience and conduct repeated trials. This can consume

a lot of labor and resources. However, by using the model proposed in this paper, simulations can be conducted to save significant time and effort. A manufacturer possesses an idle diamond

roller with left and right arc contact angles of \(41^{ \circ } 52^{\prime } 20^{\prime \prime }\) and \(41^{ \circ } 48^{\prime } 57^{\prime \prime }\), and left and right arc radii of

3.837 mm and 3.844 mm, respectively. Its pitch circle radius is 57 mm, and its lead is 12 mm. To maximize its utilization, the customer intends to employ this roller for machining products

of the same model requiring a contact angle of \(45{ ^\circ } _{{ - 3^\circ }}^{{0^\circ }}\) and a radius of \(3.858 \pm 0.03\;{\text{mm}}\). If the wheel tilt angle remains set at \(3.8^{

\circ }\), it certainly won’t meet the requirement. Based on the parameters of the diamond roll, establish the profile in the normal plane. Set the lead angle of the model to the desired

wheel tilt angle. Then, simulate by repeatedly calculating using the normal-to-axial mapping matrix (Eq. 7) and the axial-to-normal mapping matrix (Eq. 14). Ultimately, it is determined that

setting the wheel tilt angle to \(2^{ \circ }\) achieves the desired groove profile. In this case, without considering machining errors, the left and right arc contact angles are \(42^{

\circ } 25^{\prime } 44^{\prime \prime }\) and \(42^{ \circ } 6^{\prime } 41^{\prime \prime }\) respectively, with left and right arc radii of 3.836mm and 3.843mm, meeting the requirements.

The same worker was instructed to first adjust the wheel tilt angle empirically through trial machining until the inspection results met the requirements. After four rounds of adjustment,

the worker set the grinding wheel angle to \(2^{ \circ } 18^{\prime }\), a value very close to our calculated value. Subsequently, to evaluate the proposed method, the wheel tilt angle was

directly set to \(2^{ \circ }\) based on our calculation results, and the machining outcomes were then inspected. The comparative results are presented in Fig. 17. As shown in Fig. 17, both

the proposed solution and the empirical solution successfully met the requirements, with comparable results. However, the time required to obtain the first quality part using proposed

solution was only 25% of that needed by the empirical solution. This significant difference is because the empirical solution necessitates iterative machine tool adjustments based on

inspection results, where the number of adjustments is dependent on the operator’s individual experience. In contrast, the proposed solution simulated this process on a computer using the

mapping model presented in this paper. Consequently, we achieved a qualified product with only a single machine tool adjustment. While the presence of machining errors resulted in minor

discrepancies between the actual contact angle and raceway radius and the calculated values, these results still met the design tolerances. Furthermore, each machine tool adjustment

contributes to the deterioration of its precision, potentially leading to a complete loss of machine accuracy after a certain number of adjustments. Therefore, the proposed method not only

effectively reduces material waste and improves machining efficiency but also extends the service life of the machine tool. CONCLUSIONS This paper establishes a mutual mapping model of

cross-sections between the axial and normal plane of ball screws, which effectively addresses various design, manufacturing, and inspection issues encountered in practical engineering

applications. The contributions and conclusions are as follows: * (1) Based on the formation of the ball track on the ball screw, the ball track is described by a cluster of helical lines

with the same lead. By applying coordinate transformations, these helical lines are described separately in the coordinate systems of both the normal plane and the axial plane, thereby

establishing the mutual mapping model. * (2) Derived from practical engineering needs, the core issues in the mutual mapping model are addressed and solved. Firstly, employing the

“ball-dropping method” to determine the intersection between the axial plane and the normal plane, ensures that the maximum absolute error is less than \(1 \times 10^{ - 9}\) mm. Then, to

enhance the precision of circular fitting, Pratt’s least squares fitting algorithm is adopted as an alternative to the widely used Kåsa’s least squares algorithm in the industry. This not

only eliminates the influence of radius but also expands its applicability, enabling automated fitting of groove profiles. Finally, when solving the transcendental equation for the helical

travel angle, approximate solutions at both ends are taken as the bracketing interval and the bisection method is used for solving. This approach achieves an absolute error of less than \(1

\times 10^{ - 9}\) degrees, with the computation time per ten thousand points being only about 5ms. Among the 12 algorithms, this approach achieves the highest computational efficiency with

\(eps < 10^{ - 9}\). * (3) The accuracy of the proposed mutual mapping model has been verified through experiments using both the 3D model and actual products. In the actual product

testing, compared with the profile of the model 1616 ball screw directly detected from the normal plane, the proposed algorithm’s relative errors in contact angle and groove radius were both

less than 0.5%. In contrast, the commercial profilometer showed a contact angle relative error exceeding 5%, with an absolute error reaching \(2^{ \circ } 30^{\prime }\). This demonstrates

that the commercial profilometer’s algorithm has limitations, being suitable only for the detection of groove profiles in ball screws with small lead angles. * (4) This paper provides

guidance for the mutual mapping model’s application in design, manufacturing, and inspection, with an example in each area. The commercial profilometer’s applicable range is specified,

aiding enterprises in achieving more precise detection results. The maximum usable ball diameter for common leads is provided, assisting in the design of heavy-load ball screw assemblies. In

manufacturing, the model’s computational simulations enable the maximized utilization of a single diamond roller, reducing labor and material waste in factories. The model established in

this paper holds significant importance for engineering applications. Additionally, the raceway mathematical model developed during the modeling process can be utilized to create a digital

twin of the ball screw. Furthermore, this model is not limited to ball screws. It can be applied to any helical component, such as triangular threads, trapezoidal screws, and planetary

roller screws, by simply designing the corresponding profile cross-section in the normal plane. DATA AVAILABILITY The datasets generated during and/or analyzed during the current study are

not publicly available, but are available from the corresponding author on reasonable request. CODE AVAILABILITY The algorithm presented in this paper is not proprietary and can be

implemented by following the descriptions provided. As we currently lack the resources to maintain a dedicated code repository, interested readers who require the corresponding algorithm

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balls, ISO 3290-1, (2008). Download references ACKNOWLEDGEMENTS We would like to express our gratitude to Kede Numerical Control Co., Ltd. and Lianyungang Screws Robot Technology Co., Ltd.

for reporting the engineering challenges they encountered and for providing us with ball screw samples and testing equipment. This work is supported by the National Science and Technology

Major Projects of China (No. TC230H0AG-70). AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094,

China Chaoqun Qian, Yi Ou, Hutian Feng & Jian Wu Authors * Chaoqun Qian View author publications You can also search for this author inPubMed Google Scholar * Yi Ou View author

publications You can also search for this author inPubMed Google Scholar * Hutian Feng View author publications You can also search for this author inPubMed Google Scholar * Jian Wu View

author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS C.Q.: Methodology, Conceptualization, Formal analysis, Investigation, Data Curation, Writing,

and Visualization. Y.O.: Validation, Resources, Writing—Review & Editing, Supervision, and Funding acquisition. H.F.: Writing—Review & Editing, and Supervision. J.W.: Data Curation.

CORRESPONDING AUTHOR Correspondence to Yi Ou. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. ADDITIONAL INFORMATION PUBLISHER’S NOTE Springer Nature

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http://creativecommons.org/licenses/by-nc-nd/4.0/. Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Qian, C., Ou, Y., Feng, H. _et al._ A mutual mapping model of cross-sections

between the axial and normal planes of ball screws based on the forming of helical ball track. _Sci Rep_ 15, 17205 (2025). https://doi.org/10.1038/s41598-025-02034-7 Download citation *

Received: 01 August 2024 * Accepted: 09 May 2025 * Published: 17 May 2025 * DOI: https://doi.org/10.1038/s41598-025-02034-7 SHARE THIS ARTICLE Anyone you share the following link with will

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content-sharing initiative KEYWORDS * Ball screw * Ball track * Spiral mapping * Least square fitting * Engineering application