Simple and High‐Precision Hand–Eye Calibration for 3D Robot Measurement Systems

Three dimensional (3D) robot measurement systems, with binocular planar structured light cameras (3D cameras) installed at the terminal flange of robots, are widely used to measure complete workpieces by stitching point clouds obtained from various sampled poses. To ensure accurate stitching, hand–eye calibration is necessary. However, 3D cameras often demonstrate low precision in measuring jumping edges and vertices, resulting in challenges when selecting appropriate calibrators and calibration methods for hand–eye calibration. We propose a simple and high‐precision hand–eye calibration method for 3D robot measurement systems. Initially, a single sphere is utilized as the calibrator, and its poses are observed using specific robotic motions. Subsequently, the hand–eye calibration problem is formulated as a well‐known equation, AX=XB$A X = X B$ , relying solely on the high‐precision relative pose of the robot. Finally, a novel simultaneous solution for AX=XB$A X = X B$ and the corresponding closed‐form initial solution are introduced. Simulations and experiments confirm that the proposed method exhibits high noise resistance and achieves high‐precision calibration, even with a limited calibration data quantity. Compared with traditional methods, the fitting error of the proposed method can be reduced from over 0.9 mm to less than 0.6 mm.

for their hand-eye calibration.Therefore, hand-eye calibration methods of RGB cameras are also not directly applicable to 3D cameras.Calibrators with circular features can be used as substitutes for checkerboard grids and QR codes for 3D cameras.Calibration plates with multiple standard holes were used for calibration by identifying the circle centers and obtaining the poses of the plates. [22,23]Nevertheless, accurately identifying the circle centers become challenging when the camera's line of sight deviates from the normal direction of the calibration plate.This issue can be avoided using a spherical calibrator.However, the symmetrical shape of a sphere makes it difficult to identify its pose, rendering the construction of the matrix equation AX ¼ XB infeasible.Limited by this, some methods performed handeye calibration based on the position reprojection of the sphere center. [24,25]However, these calibration processes are related to the absolute robot pose, and the orthogonality of the hand-eye matrix is not guaranteed, which reduces the accuracy of the hand-eye calibration.In addition, iterative closest point methods were used to obtain the relative pose of the calibrator between different sampling poses, which require well-characterized calibrators for registration and are time-consuming. [26,27]In summary, choosing a sphere as the calibrator and constructing the matrix equation AX ¼ XB is suitable for the hand-eye calibration of 3D robot measurement systems once the pose of the sphere can be obtained.Therefore, in this study, a single sphere was employed and its pose was actively constructed using point clouds acquired from specific sampled poses.
The approaches employed to solve the equation AX ¼ XB substantially influence the calibration accuracy and can be categorized as either stepwise or simultaneous methods depending on whether the rotation and translation components of the hand-eye matrix are solved separately or jointly, respectively.The stepwise method is commonly closed-form and efficient, in which the matrix equation AX ¼ XB is divided into a nonlinear rotation equation and a linear translation equation.The axis angle, [28,29] quaternion, [30] and Euclidean group [31] were often used to represent the rotation component and convert the nonlinear rotation equation into a linear form.Finally, the rotation and translation components of the hand-eye matrix can be solved sequentially.However, in the stepwise method, the rotation error propagates to the translation component, resulting in lower precision in the translation component.Simultaneous methods have been employed to address this issue.The dual quaternion, [32] Kronecker product, [33] and screw motion [34] were often used to transform the nonlinear equation AX ¼ XB into a linear equation, enabling a closed-form solution for the rotation and translation components of the hand-eye matrix.[37] Horaud et al. [35] proposed a widely recognized optimization method for hand-eye calibration.They combined the rotation and translation errors in the equation AX ¼ XB by formulating a unified objective function and deriving a concise expression.To ensure the orthogonality of the hand-eye matrix, they introduced a penalty term in their study.However, the optimization problem they formulated was nonconvex, requiring alternative methods to obtain suitable initial values.Moreover, their method exhibited poor convergence even with reasonably chosen external initial values.To achieve global convergence, heuristic algorithms, such as the branch-andbound algorithm, have been employed to solve similar nonconvex optimization problems in hand-eye calibration. [38]owever, heuristic methods are inefficient.Alternatively, the nonconvex optimization problem was converted into a convex secondorder cone programming problem, thereby achieving fast global convergence. [39]However, the orthogonality of the rotation matrix was not adequately bounded in their study, reducing the handeye calibration accuracy.The primary goal of hand-eye calibration is to enhance measurement accuracy.Horaud's method, a simultaneous optimization method, rigorously enforces the orthogonality of the rotation matrix, theoretically offering substantial calibration precision.Thus, this study advanced Horaud's method by integrating quaternion and Kronecker product techniques to enhance its convergence.Additionally, a closed-form initial value was introduced within the same framework.
The three primary contributions of this study are as follows: 1) A simple and high-precision hand-eye calibration method was proposed for 3D robot measurement systems, including a method for constructing AX ¼ XB using a single spherical calibrator and an improved simultaneous solution for AX ¼ XB.
2) The pose of a single spherical calibrator was actively constructed using point clouds from specially sampled poses, allowing the pose of the single sphere to be measured using a 3D camera.
3) The improved simultaneous solution for AX ¼ XB addresses the shortcomings of the original approach, including poor convergence and dependency on external methods.
The remainder of the paper is organized as follows.Section 2 illustrates the principle of hand-eye calibration, including the method for constructing AX ¼ XB and the improved simultaneous solution for AX ¼ XB.Section 3 and 4 demonstrate the performance of the proposed method using simulations and experiments, respectively.Finally, Section 5 presents the conclusions of this study.

Method
Before introducing the calibration method, it is necessary to explain the symbols commonly used in this study.As illustrated in Figure 1, the robot base, robot end, camera, and workpiece coordinate systems are denoted by {B}, {E}, {C}, and {W}, respectively.The transformation relationships between these coordinate systems are represented using homogeneous matrices.For instance, B E T represents the position of {E} in {B} or the transformation from {B} to {E}.R and t denote the rotation matrix and translation vector of the homogeneous matrix T, respectively, with the expression T ¼ R t 0 1 .In the matrix equation AX ¼ XB, the matrices are defined as A ¼ , where the subscripts i and j indicate the different sampled poses.The objective of hand-eye calibration is to solve for X, which represents the transformation from {E} to {C}.

Method for Constructing AX ¼ XB
In this study, a single sphere was selected as the calibrator, with its center obtained easily by fitting the point cloud on its surface.However, the pose of the single sphere cannot be obtained easily due to its symmetry.To solve this problem, multiple point clouds from special sampled poses were used to construct the pose of the sphere proactively.Figure 2 illustrates the principle of obtaining the pose of a single sphere.The coordinate system {W} is established at the sphere's center with the same pose as {B}.The pose of {W} in {B} can be expressed as The pose of {W} in the camera coordinate system fC i g can be expressed as Suppose that there is point A in {W}, then point A can be expressed in fC i g based on the transformation relationship between {W} and fC i g, as follows By moving the robot a distance kWA !k in the opposite direction of WA !, where the camera coordinate system after movement is defined as fC j g, the pose of {W} in fC j g and the coordinate of point W in fC j g can be described as W t can be obtained using Equation ( 4) and ( 6) when B RÞ.This demonstrates that the coordinates of any point in fC i g can be represented by the coordinates of point W in fC j g if the robot is translated in the opposite direction at an equal distance.Therefore, the coordinates of points B, C, and D on the X, Y, and Z axes of {W} can be obtained.The unit axes of {W } are denoted as and the posture of {W} in fC i g is represented by . Gram-Schmidt orthogonalization [40] is used to ensure the orthogonality of C i W R. Finally, the pose of {W} in fC i g can be expressed as Upon acquiring the poses of the calibrator, the matrix equation AX ¼ XB can be formulated, thereby eliminating the need for absolute robot poses.This pose acquisition method eliminates the need for complex calibrators, such as multiple spheres or calibration plates with multiple standard holes.Additionally, subsequent experiments demonstrated the insensitivity of the calibration outcomes to the accuracy of the calibrator.Using a table tennis as the calibrator yielded a calibration accuracy comparable to that achieved with a high-precision standard sphere.

Simultaneous Solution Method for AX ¼ XB
In this section, a simultaneous method for solving the equation AX ¼ XB is introduced, which is regarded as an improvement over Horaud's method.The rotation components of A, B and C are represented by quaternion that incorporates the rotation angle information, while the translation error is simplified using Kronecker product.These modifications contribute to the enhanced convergence of Horaud's method.Additionally, within the same framework, a method for acquiring the closed-form Pose acquisition of a single sphere.
www.advancedsciencenews.com www.advintellsyst.cominitial value is proposed, effectively eliminating the reliance on the external initial value.The principle of the method is as follows.
The equation AX ¼ XB is divided into rotation and translation components, as follows The problem of solving the matrix equation AX ¼ XB is then transformed into an optimization problem of the form min f ¼ min , where kR i k 2 and kt i k 2 represent the rotation and translation errors in the ith sampled pose, respectively.Equation ( 6) can be rewritten using a quaternion as The symbols related to quaternion are bolded in this paper.
In Equation ( 7), q A , q B , and q X ¼ ½q 0 q 1 q 2 q 3 T are quaternions representing R A , R B , and R X , respectively.The conjugate quaternion of q X is denoted as q X .Additionally, t A , t B , and t X ¼ ½0 t 1 t 2 t 3 T are pure quaternions in the form of ½0 t A T , ½0 t B T , and ½0 t X T , respectively.The matrix R A is given as 1 0 0 R A .The symbol Ã represents quaternion multiplication, which can be expressed as matrix multiplication in the form The symbols used in Equation ( 8) are for explanatory purposes and hold no significance elsewhere in this paper.
By utilizing the properties of the quaternion and Kronecker product, the rotation and translation errors in the ith sampled pose can be simplified as follows The variables involved in Equation ( 10) as represented as Function vec() refers to the vectorization operation of a matrix, as follows where col denotes the column vector of the matrix.The Kronecker product is denoted as ⊗, and its expression is The symbols used in Equation ( 12) and ( 13) are explanatory and hold no significance elsewhere in this paper.
Finally, the problem is transformed into an optimization problem with a constraint, as follows where 14) allows for the simultaneous solution of the matrix equation AX ¼ XB.To avoid an order of magnitude difference between the rotation and translation errors and prevent one of the errors from being ignored in the optimization process, the two error components were normalized based on their corresponding initial error values.This normalization was performed using α 1 and α 2 as normalization coefficients.
The optimization problem is solved iteratively using the interior point method. [41]To facilitate this, the Lagrange multiplier μ is introduced to construct the Lagrange function, as follows where the parameter vector x ¼ ½q X , t X , μ T represents the variables to be solved.
In each iteration, the incremental change in x, denoted by Δx ¼ ½Δq X , Δt X , Δμ T , can be obtained by solving the following equation Δq X Δt X Δμ where , and the subscripts represent the elements within the corresponding matrix.
The partial derivatives in Equation ( 16) can be expressed as where the subscripts represent the elements within the corresponding matrix.
The solution to Equation ( 16) is obtained as Δx ¼ ðH T HÞ À1 H T g.The parameter x ¼ ½q X , t X , μ T is then updated using a step length β, which is determined using a backing line search (if β tends to 0, a trust region step is employed instead). [41]The iteration continues until kΔxk becomes smaller than a predefined threshold ξ thresh or the maximum iteration limit L max is reached.
The optimization problem presented in Equation ( 14) is nonconvex, making it necessary to obtain a reliable initial value.To address this issue, we proposed a method for determining the closed-form initial values q 0 and t 0 based on Equation ( 14), thereby eliminating the need to rely on other methods.According to Equation ( 6), the true hand-eye calibration matrix X can simultaneously minimize the rotation and translation errors (min or minimize the rotation error alone (min Consequently, the solution to the latter problem min P N i¼1 ðkR i k 2 Þ should serve as a suitable initial value for the former problem Equation (14).Referring to Equation ( 9), the optimization problem min where the global solution q 0 can be effectively utilized as a suitable initial value for Equation (14).
The Lagrange multiplier method is employed to solve this optimization problem, resulting in the formulation of a new objective function, as follows where λ represents the Lagrange multiplier.The partial derivatives of Equation ( 20) with respect to q X and λ are then obtained as where λ and q X are the eigenvalue and eigenvector of matrix A, respectively.The minf ðq X Þ ¼ λ can be obtained by substituting Equation ( 21) into (20).Therefore, the global solution q 0 for Equation ( 19) can be represented by the eigenvector corresponding to the smallest eigenvalue of matrix A, as follows Because matrix A is at least a positive semi-definite matrix, q 0 can always be obtained.
By substituting q X in Equation ( 14) with q 0 , we derive where the global solution t 0 serves as a good initial value for Equation (14), according to the theory of step-by-step iteration. [19]he optimization problem represented by Equation ( 23) is convex because matrix C is a positive semi-definite matrix.
The global solution t 0 , obtained by differentiating Equation (23), is demonstrated as In summary, the proposed method for solving AX ¼ XB involves two main steps.First, the closed-form initial value is obtained using Equation ( 22) and (24).Second, this initial value is substituted into Equation ( 14) to derive the final solution.The pseudo-code for the calibration process is presented in Algorithm 1.The method for acquiring the closed-form initial value is based on the simultaneous optimization framework, eliminating the need for external initial values.Moreover, the initial value within the same framework closely approximates the final solution, resulting in a significantly reduced solution time, as demonstrated by subsequent simulations and experiments.

Simulation Process
Simulations were conducted to verify the effectiveness of the proposed method.The simulations were composed of three steps: simulation data generation, hand-eye calibration, and result assessment, as depicted in Figure 3.

Data Generation
The data generation process is illustrated in the upper part of Figure 3.During the actual calibration, errors existed in the values of B E T provided by the robot and C W T measured by the camera.To closely approximate a real scene, this study employed the following methods to represent robot and camera errors.Robot errors arising from geometric parameter errors in the manufacturing process were represented by the improved Denavit-Hartenberg method. [9]The measurement error of a 3D camera, arising from the baseline, tilt, focal length, and image resolution errors, [42] has not yet been sufficiently studied.Some studies have used polynomials and Gaussian processes to fit the relationship between the camera error and measured values. [43,44]Similarly, this study proposed an m-order error model for the camera with reference to the measurement principle and empirical error distribution of the 3D camera, [11,12,42] as follows 8 > > > < > > > : where m signifies the order of error models.x, y, and z represent the measured coordinate values, while Δx, Δy, and Δz correspond to the errors along each axis.k is the error coefficient, and i, j, u, v, w, r, and s are nonnegative integers.The camera error was added to the unit axis of the coordinate system {W}, which in turn affected C W T. The ideal B E T 0 was calculated using the ideal robot forward kinematic model, followed by the generation of the ideal C W T 0 using The actual B E T and C W T were then generated by incorporating errors from both the robot and camera.In addition to these two errors, the actual measurements included noise.Rotation random noise was added as angleðRÞ þ e rot , e rot ∼ Nð0, σ rot Þ, and e rot was random noise obeying a Gaussian distribution Nð0, σ rot Þ.The function angle() was used to extract the rotation angle from the rotation matrix.Translation random noise was added as t þ e tran , e tran ∼ Nð0, σ tran Þ, where R and t represent the rotation matrix and translation vector of T, respectively.

Hand-Eye Calibration and Methods for Comparison
After obtaining the simulation data, B E T and C W T were substituted into Algorithm 1 to solve the hand-eye matrix E C T. To evaluate the performance of the proposed method, several methods for solving AX ¼ XB were chosen for comparison, such as Andreff 's simultaneous closed-form method using Kronecker product (Andreff 's method), [33] Horaud's simultaneous nonlinear optimization method (Horaud's method), [35] Zhao's simultaneous convex optimization method (Zhao's AX ¼ XB method), [39] and Ali's simultaneous nonlinear optimization method (Ali's method). [37]In addition, Fu's simultaneous closed-form initial solution using Kronecker product for reprojection error equation (Fu's method), [24] Zhao's simultaneous convex optimization method for equation AX ¼ ZB (Zhao's AX ¼ ZB method), [19] and Wang's simultaneous nonlinear optimization method for Calculate q 0 and t 0 using Equation ( 22) and ( 24); Set q 1 ¼ q 0 , t 1 ¼ t 0 , and Set the number of iterations l ¼ 1; While l < L max ∩ kΔxk < ξ thresh , do: Calculate Δx ¼ ðH T HÞ À1 H T g; ideal camera data equation AX ¼ ZB (Wang's method) [18] were employed for comparison to verify the necessity of formulating AX ¼ XB.Gram-Schmidt orthogonalization was employed to ensure the orthogonality condition of hand-eye matrices solved by Andreff 's, Zhao's AX ¼ XB, Ali's, Fu's, and Zhao's AX ¼ ZB methods.

Evaluation Criteria
The evaluation criteria for the solution comprised three key metrics: solution time, rotation error, and translation error.The rotation error was quantified as absðangleð

Solving Efficiency and Accuracy
The results of each method are presented in Table 2, and the iterative processes of the optimization methods are shown in Figure 4.When employing error-free data for calibration, the proposed method successfully derived the accurate hand-eye matrix, thereby affirming the correctness of the method's principle.When using data with errors for calibration, several conclusions can be drawn from the analysis presented in Table 2. First, the optimization methods (including the proposed, Horaud's, Zhao's AX ¼ XB, Ali's, Zhao's AX ¼ ZB, and Wang's methods) generally required more solution time than the closed-form methods (Andreff 's and Fu's methods).However, the proposed method outperformed the other optimization methods in terms of solution time, except for Wang's method, which benefitted from an initial value that closely approximated the final solution.This is evident in Figure 4a, where the final function value closely aligned with that of the 1st generation.Second, the proposed method demonstrated superior accuracy compared to the other methods in terms of both rotation and translation errors.The rotation error was significantly smaller than those of the other methods, and the translation error was slightly worse than that of Wang's method, on par with that of Ali's method, and smaller than those of the other methods.In contrast, Horaud's method failed to achieve an accurate calibration, as shown in Table 2 and Figure 4b.This indicates that the proposed method effectively improved the convergence of Horaud's method.Third, although Wang's method exhibited a slightly smaller translation error compared to the proposed method, the latter demonstrated distinct advantages in rotation accuracy.The overall performances of Fu's, Zhao's AX ¼ ZB, and Wang's methods were weaker than that of the proposed method.This underscores the superiority of the proposed method for leveraging the relative pose of a robot using the AX ¼ XB equation.Finally, the methods that ignore or relax the orthogonality condition of the rotation matrix (Andreff 's, Zhao's AX ¼ XB, Ali's,  Fu's, and Zhao's AX ¼ ZB methods) did not yield satisfactory results.In summary, in the proposed method, the combination of orthogonality preservation and the use of the relative robot poses resulted in an improved calibration performance compared to the other methods evaluated.

Data Quantity Requirement
A minimum of three sets fA, Bg was required for calibration, and the calibration accuracy increased as more data were used. [28,29]o investigate the impact of the data quantity on the accuracy of each method, we observed the error variation for each method by gradually increasing the data quantity while setting the rotation and translation noises to σ rot ¼ 0.1 and σ tran ¼ 0.1, respectively.The results are shown in Figure 5. Ali's method was found to require a minimum of four sets fA, Bg to achieve calibration.
As the data quantity increased, the rotation and translation errors of all methods, except for Horaud's method, showed a notable decrease.With respect to the rotation error, the proposed method consistently outperformed the other methods, even with less calibration data.Regarding the translation error, the proposed method was slightly inferior to Wang's method, and slightly superior to Ali's method, but significantly outperformed all other methods.This outstanding performance can be attributed to the incorporation of additional orthogonal constraints in the proposed method, resulting in reduced data requirements.

Noise Tolerance
To investigate the noise tolerance of the proposed method, random noise was gradually increased by multiplying a factor (called noise level) with the rotation and translation noise parameters.
The resulting error variances for each method are shown in Figure 6.Regarding the rotation error, the proposed method demonstrated the smallest slope, indicating superior noise tolerance compared to the other methods.Regarding the translation error, the performance of the proposed method was similar to that of Wang's method while still surpassing that of the other methods.In contrast, Horaud's method consistently failed to achieve accurate calibration, highlighting the significant improvement of the proposed method on Horaud's method.The ABB IRB1410 robot, with a repeatable positioning accuracy of AE0.05 mm, was utilized along with the Surface HD50 3D camera, which has a repeatable accuracy of AE0.15 mm.Table tennis with a diameter of %40 mm was selected as the calibrator, while a ceramic standard sphere with a diameter of 38.1 mm and a diameter deviation of no more than 1 μm was used for comparison.An Intel(R) Core(TM) i5-9400 F CPU and 16 G RAM were used to perform all the calculations.The calibration system and calibrators are shown in Figure 7a-c.The robot was controlled to capture the point clouds of the table tennis and standard sphere from various sampled poses, with the robot poses B E T fed back by the robot controller.Subsequently, these point clouds were segmented, and the position of the sphere center was determined using spherical fitting, as shown in Figure 7d.The method described in Section 2.1 was then employed to calculate the poses C W T of the table tennis and standard sphere.

Evaluation Criteria
In the actual experiment, the true E C T 0 was unavailable, rendering the direct application of the evaluation criteria used in Section 3.1 unfeasible.The purpose of hand-eye calibration was to improve the accuracy of the measurement system.Therefore, we employed the 3D reconstruction error [17] (3DRE) and distance measurement error (DME) of the measurement system as indirect criteria of the hand-eye calibration accuracy.The 3D reconstruction error was determined as follows.After hand-eye calibration, the standard sphere was placed in a different position, and the robot measurement system captured its sphere center position from various poses.These sphere center positions in the camera coordinate system {C}, denoted as C W t, were then transformed into the robot base coordinate system {B} by The repeated position accuracy of B W t served as a measure of the system's 3D reconstruction error.The distance measurement error was calculated as follows.After hand-eye calibration, the standard sphere was fixed on a linear slide table with a distance accuracy of AE0.03 mm, as shown in Figure 7c.C W t 1 was obtained from different sampled poses and then converted to B W t 1 .The linear slide table was then moved by a known distance, and the process was repeated to obtain B W t 2 .The mean value of absðk B W t 1 À B W t 2 k À distanceÞ was used as a measure of the system's distance measurement error.Furthermore, the solution time was also considered to evaluate the solution efficiency of each method.In addition to these metrics, the performance of each method on actual measurements was studied,  and the fitting error (FE) was used to evaluate each method, which is discussed at the end of this section.

Solving Efficiency and Accuracy
In the experiment, 15 sets fA, Bg were utilized for each method to solve the hand-eye matrix, and 50 sets of f B W t, B W t 1 , B W t 2 g were employed to evaluate the 3D reconstruction and distance measurement errors.The evaluation results are presented in Table 3 and the iterative processes are shown in Figure 8.Using a table tennis vs. a standard sphere as the calibrator had minimal impact on the calibration results.This may be because the measurement process was influenced by the comprehensive accuracy of the robot, camera, and calibrator.The overall measurement accuracy was primarily limited by the least accurate component, making the use of a high-precision calibrator less influential.Regarding the solution efficiency, the proposed method remained significantly faster than the other methods, except for Wang's method and the closed-form methods.In terms of the 3D reconstruction accuracy, the proposed method performed the best, followed by Ali's, whereas the other methods demonstrated poor performance.Similar patterns were observed for the distance measurement accuracy results.Horaud's method yielded a hand-eye matrix that was noticeably unrealistic, resulting in the largest error.These findings indicate that the proposed method is a highly accurate calibration method with medium efficiency.The characteristics of the iterative process were consistent with those observed in the simulation.

Data Quantity Requirement
Figure 9 depicts the error variation for each method as more data are included in the calibration process.The choice of the calibrator had a limited impact on the results.As the data quantity increased, all methods, except for Horaud's method, showed a gradual improvement in accuracy.The proposed method consistently outperformed the other methods with equivalent data quantity.Remarkably, the proposed method achieved superior results even with limited data, surpassing the performance of other methods relying on larger data sets.These results align with the simulation findings and further validate the capability Figure 8. Iterative process with 15 sets fA, Bg, using the table tennis and standard sphere as the calibrator, respectively.a) Proposed method, b) Horaud's method, c) Zhao's AX ¼ XB method, d) Ali's method, e) Zhao's AX ¼ ZB method, and f ) Wang's method. (a) Impact of increasing data quantity on 3DRE and DME, respectively (images within the same row share the same Y-axis label).a) 3DRE using the table tennis of methods (except for Horaud's method) and b) Horaud's and Fu's methods.e) DME using the table tennis of methods (except for Horaud's method) and f ) Horaud's and Fu's methods.c) 3DRE using the standard sphere of methods (except for Horaud's method) and d) Horaud's and Fu's methods.g) DME using the standard sphere of methods (except for Horaud's method) and h) Horaud's and Fu's methods.
of the proposed method to achieve a highly accurate calibration, even with limited data.

Performance in Practical Application
Hand-eye calibration plays a critical role in improving the measurement accuracy of 3D robot measurement systems, enabling more precise measurements of workpieces.For practical applications, evaluating the performance of each calibration method is crucial.To achieve this, a standard sphere was used as the workpiece to be measured.The robot was controlled to capture local point clouds of the standard sphere from different sampled These point clouds were then stitched together using the hand-eye matrix obtained from each calibration method.The performance of each method was assessed based on the stitching quality of the point clouds.
Figure 10 and 11 illustrate the stitching results using the table tennis and standard sphere as the calibrator, respectively.To demonstrate the stitching quality clearly, point clouds from the three sampled poses were stitched together.The stitching result of Horaud's method is not presented because it exceeded the solving range of the computer.Moreover, the choice of calibrator had a minimal impact on the stitching results.Regarding the stitching quality, the proposed method exhibited the smallest error in the stitching point clouds, followed by Ali's and Wang's methods, which performed slightly worse.In contrast, the other methods yielded poor stitching results.These visual results provide graphical evidence that the proposed method achieved the highest accuracy in solving a hand-eye matrix.To compare the performance of each method quantitatively, additional point clouds from multiple sampled poses were stitched and fitted into spheres.The deviation between the fitted diameter and ideal value (fitting error) was used to quantify the accuracy of each  method.Remarkably, the point clouds stitched using the proposed method, Ali's method, and Wang's method were successfully fitted into spheres, whereas the other methods failed.
The fitting errors of each method are listed in Table 3, and the proposed method exhibited the smallest fitting errors.The stitching and fitting results of the proposed, Ali's, and Wang's methods are shown in Figure 12.Some flaws can be observed, particularly in the results of Ali's, and Wang's methods, further supporting the superiority of the proposed method for achieving highly accurate calibration results.

Conclusion
Hand-eye calibration is crucial for the accurate utilization of 3D robot measurement systems.Nevertheless, 3D cameras have inadequate accuracy when measuring jumping edges and vertices, making it difficult to refer to hand-eye calibration methods for RGB cameras.To address this issue, a simple and highprecision hand-eye calibration method was proposed in this study.In this method, only a single sphere is used as the calibrator, eliminating the dependence on complex calibrators.The poses of a single sphere were obtained through specific robot motions, and subsequently, the equation AX ¼ XB was formulated, relying on high-precision relative robot poses.A simultaneous solution for AX ¼ XB was proposed in which the orthogonality of the hand-eye matrix was ensured.
Simulations and experiments demonstrate that the method has a high calibration accuracy, great noise resistance, and minimal data quantity requirements.In practical applications, point clouds stitched using the proposed method exhibited superior accuracy compared with traditional methods.However, flaws remain in the stitched point clouds, which can be attributed to the omission of robot and camera errors in this study.Calibrating these errors is crucial to further enhance the measurement accuracy of the system.Stitching results (more sampled poses) and fitting results using the table tennis as the calibrator.a) Proposed method, b) the fitting result of (a); c) Ali's method, d) the fitting result of (c); e) Wang's method, f ) the fitting result of (e).Stitching results (more sampled poses) and fitting results using the standard sphere as the calibrator.g) Proposed method, h) the fitting result of (g); i) Ali's method, j) the fitting result of (i); k) Wang's method, l) the fitting result of (k).

Algorithm 1 .
Calibration process pseudo-code.Input: N sets of A and B (N þ 1 sets of B E T and C W T) Output: Hand-eye matrix E C T Calculate A, B, C, D, and E as per Equation (14);

Figure 3 .
Figure 3. Data flow of the simulation.

Figure 6 .
Figure 6.Impact of increasing noise on calibration error using 15 sets fA, Bg. a) Rotation error of methods (except for Horaud's method), b) Translation error of methods (except for Horaud's method), and c) Rotation and translation errors of Horaud's method.

Figure 5 .
Figure 5. Impact of increasing data quantity on calibration error with rotation and translation noise parameters set to σ rot ¼ 0.1 and σ tran ¼ 0.1, respectively.a) Rotation error of methods (except for Horaud's method), b) Rotation error of Horaud's and Fu's methods, c) Translation error of methods (except for Horaud's method), d) Rotation error of Horaud's and Fu's methods, and e) The magnified view of (c).

Figure 7 .
Figure 7.The calibration system and the method of obtaining sphere center coordinates.a) The calibration system, b) calibrator (table tennis), c) standard sphere installed on a linear slide, and d) method of obtaining sphere center coordinates.

Figure 12 .
Figure 12.Stitching results (more sampled poses) and fitting results using the table tennis as the calibrator.a) Proposed method, b) the fitting result of (a); c) Ali's method, d) the fitting result of (c); e) Wang's method, f ) the fitting result of (e).Stitching results (more sampled poses) and fitting results using the standard sphere as the calibrator.g) Proposed method, h) the fitting result of (g); i) Ali's method, j) the fitting result of (i); k) Wang's method, l) the fitting result of (k).

Table 1 .
Parameters of the ABB IRB1410 robot.

Table 2 .
Evaluation results for each method in the simulation.

Table 3 .
Evaluation results for each method in the experiment.