A Self-Calibration Method for Robot End-Effector Using Spherical Constraints

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Using a calibration sphere with a known diameter allows for calibration of a robot that includes both geometric errors and end-effector deformation errors.

Abstract

A self-calibration method utilizing spherical constraints is proposed for calibration of robot end-effectors. The method establishes a mathematical model to account for both the geometric errors of the robot and the deformation errors of the end-effector. A nonlinear least-squares parameter identification technique based on spherical constraints is employed to achieve autonomous calibration of the end-effector. Contrasted with methodologies relying on point plane or distance constraints, this novel technique delivers superior positioning accuracy, streamlined operational procedures and enhanced efficiency. Both simulation and experimental validation confirm that the self-calibration method using spherical constraints improves the positioning accuracy of the robot end-effector from 3 mm to 0.3 mm, showing the effectiveness of the method.

1. Introduction

Due to factors such as the robot’s inherent precision limitations and end-effector deformation, there is always a deviation between the theoretical and actual poses of the robot’s end-effector coordinate system. In many industrial applications involving robots, high positioning accuracy of the end-effector is essential. Therefore, it is crucial to develop a fast and efficient method for autonomous calibration of the robot end-effector [1,2].

Calibrating the robot end-effector coordinate system using physical constraints offers advantages in terms of simplicity and speed. The core concept of this method is to constrain the robot’s end-effector in various poses at specific points or planes and establish constraint equations based on known physical parameters. By comparing the robot’s theoretical pose with its actual pose, errors in different configurations can be identified. Currently, most calibration methods that utilize physical constraints are primarily classified into plane constraints, point constraints and distance constraints. Based on plane constraints, Li et al. [3] developed a calibration model using a contact measurement probe to measure three planes of the calibration block, thereby completing the identification of the robot’s kinematic parameters. Zhong et al. [4] extended the manipulator arm with a trigger probe to record the joint sensor values when the manipulator endpoint touched the constraint plane, enabling kinematic error identification. Xia [5] developed a point constraint method by mounting a laser on the robot’s end-effector and ensuring that the laser’s projection remained fixed on the observation plane, constructing a virtual kinematic chain. This method offers a simple experimental setup and high accuracy in parameter identification. Li et al. [6], focusing on large industrial robots in the aerospace sector, used a laser tracker to measure the actual position of a target ball fixed to the robot’s end-effector, compared it with the theoretical position and calculated error parameters. Wan et al. [7] used a 3D displacement measurement device to constrain the center point of the robot’s end-effector to a fixed point. By altering the robot’s poses and recording the corresponding joint angles, they achieved an average absolute positioning error of 0.339 mm after robot calibration. Based on distance constraints, Zou et al. [8] achieved TCP position calibration using distance constraints between the center of an optical target ball and the origin of the end-effector flange. Gu et al. [9] designed a self-calibrating movable ball rod comprising two G3 steel balls with known radius and center-to-center distance. By measuring the ball center positions with the robot’s end-effector measuring device and comparing these measurements to the theoretical distances, they established distance constraints to perform calibration.

End-effector positioning errors in robots can generally be categorized into kinematic errors and non-kinematic errors. Kinematic errors are parameter inaccuracies that occur during the kinematic modeling of the robot. These include errors resulting from joint wear and link deformation due to prolonged use, as well as deformation of robot components caused by environmental factors such as temperature and humidity. In contrast, non-kinematic errors arise from factors external to the robot itself. This category encompasses errors due to flexible deformations of robot links caused by payload effects and deformation errors of the end-effector tool resulting from insufficient stiffness. Qi et al. [10], addressing the effects of the robot’s own weight and external loads, developed a flexible error model for the primary stressed joints to enhance the robot’s positioning accuracy. Zhang [11], considering both joint and link deformation as well as the impact of the robot’s weight, established an end-effector deformation model. They applied forces of varying magnitudes and directions to the end-effector and measured the resulting deformation coefficients. Jiao et al. [12] examined the combined effects of static errors, link flexibility and joint flexibility on the robot’s end-effector pose, validating the model’s feasibility through simulation. Dini et al. [13] proposed a non-linear control technique that investigates the effects of torque oscillations in a permanent magnet synchronous motor (PMSM) on control precision. This method was applied to a two-degree-of-freedom robotic arm, leading to enhanced position control accuracy for the robot’s end-effector. In parallel, Zu et al. [14] conducted a thorough analysis of errors in robotic kinematics and dynamics parameters, considering joint angle errors induced by end loads. Their research significantly improved the robot’s positioning accuracy, reducing it from 7.52 mm to 1.29 mm.

Among the three constraint methods discussed, calibration methods based on distance and plane constraints are typically large, cumbersome and involve complex processes requiring numerous data points. The point constraints calibration method, which necessitates manually aligning the robot end-effector to a specific point, introduces significant uncertainty due to the difficulty of ensuring consistent alignment. This results in low calibration efficiency and longer time consumption. In real industrial robot applications, various end-effectors are mounted on the manipulator to address different operational needs. The deformation of the robot end-effector is inevitable and can lead to a reduction in positioning accuracy. To address this issue, this paper proposes a self-calibration method based on spherical constraints that accounts for end-effector deformation. This method features a compact and portable calibration device, eliminates the need for manual alignment and offers high calibration accuracy and efficiency. The remainder of this paper is organized as follows. Section 2 derives the error models for geometric errors of the robot and deformation errors of the end-effector. Section 3 presents a method for identifying error parameters using spherical constraints. Section 4 describes the design of simulation calibration experiments conducted in MATLAB. Section 5 details the implementation of error calibration and compensation on a KUKA robot. Conclusions and future research are addressed in Section 6.

2. The Error Model

This paper primarily addresses the robot’s geometric errors and end-effector deformation errors. Establishing a model is the initial step in analyzing the positioning errors of the robot’s end-effector. The Denavit–Hartenberg (D–H) model [15] is a widely used method for robot kinematic modeling. In the D–H model, each link is represented by a coordinate system with its origin at the corresponding joint, and the orientation of the coordinate system defines the link’s motion direction. The D–H parameters facilitate the description of the motion relationships between adjacent links.

The standard D–H parameters typically consist of four values: link length ($a$), joint offset ($d$), twist angle ($\alpha$) and joint angle ($\theta$). When two adjacent axes of a robot are parallel or nearly parallel, the traditional D–H model can produce substantial changes in geometric parameters, leading to a singular Jacobian matrix. To address the singularity issues inherent in the traditional model, Hayati [16] introduced a new parameter, β, which represents the rotation angle around the y-axis. The pose transformation between two consecutive coordinate frames, $O_{i-1}{X}_{i-1}{Y}_{i-1}{Z}_{i-1}$ and $O_i{X}_i{Y}_i{Z}_i$, can then be represented using a homogeneous transformation matrix, $T_i^{i-1}$, as shown in Equation (1).

$$T_i^{i-1}=R_z\left({\theta }_i\right)T_z\left(d_i\right)T_x\left(a_i\right)R_x\left({\alpha }_i\right)R_y\left({\beta }_i\right)=\left[\begin{array}{cccc}c{\theta }_{i}c{\beta }_i-s{\beta }_{i}s{\alpha }_{i}s{\theta }_i& -c{\alpha }_{i}s{\theta }_i& s{\beta }_{i}c{\theta }_i+c{\beta }_{i}s{\alpha }_{i}s{\theta }_i& a_{i}c{\theta }_i\\ s{\theta }_{i}c{\beta }_i+s{\beta }_{i}s{\alpha }_{i}c{\theta }_i& c{\alpha }_{i}c{\theta }_i& s{\beta }_{i}s{\theta }_i-c{\beta }_{i}s{\alpha }_{i}c{\theta }_i& a_{i}s{\theta }_i\\ -s{\beta }_{i}c{\alpha }_i& s{\alpha }_i& c{\beta }_{i}c{\alpha }_i& d_i\\ 0& 0& 0& 1\end{array}\right]$$

(1)

This paper focuses on the KR5 R1400 robot, manufactured by KUKA. The robot is a six degree-of-freedom (DOF) serial manipulator, with its geometric parameters displayed in Figure 1. The D–H parameters are shown in Figure 2.

The pose of the end-effector coordinate frame relative to the base coordinate frame can be expressed as Equation (2). Here, $T_t^0$ represents the homogeneous transformation matrix from the robot’s base coordinate system to the end-effector, while $T_t^6$ denotes the homogeneous transformation matrix from the sixth link of the robot to the end-effector.

$$T_t^0=T_1^0{T}_2^1{T}_3^2{T}_4^3{T}_5^4{T}_6^5{T}_t^6$$

(2)

2.1. Geometric Error

Robot geometric errors stem from a variety of factors. During manufacturing, insufficient machining precision in the mechanical structure can result in dimensional and shape deviations in components such as links and joints. Transmission errors may also occur due to clearances or deformations in the drive system, including reducers or gears. Over time, prolonged use or the impact of end-effector loads can lead to slight loosening or deformation of joints, affecting their positional accuracy. Improper handling or inaccurate calibration during installation can introduce pose deviations. Additionally, environmental factors such as fluctuations in temperature and humidity can cause variations in component dimensions, further compromising the robot’s accuracy. Thus, it is crucial to develop a geometric error model and identify kinematic parameters to ensure the end-effector’s positioning accuracy.

In the D–H model, geometric errors between two adjacent links can be represented by $∆{\theta }_i$, $∆d_i$, $∆{\beta }_i$, $∆a_i$ and $∆{\alpha }_i$. Typically, $∆{\beta }_i$ does not exist and $∆d_i\ne 0$. However, when the axes of two adjacent joints are nearly parallel, $∆d_i$ becomes zero while $∆{\beta }_i$ becomes non-zero. The theoretical transformation matrix between two adjacent coordinate systems differs from the actual transformation matrix by a deviation $\mathrm{d}{T}_i$. Assuming the theoretical homogeneous transformation matrix is $T_i^N$, the actual homogeneous transformation matrix is given by Equation (3).

$$T_i^R=T_i^N+\mathrm{d}{T}_i$$

(3)

By performing a total differential on $\mathrm{d}{T}_i$, Equation (4) is derived.

$$\mathrm{d}{T}_i={\displaystyle \frac{\partial T_i^N}{\partial {\theta }_i}}\mathrm{d}{\theta }_i+{\displaystyle \frac{\partial T_i^N}{\partial d_i}}\mathrm{d}{d}_i+{\displaystyle \frac{\partial T_i^N}{\partial a_i}}\mathrm{d}{a}_i+{\displaystyle \frac{\partial T_i^N}{\partial {\alpha }_i}}\mathrm{d}{\alpha }_i+{\displaystyle \frac{\partial T_i^N}{\partial {\beta }_i}}\mathrm{d}{\beta }_i$$

(4)

The differential transformation of the coordinate frame consists of a combination of differential translation and differential rotation, which can occur in any order. Therefore, $\mathrm{d}{T}_i$ can be expressed as Equation (5), where ${∆}_i$ represents the differential operator for the i-th coordinate, accounting for the coordinate system variations caused by geometric errors.

$$\mathrm{d}{T}_i={∆}_i{T}_i^N$$

(5)

Based on differential kinematics [17], Equation (6) is obtained.

$${∆}_i=\left[\begin{array}{cccc}0& -\mathsf{\delta }z& \mathsf{\delta }y& \mathrm{d}x\\ \mathsf{\delta }z& 0& -\mathsf{\delta }x& \mathrm{d}y\\ -\mathsf{\delta }y& \mathsf{\delta }x& 0& \mathrm{d}z\\ 0& 0& 0& 0\end{array}\right]$$

(6)

By converting the differential operator into a column vector, we have Equation (7).

$${∆}_{\mathit{i}}=\left[\begin{array}{c}{\mathrm{d}x}_i\\ {\mathrm{d}y}_i\\ {\mathrm{d}z}_i\\ {\mathsf{\delta }x}_i\\ {\mathsf{\delta }y}_i\\ {\mathsf{\delta }z}_i\end{array}\right]=\left[\begin{array}{ccccc}0& 0& c{\theta }_i& -d_{i}s{\theta }_i& a_{i}s{\alpha }_{i}s{\theta }_i-d_{i}c{\alpha }_{i}c{\theta }_i\\ 0& 0& s{\theta }_i& d_{i}c{\theta }_i& -a_{i}s{\alpha }_{i}c{\theta }_i-d_{i}c{\alpha }_{i}s{\theta }_i\\ 0& 1& 0& 0& a_{i}c{\alpha }_i\\ 0& 0& 0& c{\theta }_i& -c{\alpha }_{i}s{\theta }_i\\ 0& 0& 0& s{\theta }_i& c{\alpha }_{i}c{\theta }_i\\ 1& 0& 0& 0& s{\alpha }_i\end{array}\right]·\left[\begin{array}{c}\mathrm{d}{\theta }_i\\ \mathrm{d}{d}_i\\ \mathrm{d}{a}_i\\ \mathrm{d}{\alpha }_i\\ \mathrm{d}{\beta }_i\end{array}\right]=G_i·{\mathit{X}}_{\mathit{i}}$$

(7)

In the above Equation (7), ${∆}_{\mathit{i}}$ represents the error of the i-th coordinate frame relative to the (i − 1)-th coordinate frame, $G_i$ is the geometric error transmission matrix, and ${\mathit{X}}_{\mathit{i}}$ is the geometric error vector. It is important that $\mathrm{d}{d}_i$ and $\mathrm{d}{\beta }_i$ cannot exist simultaneously, meaning that $G_i$ is a 6 × 4 matrix, and ${\mathit{X}}_{\mathit{i}}$ is a 4 × 1 vector. For the robot in this study, when $i=2$, $\mathrm{d}{d}_i=0$, and when $i\ne 2$, $\mathrm{d}{\beta }_i=0$. In practical applications, all coordinate system errors must be transformed into the base coordinate system, which can be expressed as Equation (8).

$${∆}_{\mathit{b}\mathit{a}\mathit{s}\mathit{e}}^{\mathit{i}}=\left[\begin{array}{cccccc}{n}_x& o_x& a_x& {\left(\mathit{p}\times \mathit{n}\right)}_x& {\left(\mathit{p}\times \mathit{o}\right)}_x& {\left(\mathit{p}\times \mathit{a}\right)}_x\\ n_y& o_y& a_y& {\left(\mathit{p}\times \mathit{n}\right)}_y& {\left(\mathit{p}\times \mathit{o}\right)}_y& {\left(\mathit{p}\times \mathit{a}\right)}_y\\ n_z& o_z& a_z& {\left(\mathit{p}\times \mathit{n}\right)}_z& {\left(\mathit{p}\times \mathit{o}\right)}_z& {\left(\mathit{p}\times \mathit{a}\right)}_z\\ 0& 0& 0& n_x& o_x& a_x\\ 0& 0& 0& n_y& o_y& a_y\\ 0& 0& 0& n_z& o_z& a_z\end{array}\right]·{∆}_{\mathit{i}}=K_i·{∆}_{\mathit{i}}$$

(8)

Here, $K_i$ denotes the coordinate system error transmission matrix, and the elements $\mathit{n},\mathit{o},\mathit{a},\mathit{p}$ in $K_i$ correspond to the elements $\mathit{n},\mathit{o},\mathit{a},\mathit{p}$ of the homogeneous transformation matrix $T_i^0$, which describes the i-th coordinate frame relative to the base coordinate frame. For typical industrial robots, the deformations in links and joints are relatively small. Within a limited range, the differential transformations of coordinate frames can be linearly superimposed [17]. Consequently, the errors of all coordinate frames can be accumulated in the base coordinate frame as Equation (9).

$${∆}_{\mathit{b}\mathit{a}\mathit{s}\mathit{e}}=\sum _{i=1}^6{∆}_{\mathit{b}\mathit{a}\mathit{s}\mathit{e}}^{\mathit{i}}=\sum _{i=1}^6{K}_i·{∆}_{\mathit{i}}$$

(9)

$K_i$ is the coordinate system error transmission matrix, and $K_1$ is the identity matrix. By combining Equations (7) and (9), we obtain the following Equation (10).

$${∆}_{\mathit{b}\mathit{a}\mathit{s}\mathit{e}}=\sum _{i=1}^6{K}_i·{∆}_{\mathit{i}}=\sum _{i=1}^6{K}_i·G_i·{\mathit{X}}_{\mathit{i}}=J_G·{\mathit{X}}_{\mathit{G}}$$

(10)

$J_G$ is the geometric error transmission Jacobian matrix, a 6 × 24 matrix composed of known parameters once the robot’s pose is determined. ${\mathit{X}}_{\mathit{G}}={\left[{{\mathit{X}}_1}^{\mathit{T}}{{\mathit{X}}_2}^{\mathit{T}}\cdots {{\mathit{X}}_6}^{\mathit{T}}\right]}^{\mathit{T}}$ contains all the geometric errors between adjacent links of the robot, forming a 24 × 1 vector. These parameters are the ones that need to be identified.

2.2. End-Effector Deformation Error

In practical industrial applications, robots are often equipped with various end-effectors to meet the requirements of different operational scenarios. Typically, these end-effectors are connected to the robot’s end flange via a link, and the connection structure can be regarded as a “cantilever beam”. Due to the substantial mass of the end-effectors and their extended reach, the link experiences deformation, which reduces the positioning accuracy of the robot. Although most industrial robots can achieve positioning accuracy within 0.5 mm, the addition of end-effectors can significantly diminish this precision. The geometric errors discussed earlier only account for the robot’s intrinsic parameters and do not consider the influence of the end-effectors. Therefore, it is essential to develop a model that includes the deformation errors introduced by the robot’s end-effectors.

A cantilever beam is a common structural element in engineering, characterized by a fixed support at one end and a free, unsupported section at the other, forming a cantilever structure. In the case of a robot equipped with an end-effector, the connecting link has one end fixed to the robot’s end flange and the other attached to the end-effector, making it analogous to a cantilever beam. For a cantilever beam subjected to an end load of magnitude $P$, the deflection at the tip $z_t$ and the rotation angle at the end cross-section ${\theta }_t$ can be expressed by Equations (11) and (12).

$$z_t=-{\displaystyle \frac{F{l}^3}{3EI}}=-{\displaystyle \frac{1}{3}}·c_t·l^3·P$$

(11)

$${\theta }_t=-{\displaystyle \frac{F{l}^2}{2EI}}=-{\displaystyle \frac{1}{2}}·c_t·l^2·P$$

(12)

Here, $l$ is the length of the beam, $E$ is the elastic modulus and $I$ is the moment of inertia of the cross-section. The parameter $c_t$ represents the bending stiffness of the material, which needs to be identified. Based on the D–H model established earlier, the $z_6$ axis of the robot’s end flange coordinate system $O_6$ indicates the direction of the end-effector extension. Therefore, the effective load on the end-effector can be represented by Equation (13).

$$F=G·sin〈{\mathit{z}}_0,{\mathit{z}}_6〉$$

(13)

$G$ represents the gravitational force acting on the end-effector, and $〈z_0,z_6〉$ denotes the angle between the vectors $z_0$ and $z_6$. The end-effector, influenced by gravity, will deform only in the vertical plane. The normal vector of this plane is shown in Equation (14).

$$\mathit{n}={\mathit{z}}_0\times {\mathit{z}}_6$$

(14)

As shown in Figure 3, the deflection angle of the robot flange coordinate system is defined as Equation (15).

$${\theta }_6=〈\mathit{n},{\mathit{y}}_6〉$$

(15)

Thus, the differential displacement of the end-effector’s coordinate system can be expressed as Equations (16) and (17).

$${\mathrm{d}x}_t=z_t·cos{\theta }_6=-{\displaystyle \frac{1}{3}}F{l_t}^3·cos{\theta }_6·c_t$$

(16)

$${\mathrm{d}y}_t=-z_t·sin{\theta }_6={\displaystyle \frac{1}{3}}F{l_t}^3·sin{\theta }_6·c_t$$

(17)

Similarly, the differential rotational displacement is given by Equations (18) and (19).

$${\mathsf{\delta }x}_t={\theta }_t·sin{\theta }_6=-{\displaystyle \frac{1}{2}}F{l_t}^2·sin{\theta }_6·c_t$$

(18)

$${\mathsf{\delta }y}_t={\theta }_t·cos{\theta }_6=-{\displaystyle \frac{1}{2}}F{l_t}^2·cos{\theta }_6·c_t$$

(19)

Here, $l_t$ represents the length of the end-effector, which can be determined through actual measurement. Expressing the above equations as a column vector of differential operators, we can obtain Equation (20).

$$\left[\begin{array}{c}\mathrm{d}{x}_t\\ {\mathrm{d}y}_t\\ {\mathrm{d}z}_t\\ {\mathsf{\delta }x}_t\\ {\mathsf{\delta }y}_t\\ {\mathsf{\delta }z}_t\end{array}\right]=\left[\begin{array}{c}-{\displaystyle \frac{1}{3}}F{l_t}^{3}cos{\theta }_6\\ {\displaystyle \frac{1}{3}}F{l_t}^{3}sin{\theta }_6\\ 0\\ -{\displaystyle \frac{1}{2}}F{l_t}^{2}sin{\theta }_6\\ -{\displaystyle \frac{1}{2}}F{l_t}^{2}cos{\theta }_6\\ 0\end{array}\right]·c_t=G_t·c_t$$

(20)

The deformation error of the end-effector is relative to the coordinate system $O_6$, so it must be transformed into the base coordinate system. According to Equation (8), we have Equation (21).

$${∆}_{\mathit{b}\mathit{a}\mathit{s}\mathit{e}}^{\mathit{t}}=K_t{·G}_t·c_t=J_t·c_t$$

(21)

The total error of the end-effector relative to the base coordinate system can be expressed as Equation (22).

$${∆}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}=J·\mathit{X}$$

(22)

The total error transmission Jacobian matrix $J=\left[J_G J_t\right]$ is a 6 × 25 matrix, and $\mathit{X}=\left[\begin{array}{c}{\mathit{X}}_{\mathit{G}}\\ c_t\end{array}\right]$ is a 25 × 1 vector of error parameters to be determined.

3. Parameter Identification

3.1. Spherical Constraint Model

For a typical 6-DOF serial robot, due to the presence of errors, the homogeneous transformation matrix of the end-effector coordinate frame relative to the base frame can be expressed as Equation (23).

$${}_t{}^0{T}^R=\left(T_1^N+\mathrm{d}{T}_1\right)\left(T_2^N+\mathrm{d}{T}_2\right)\cdots \left(T_6^N+\mathrm{d}{T}_6\right)T_t$$

(23)

By ignoring higher-order terms, we obtain Equation (24).

$${}_t{}^0{T}^R={}_t{}^0{T}^N+\sum _{i=1}^6\left[{}_{i-1}{}^0{T}^N{∆}_i{\left({}_{i-1}{}^0{T}^N\right)}^{-1}\right]{}_t{}^0{T}^N={}_t{}^0{T}^N+∆{}_t{}^0{T}^N$$

(24)

The above equation can also be expressed as follows as Equation (25):

$$\left[\begin{array}{cc}{R}^R& {\mathit{p}}^{\mathit{R}}\\ 0^{\mathit{T}}& 1\end{array}\right]=\left[\begin{array}{cc}{R}^N& {\mathit{p}}^{\mathit{N}}\\ 0^{\mathit{T}}& 1\end{array}\right]+∆\left[\begin{array}{cc}{R}^N& {\mathit{p}}^{\mathit{N}}\\ 0^{\mathit{T}}& 1\end{array}\right]$$

(25)

where $R$ is the rotation matrix, $p$ is the translation vector and $∆$ represents the differential operator matrix. The self-calibration method proposed in this paper establishes constraint equations based on spherical constraints, requiring only the positional information of the end-effector coordinate frame without the need for orientation data. The actual position of the end-effector coordinate frame can be expressed as Equation (26).

$$\left[\begin{array}{c}{p}_x^R\\ p_y^R\\ p_z^R\end{array}\right]=\left[\begin{array}{cccccc}1& 0& 0& 0& p_z^N& -p_y^N\\ 0& 1& 0& -p_z^N& 0& p_x^N\\ 0& 0& 1& p_y^N& -p_x^N& 0\end{array}\right]\left[\begin{array}{c}\mathrm{d}x\\ \mathrm{d}y\\ \mathrm{d}z\\ \mathsf{\delta }x\\ \mathsf{\delta }y\\ \mathsf{\delta }z\end{array}\right]+\left[\begin{array}{c}{p}_x^N\\ p_y^N\\ p_z^N\end{array}\right]$$

(26)

Substituting Equation (22) into this, we obtain Equation (27).

$${\mathit{p}}^{\mathit{R}}=A\mathit{X}+{\mathit{p}}^{\mathit{N}}$$

(27)

The matrix $A$ represents the pose error at the robot’s end-effector and is a 3 × 25 matrix. The physical constraint introduced in this paper is a spherical constraint. During the calibration process, a calibration sphere with a known diameter is used. When the robot’s end-effector contacts the surface of the calibration sphere, its actual position must satisfy the following Equation (28).

$${\left(p_x^R-x_c\right)}^2+{\left(p_y^R-z_c\right)}^2+{\left(p_z^R-z_c\right)}^2=R^2$$

(28)

Here, $x_c$, $y_c$ and $z_c$ represent the unknown coordinates of the calibration sphere in the base coordinate frame, while $R$ is the known radius of the calibration sphere. By substituting these variables into Equation (27) and treating $x_c$, $y_c$ and $z_c$ as parameters to be identified, we derive Equation (29).

$${\mathit{p}}^{\mathit{R}}-{\mathit{p}}_{\mathit{c}}=\tilde{A}\tilde{\mathit{X}}+{\mathit{p}}^{\mathit{N}}$$

(29)

Here, ${\mathit{p}}_{\mathit{c}}={[x_c,y_c,z_c]}^T$ represents the coordinates of the calibration sphere, $\tilde{A}=\left[\left.A\right|{-I}_{3\times 3}\right]$ is the augmented matrix of the robot’s end-effector error pose transformation matrix, and ${\tilde{\mathit{X}}}^{\mathit{T}}=\left[{\mathit{X}}^{\mathit{T}}{\mathit{p}}_{\mathit{c}}^{\mathit{T}}\right]$ contains all the parameters to be identified, including the coordinates of the calibration sphere. From the constraint Equation (28), we obtain Equation (30).

$${\left(\tilde{A}\tilde{\mathit{X}}+{\mathit{p}}^{\mathit{N}}\right)}^T\left(\tilde{A}\tilde{\mathit{X}}+{\mathit{p}}^{\mathit{N}}\right)=R^2$$

(30)

3.2. Error Parameter Identification Method

The robot’s end-effector collects positional data by making contact with the surface of the calibration sphere, where its theoretical position should describe a perfect sphere. However, due to geometric error and end-effector deformation, the actual position in three-dimensional space deviates from this ideal sphere. To address this, the parameter identification method aims to determine the appropriate error parameters such that, after correcting for these errors, the actual position of the robot’s end-effector conforms to a standard sphere. The following objective function is then optimized using the least squares method, which can be expressed as Equation (31).

$$\underset{x}{\min }F\left(x\right)={‖f\left(x\right)‖}_2^2$$

(31)

For practical calibration, it is necessary to collect $n$ points on the surface of the calibration sphere and perform fitting, with the robot assuming different poses for each point. Thus, Equation (32) holds:

$$f\left(x\right)={\left[f_1\left(\tilde{\mathit{X}}\right)f_2\left(\tilde{\mathit{X}}\right)\cdots f_n\left(\tilde{\mathit{X}}\right)\right]}^T$$

(32)

For each point, we can express it as shown in Equation (33).

$$f_i\left(\tilde{\mathit{X}}\right)={\left({\tilde{A}}_i\tilde{\mathit{X}}+{\mathit{p}}_{\mathit{i}}^{\mathit{N}}\right)}^T\left({\tilde{A}}_i\tilde{\mathit{X}}+{\mathit{p}}_{\mathit{i}}^{\mathit{N}}\right)-R^2,i=\mathrm{1,2},\cdots ,n$$

(33)

Simplifying, we obtain Equation (34).

$$f_i\left(\tilde{\mathit{X}}\right)={\tilde{\mathit{X}}}^{\mathit{T}}{{\tilde{A}}_i}^T{\tilde{A}}_i\tilde{\mathit{X}}+2{\left({\mathit{p}}_{\mathit{i}}^{\mathit{N}}\right)}^T{\tilde{A}}_i\tilde{\mathit{X}}+{\left({\mathit{p}}_{\mathit{i}}^{\mathit{N}}\right)}^T{\mathit{p}}_{\mathit{i}}^{\mathit{N}}-R^2,i=\mathrm{1,2},\cdots ,n$$

(34)

By performing a first-order Taylor expansion of the function $f\left(x\right)$ at $\tilde{\mathit{X}}={\tilde{\mathit{X}}}_{\mathit{k}}$ and neglecting higher-order terms, we obtain Equation (35).

$$F\left({\tilde{\mathit{X}}}_{\mathit{k}}+∆{\tilde{\mathit{X}}}_{\mathit{k}}\right)={‖f\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)+{J\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)}^T∆{\tilde{\mathit{X}}}_{\mathit{k}}‖}_2$$

(35)

Moreover, as shown in Equation (36), the Jacobian matrix can be obtained by taking the partial derivatives.

$$J_i\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)={\displaystyle \frac{\partial f}{\partial {\tilde{\mathit{X}}}_{\mathit{k}}}}=2{{\tilde{A}}_i}^T{\tilde{A}}_i{\tilde{\mathit{X}}}_{\mathit{k}}+2{\left({\mathit{p}}_{\mathit{i}}^{\mathit{N}}\right)}^T{\tilde{A}}_i$$

(36)

Therefore, we can obtain Equation (37).

$$J\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)={\left[J_1\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)J_2\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)\cdots J_n\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)\right]}^T$$

(37)

For $n$ points, $J\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)$ is an $n\times 28$ matrix, where all parameters are known. The solution to this problem is approached using nonlinear least squares. However, during the computation process, $J\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)$ may not be full rank, which could result in errors due to matrix singularity when using the Gauss–Newton algorithm for iterative calculations. To address this, the Levenberg–Marquardt algorithm is employed to solve the increment equation, which is shown in Equation (38).

$$∆{\tilde{\mathit{X}}}_{\mathit{k}}=-{\left[J\left({\tilde{\mathit{X}}}_{\mathit{k}}\right){J\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)}^T+\lambda I\right]}^{-1}J\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)f\left({\tilde{\mathit{X}}}_{\mathit{k}}\right)$$

(38)

$\lambda$ is a positive adjustable coefficient. In nonlinear least squares problems, the choice of the initial values is crucial. If the initial values are not chosen appropriately, it may lead to slow convergence or cause the algorithm to become trapped in a local optimum. In this calibration problem, since the robot’s geometric errors are small, the initial values can be set to zero. The initial values for the end-effector deformation error parameters can be derived based on their physical meaning. The coordinates of the calibration sphere’s center, ${\mathit{p}}_{\mathit{c}}={[x_c,y_c,z_c]}^T$, can be determined by fitting the actual positions measured by the end-effector using the least squares method [18]. Once the initial values are established, the step size $∆{\tilde{\mathit{X}}}_{\mathit{k}}$ is calculated using Equation (38) and the parameters are iteratively updated, gradually minimizing the objective function. The iteration stops when the step size becomes smaller than a specified threshold, at which point the parameter values represent the optimal solution to the nonlinear least squares problem.

4. Simulation Error Calibration Experiment

4.1. Simulation Experiment Design

To verify the effectiveness of the proposed self-calibration algorithm, a robot self-calibration simulation experiment was designed. The steps for this simulation experiment are shown in Figure 4.

4.2. Simulation Experiment

Based on the D–H convention and the geometric parameters of the KUKA KR5 R1400 robot, the D–H parameter table was constructed, as shown in

Table 1. KUKA KR5 R1400 robot D–H parameters.

Link	$\mathit{\theta } (°)$	$\mathit{d} \left(\mathbf{m}\mathbf{m}\right)$	$\mathit{a} \left(\mathbf{m}\mathbf{m}\right)$	$\mathit{\alpha } (°)$	$\mathit{\beta } (°)$
1	$-{\theta }_1$	400	180	−90	-
2	${\theta }_2$	-	600	0	0
3	${\theta }_3-90$	0	40	−90	-
4	$-{\theta }_4$	620	0	90	-
5	${\theta }_5+180$	0	0	90	-
6	${-\theta }_6$	80	0	0	-

.

The kinematic model of the robot was simulated using the Robotics Toolbox in MATLAB. The simulation results, as shown in the Figure 5 below, confirmed the validity of the established robot model.

According to the simulation experiment procedure, the first step was to preset the robot’s geometric error parameters and the end-effector deformation error parameter. These errors were introduced to simulate real-world conditions. The preset geometric error parameters are listed in

Table 2. Robot geometric error parameter presetting.

Link	$\mathit{\theta } (°)$	$\mathit{d} \left(\mathbf{m}\mathbf{m}\right)$	$\mathit{a} \left(\mathbf{m}\mathbf{m}\right)$	$\mathit{\alpha } (°)$	$\mathit{\beta } (°)$
1	0.001	0.001	−0.001	0.001	-
2	0.002	-	0.001	−0.001	0.001
3	0.002	0.001	−0.001	0.001	-
4	0.001	−0.001	0.001	−0.001	-
5	0.002	0.001	−0.001	0.001	-
6	0.001	−0.001	0.001	−0.001	-

, and the deformation error parameter of the robot’s end-effector $c_t$ was set to 0.002.

Considering the robot’s workspace, the center of the calibration sphere was positioned at $x=1.34\mathrm{m},y=-0.24\mathrm{m},z=0.35\mathrm{m}$ in the base coordinate frame, with the calibration sphere having a diameter of 0.06 m. The proposed self-calibration method involves a total of 28 error parameters, necessitating at least 28 points to identify these errors. In this study, 50 points were randomly generated on the calibration surface, and the inverse kinematics of the robot was used to compute the joint angles with errors, simulating the scenario where the robot’s end-effector touches the calibration surface. Additionally, care was taken to ensure no interference occurred between the robot body and the calibration sphere, closely replicating real-world conditions.

After setting the initial values, a MATLAB(R2021a) program was developed to perform the iterative calculations. The iterative process was depicted in Figure 6. After 50 iterations, the data converged, and the iteration process was successfully completed.

4.3. Results Analysis

To validate the effectiveness of the simulation results, the compensated errors were applied to the robot, and the position of the end-effector was compared to its position without compensation, as shown in Figure 7. In the figure, the sphere represents the actual surface of the calibration sphere. The blue points indicate the position of the robot’s end-effector without error compensation, showing a significant deviation from the surface of the calibration sphere. In contrast, the red points represent the position of the end-effector after error compensation, closely aligning with the surface of the calibration sphere, effectively restoring the original contact positions.

As shown in Equation (39), the positioning accuracy of the robot was evaluated by calculating the absolute difference between the distance from the robot’s end-effector to the center of the calibration sphere and the sphere’s radius.

$$\epsilon =\left|\sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2+{\left(z-z_0\right)}^2}-R\right|$$

(39)

Figure 8 presents a comparison of the robot’s positioning accuracy before and after error compensation. Following calibration and error compensation, the maximum positional error of the robot’s end-effector was reduced from 5.559 mm to 1.104 mm, resulting in an 80.1% improvement in accuracy. Similarly, the average error decreased from 3.657 mm to 0.665 mm, reflecting an 81.8% improvement in accuracy.

5. Simulation Experiment

5.1. Experimental Validation

In practical applications, the self-calibration method proposed in this paper requires only a high-precision calibration sphere with a known diameter. The calibration device is portable, allowing for fast calibration at low cost. The schematic diagram of the calibration system is shown in Figure 9. The calibration sphere is first positioned within the robot’s workspace, enabling the end-effector to contact it from various directions and angles. A PC then controls the robot to touch the surface of the sphere at different orientations, collecting and recording the robot’s joint angles at each point of contact. The entire data collection process can be fully automated through programming, removing the need for manual intervention.

5.2. Experimental Equipment

The robot used in this experiment is a KUKA KR5 R1400 industrial robot, manufactured by KUKA in Augsburg, Bavaria, Germany. This versatile, lightweight robot was designed for a range of industrial manufacturing and automation applications. In the experiment, the end-effector was a welding gun with a tip carrying a 3.3 V voltage. A metal sphere with a known diameter was mounted on a test stand, and its shell was grounded. A microcontroller detected contact between the welding gun tip and the metal sphere. The welding gun was connected to the microcontroller via a wire, and when the microcontroller detects a voltage change, it sends a signal to the computer through a serial port. The computer then sends a stop command to the KUKA robot, allowing for the automatic collection and storage of the robot’s joint angle data. The complete experimental setup is shown in Figure 10 below.

5.3. Calibration of End-Effector Coordinate System

The end-effector coordinate system refers to determining the position $O_t{X}_t{Y}_t{Z}_t$ relative to the robot end flange coordinate system $O_6{X}_6{Y}_6{Z}_6$. This relationship can typically be derived from the robot’s 3D model design drawings or using the calibration programs provided by KUKA. In this experiment, the 4-point calibration program of the KUKA robot was used to obtain the transformation matrix between the end-effector coordinate system and the robot end flange coordinate system, as shown below.

$$T_t^6=\left[\begin{array}{cccc}0.7773& 0.6273& -0.0489& -0.0599\\ -0.6260& 0.7788& 0.0393& 0.0505\\ 0.0627& 0.0007& 0.9980& 0.4103\\ 0& 0& 0& 1\end{array}\right]$$

(40)

5.4. Data Measurement

Before the experiment commenced, the actual diameter of the metal sphere was measured to be 61.4 mm, and the end-effector load weight was 1.535 kg. In accordance with the calibration method described earlier, two sets of data were collected to ensure both accuracy and speed in parameter identification. The first set was utilized for robot parameter error identification and calibration, while the second set was employed to compare the robot’s end-effector positioning accuracy before and after error compensation, thereby validating the model’s accuracy and the effectiveness of the compensation. During data collection, it was essential to ensure that the data points were evenly distributed across the surface of the sphere. After the robot’s end-effector contacted the sphere, the joint angle data for each group were recorded. A MATLAB program was then written, using the parameter identification method described earlier, to identify the error parameters. The results of the robot’s geometric error parameter identification are presented in

Table 3. Robot geometric error parameter identification results.

Link	$\mathit{\theta } (°)$	$\mathit{d} \left(\mathbf{m}\mathbf{m}\right)$	$\mathit{a} \left(\mathbf{m}\mathbf{m}\right)$	$\mathit{\alpha } (°)$	$\mathit{\beta } (°)$
1	0.021	0.816	−0.231	−0.055	-
2	−0.019	-	−0.919	−0.035	−0.009
3	0.037	−0.280	1.551	−0.013	-
4	0.031	−0.413	1.234	0.029	-
5	−0.014	−0.001	0.001	0.052	-
6	−0.020	0.685	−0.245	0.038	-

, with the end-effector deformation error $c_t=0.00036$. Based on the material and dimensional parameters of the end-effector, the identified deformation error $c_t$ of the end-effector aligns with its physical significance.

Using the second set of 30 data points, the geometric and end-effector deformation errors of the robot were corrected based on the identified error parameters. The positioning accuracy was then evaluated using the same criterion as defined in Equation (39). Figure 11 illustrates the positioning accuracy of the robot’s end-effector before and after error compensation. The KUKA robot examined in this paper exhibits an absolute positioning accuracy of approximately 0.5 mm. However, this accuracy significantly diminishes when the end-effector was attached. After implementing the proposed algorithm for error calibration and compensation, the maximum positional error of the end-effector was reduced from 4.715 mm to 1.153 mm, representing an accuracy improvement of 75.5%. Additionally, the average error decreased from 1.923 mm to 0.383 mm, resulting in an accuracy enhancement of 80.1%. This level of positioning accuracy meets the requirements of most industrial applications, thereby validating the effectiveness of the self-calibration method introduced in this paper.

6. Conclusions

This paper introduces a method for calibration of the robot end-effector coordinate system using spherical constraints. Initially, a mathematical model incorporating both robot geometric errors and end-effector deformation errors was established. Various errors were propagated to the base coordinate system via an error propagation matrix. Utilizing the spherical constraints of the calibration sphere, an error parameter identification model was developed, and the Levenberg–Marquardt algorithm was applied to solve the nonlinear least squares problem. Results from MATLAB simulations and actual robot calibration experiments demonstrate that the proposed self-calibration method converges rapidly and significantly enhances the positioning accuracy of the robot end-effector.