3.1 The theoretical kinematic model of the measurement robot

As shown in Fig. 4, the measurement robot is illustrated, where coordinate systems are established for the robot base, flange, laser sensor, and workpiece. In this figure, {base1} is the base coordinate system of the measurement robot. {end1} denotes the flange coordinate system. {laser} refers to the coordinate system of the laser sensor, with its origin at the light-emitting port of the sensor. {work} indicates the workpiece coordinate system, and its origin is at the geometric center of the square fixture.

Figure 4.

The theoretical kinematic model of the measurement robot.

$\,_{work}^{base1}T$ denotes the transformation matrix from {base1} to {work} and is defined as

(1) \begin{equation}_{work}^{base1}T={}_{work}^{base1}T\_ P\cdot Rot_{z}(R_{z})\cdot Rot_{y}(R_{y})\cdot Rot_{x}(R_{x})\end{equation}

In this equation, $\,_{work}^{base1}T\_ P$ is defined as

(2) \begin{equation}_{work}^{base1}T\_ P=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & X_{work}\\ 0 & 1 & 0 & Y_{work}\\ 0 & 0 & 1 & Z_{work}\\ 0 & 0 & 0 & 1 \end{array}\right]\end{equation}

$\,_{work}^{base1}T$ contains three rotation variables $R_{x}$ , $R_{y}$ , $R_{z}$ and three position variables $X_{work}$ , $Y_{work}$ , $Z_{work}$ .

$\,_{\textit{laser}}^{end1}T$ denotes the homogeneous transformation matrix from {end1} to {laser} and is defined as

(3) \begin{equation}_{\textit{laser}}^{end1}T={}_{\textit{laser}}^{end1}T\_ P\cdot Rot_{z}(r_{z})\cdot Rot_{y}(r_{y})\cdot Rot_{x}(r_{x})\end{equation}

In this equation, $\,_{\textit{laser}}^{end1}T\_ P$ represents the position of the laser sensor’s origin in {end1} and is defined by the following homogeneous matrix:

(4) \begin{equation}_{\textit{laser}}^{end1}T\_ P=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & X_{\textit{laser}}\\ 0 & 1 & 0 & Y_{\textit{laser}}\\ 0 & 0 & 1 & Z_{\textit{laser}}\\ 0 & 0 & 0 & 1 \end{array}\right]\end{equation}

$\,_{\textit{laser}}^{end1}T$ contains three rotation variables $r_{x}$ , $r_{y}$ , $r_{z}$ and three position variables $X_{\textit{laser}}$ , $Y_{\textit{laser}}$ , $Z_{\textit{laser}}$ . Using $\,_{\textit{laser}}^{end1}T$ , the homogeneous transformation matrix from {base1} to {laser} can be obtained as

(5) \begin{equation} _{\textit{laser}}^{base1}T={}_{end1}^{base1}T\cdot {}_{\textit{laser}}^{end1}T\end{equation}

where $_{end1}^{base1}{T}{}$ represents the robot forward kinematics model. Using the D-H (Denavit-Hartenberg) parameters provided by the robot manufacturer, $_{end1}^{base1}{T}{}$ can be constructed.

According to the mechanical parameters of the LK-H050 laser sensor, the sensor’s zero point is located 50 mm away from the light-emitting port. If the direction away from the light-emitting port is considered positive, the sensor’s return value $d$ represents the distance from the zero point to the intersection of the laser and the contact surface. As illustrated in Fig. 5, the measured point can be described as

Figure 5.

The model of the laser sensor and a standard sphere.

(6) \begin{equation}_{\textit{touch}}^{base1}T={}_{\textit{laser}}^{base1}{T}{}\cdot {}_{\textit{touch}}^{\textit{laser}}{T}{}\_ P\end{equation}

where $_{\textit{touch}}^{\textit{laser}}{T}{}\_ P$ represents the position of the measured point in {laser} and can be expressed as

(7) \begin{equation}_{\textit{touch}}^{\textit{laser}}{T}{}\_ P=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 50+d\\ 0 & 0 & 0 & 1 \end{array}\right]\end{equation}

Thus, the position of the measured point in the base coordinate system of the measurement robot, $\,_{\textit{touch}}^{base1}P=[x,y,z]^{T}$ , can be obtained, which is contained within $\,_{\textit{touch}}^{base1}T$ .

3.2 The theoretical kinematic model of the machining robot

Figure 6 shows the machining robot. Corresponding coordinate systems are established for its base, flange, ball-end milling cutter, and workpiece. {base2} denotes the base coordinate system of the machining robot. {end2} refers to the flange coordinate system of the measurement robot. {tool} represents the coordinate system of the ball-end milling cutter, with its origin located at the tool tip. {work} corresponds to the workpiece coordinate system and matches {work} in Section 3.1. Figure 7 shows the logical relationships between the established coordinate systems.

Figure 6.

The theoretical kinematic model of the measurement robot.

Figure 7.

The logical relationships between the established coordinate systems.

Since the constructure of the machining robot are identical to that of the measurement robot, the homogeneous transformation matrices between the coordinate systems are familiar. The process for obtaining the target matrices $_{work}^{base2}T$ and $_{end2}^{base2}T$ is also identical to that of the measurement robot. The matrix $_{tool}^{end2}T$ represents the homogeneous transformation matrix of {tool} relative to {end2} and can be expressed as

(8) \begin{equation}_{tool}^{end2}T={}_{tool}^{end2}T\_ P\cdot Rot_{z}(r\mathrm{'}_{z})\cdot Rot_{y}(r\mathrm{'}_{y})\cdot Rot_{x}(r\mathrm{'}_{x})\end{equation}

In this equation, $_{tool}^{end2}{T}{}\_ P$ represents the position of the tool end in {end2}, expressed by the following homogeneous transformation matrix:

(9) \begin{equation}_{tool}^{end2}T\_ P=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & X_{tool}\\ 0 & 1 & 0 & Y_{tool}\\ 0 & 0 & 1 & Z_{tool}\\ 0 & 0 & 0 & 0 \end{array}\right]\end{equation}

Equations and involve three rotational variables, $r'_{x}$ , $r'_{x}$ , $r'_{\mathrm{z}}$ and three positional variables, $X_{tool}$ , $Y_{tool}$ , and $Z_{tool}$ . Consequently, the position of the tool tip in {base2} is denoted as $_{tool}^{base2}P=[x,y,z]^{T}$ . This position is encapsulated within the matrix $_{tool}^{base2}{T}{}$ , which is calculated using the following formula:

(10) \begin{equation}_{tool}^{base2}{T}{}={}_{end2}^{base2}T\cdot {}_{tool}^{end2}T\end{equation}

3.3 The theoretical kinematic model of the dual-robot system

The schematic diagram illustrating the logical relationships between the established coordinate systems is as follows:

Based on the established kinematic relationships of each robot, it is necessary to obtain the kinematic relationship between the robots, thus forming a dual-robot system. The matrix $\,_{base1}^{base2}T$ defines the relationship between the base coordinate systems of the measurement robot and machining robot. Considering that both robots share a common workpiece coordinate system, the matrix $\,_{base1}^{base2}T$ can be expressed as

(11) \begin{equation}_{base1}^{base2}T={}_{work}^{base2}{T}{}\cdot \left(_{work}^{base1}{T}{}\right)^{-1}\end{equation}

$_{work}^{base1}{T}{}$ and $_{work}^{base2}{T}{}$ denote the calibrated homogeneous transformation matrices of the {work} relative to {base1} and {base2}, respectively.

4.1 The calibration stage

As both robots are of the same model, the following discussion will focus on the calibration process of the measurement robot. The same method could be applied for optimizing the model of the machining robot.

The homogeneous transformation matrix $_{end1}^{base1}T$ is obtained by multiplying six individual transformation matrices. According to the D-H parameter, each transformation matrix $_{i}^{i-1}T$ contains four variables ( $\alpha _{i-1}$ , $a_{i-1}$ , $d_{i}$ , $\theta _{i}$ ), which can be represented as

(12) \begin{equation}_{i}^{i-1}T=Rot_{x}(\alpha _{i-1})\cdot Trans_{x}(a_{i-1})\cdot Rot_{z}(\theta _{i}+q_{i})\cdot Trans_{z}(d_{i})\end{equation}

where $q_{i}$ represents the drive joint angle. By adding deviation quantities $\Delta \alpha _{i-1}$ , $\Delta a_{i-1}$ , $\Delta d_{i}$ , and $\Delta \theta _{i}$ to their theoretical values, we obtain the error-considered coordinate transformation matrix $_{i}^{i-1}T_{e}$ , as expressed in the following equation:

(13) \begin{equation}_{i}^{i-1}T_{e}=Rot_{x}(\alpha _{i-1}+\Delta \alpha _{i-1})\cdot Trans_{x}(a_{i-1}+\Delta a_{i-1})\cdot Rot_{z}(\theta _{i}+\Delta \theta _{i}+q_{i})\cdot Trans_{z}(d_{i}+\Delta d_{i})\end{equation}

Thus, the overall kinematic model of the robot is represented by the following formula:

(14) \begin{equation}_{end1}^{base1}{T_{e}}{}={}_{6}^{0}{T_{e}}{}={}_{1}^{0}{T_{e}}{}\cdot {}_{2}^{1}{T_{e}}{}\cdot {}_{3}^{2}{T_{e}}{}\cdot {}_{4}^{3}{T_{e}}{}\cdot {}_{5}^{4}{T_{e}}{}\cdot {}_{6}^{5}{T_{e}}{}\end{equation}

Considering position-pose errors of $_{\textit{laser}}^{end1}{T}{}$ , the $_{\textit{laser}}^{end1}T_{e}$ is established and is expressed by adding deviation quantities $\Delta r_{x}$ , $\Delta r_{y}$ , $\Delta r_{z}$ , $\Delta X_{\textit{laser}}$ , $\Delta Y_{\textit{laser}}$ , and $\Delta Z_{\textit{laser}}$ as follows:

(15) \begin{equation}_{\textit{laser}}^{end1}T_{e}={}_{\textit{laser}}^{end1}T\_ P_{e}\cdot Rot_{z}(r_{z}+\Delta r_{z})\cdot Rot_{y}(r_{y}+\Delta r_{y})\cdot Rot_{x}(r_{x}+\Delta r_{x})\end{equation}

where $_{\textit{laser}}^{end1}T\_ P_{e}$ can be expressed as

(16) \begin{equation}_{\textit{laser}}^{end1}T\_ P_{e}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & X_{\textit{laser}}+\Delta X_{\textit{laser}}\\ 0 & 1 & 0 & Y_{\textit{laser}}+\Delta Y_{\textit{laser}}\\ 0 & 0 & 1 & Z_{\textit{laser}}+\Delta Z_{\textit{laser}}\\ 0 & 0 & 0 & 1 \end{array}\right]\end{equation}

Considering the deviation quantities, the coordinates of the measurement point in {base1} are given by

(17) \begin{equation}_{\textit{touch}}^{base1}{T\_ P_{e}}{}={}_{end1}^{base1}{T_{e}}{}\cdot {}_{\textit{laser}}^{end1}{T_{e}}{}\cdot {}_{\textit{touch}}^{\textit{laser}}{T}{}\_ P\end{equation}

$_{\textit{touch}}^{base1}{T\_ P_{e}}{}$ denotes the homogeneous transformation matrix representing the position of the measured point in {base1}. The coordinate vector of $_{\textit{touch}}^{base1}{T\_ P_{e}}{}$ is denoted as $_{\textit{touch}}^{base1}{P_{e}}{}=[x,y,z]^{T}$ .

The error-considered transformation matrix $_{work}^{base1}T_{e}$ from {base1} to {work} is established by introducing deviation quantities $\Delta R_{x}$ , $\Delta R_{y}$ , $\Delta R_{z}$ , $\Delta X_{work}$ , $\Delta Y_{work}$ , and $\Delta Z_{work}$ into the six variables $R_{x}$ , $R_{y}$ , $R_{z}$ , $X_{work}$ , $Y_{work}$ , and $Z_{work}$ . This results in the error-considered transformation matrix $_{work}^{base1}{T_{e}}{}$ :

(18) \begin{equation}_{work}^{base1}T_{e}={}_{work}^{base1}T\_ P_{e}\cdot Rot_{z}(R_{z}+\Delta R_{z})\cdot Rot_{y}(R_{y}+\Delta R_{y})\cdot Rot_{x}(R_{x}+\Delta R_{x})\end{equation}

In this equation, $_{work}^{base1}{T\_ P_{e}}{}$ can be expressed as

(19) \begin{equation}_{work}^{base1}{T\_ P_{e}}{}=\left[\begin{array}{c@{\quad}c} \begin{array}{c@{\quad}c} 1 & 0\\ 0 & 1 \end{array} & \begin{array}{c@{\quad}c} 0 & X_{work}+\Delta X_{work}\\ 0 & Y_{work}+\Delta Y_{work} \end{array}\\ \begin{array}{c@{\quad}c} 0 & 0\\ 0 & 0 \end{array} & \begin{array}{c@{\quad}c} 1 & Z_{work}+\Delta Z_{work}\\ 0 & 1 \end{array} \end{array}\right]\end{equation}

Due to the high assembly accuracy between the standard sphere and the workpiece fixture, there is no need to introduce deviation quantities in the matrix $_{ball}^{work}{T}{}\_ O$ , which represents the coordinates of the standard sphere center in {work}. Thus, $_{ball}^{work}{T}{}\_ O$ can be expressed as

(20) \begin{equation}_{ball}^{work}{T}{}\_ O=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & X_{ball}\\0 & 1&0 & Y_{ball}\\ 0 & 0 &1 & Z_{ball} \\ 0 & 0 & 0 & 1 \end{array}\right]\end{equation}

Based on the relationship between {work} and {base1}, the homogeneous transformation matrix for the center of the standard sphere relative to {base1} can be obtained:

(21) \begin{equation}_{ball}^{base1}{T\_ O_{e}}{}={}_{work}^{base1}{T_{e}}{}\cdot {}_{ball}^{work}{T}{}\_ O\end{equation}

$_{ball}^{base1}{T\_ O_{e}}{}$ denotes the homogeneous transformation matrix representing the position of the sphere center in {base1}. The coordinate vector of $_{ball}^{base1}{T\_ O_{e}}{}$ is denoted as $_{ball}^{base1}{O_{e}}{}=[x,y,z]^{T}$ .

Therefore, in {base1}, the coordinates of the measured point (i.e., the contact point on the sphere surface where the laser line intersects) $_{\textit{touch}}^{base1}{P_{e}}{}$ and the position coordinates of the sphere center $_{ball}^{base1}{O_{e}}{}$ are obtained. The distance between the measured point and the sphere center is denoted as $R_{\textit{measured}}$ , which can be expressed as

(22) \begin{equation}R_{\textit{measured}}=\left\| _{\textit{touch}}^{base1}{P_{e}}{}- _{ball}^{base1}{O_{e}}{}\right\|\end{equation}

$R_{\textit{measured}}$ represents the actual value of the radius measured by measurement robot and $R_{real}$ represents the real value of the radius of the standard spheres provided by manufacturer. Theoretically, $R_{\textit{measured}}$ and $R_{real}$ should be equal. However, due to deviations in the kinematic model of the robot, $R_{\textit{measured}}$ may not match $R_{real}$ . Thus, the deviation between $R_{\textit{measured}}$ and $R_{real}$ can be used as an optimization target to adjust the constructed error-considered kinematic model. When the difference between $R_{\textit{measured}}$ and $R_{real}$ is minimized, the accuracy of the corresponding 36 deviation quantities is improved. The specific procedure is as follows:

In this study, $n$ sets of measurement data can be obtained by measuring the standard spheres, with each set containing values for 6 robot joint angles and $d$ . For each set of data, a deviation function $f_{i}(i\in [1,n],i\in N)$ can be derived based on 36 decision variables, expressed as follows:

(23) \begin{equation}f_{i}=({R_{i,\textit{measured}}}-{R_{real}})^{2}\end{equation}

$R_{i,\textit{measured}}$ represents the radius value calculated based on the $i$ th dataset. The objective function $G$ can be expressed as

(24) \begin{equation}G=\frac{1}{n}\sum _{i=1}^{n}f_{i}\end{equation}

The final nonlinear optimization problem can be expressed as

(25) \begin{equation}minG(\boldsymbol{x});\,s.t.\begin{cases} x_{i,\min}\leq x_{i}\leq x_{i,\max}\\ i=1,2,3,\ldots, 36 \end{cases}\end{equation}

In the formula above, $\boldsymbol{x}=[{x_{1}},{x_{2}},\ldots, {x_{36}}]^{T}\in R^{36}$ represents decision variables, which are four deviation quantities ( $\Delta \alpha _{i-1}$ , $\Delta a_{i-1}$ , $\Delta d_{i}$ , $\Delta \theta _{i}$ ) for each joint of the robot, six deviation quantities ( $\Delta r_{x}$ , $\Delta r_{y}$ , $\Delta r_{z}$ , $\Delta X_{\textit{laser}}$ , $\Delta Y_{\textit{laser}}$ , $\Delta Z_{\textit{laser}}$ ) for the position and pose of the laser sensor coordinate system, and another six deviation quantities ( $\Delta R_{x}$ , $\Delta R_{y}$ , $\Delta R_{z}$ , $\Delta X_{work}$ , $\Delta Y_{work}$ , $\Delta Z_{work}$ ) for the position and pose of the workpiece coordinate system. $x_{i,min}$ and $x_{i,max}$ denote the lower and upper bounds of $x_{i}$ , respectively. These bounds can be assigned appropriate values based on theoretical estimates of the error.

The optimization problem involves 36 variables, resulting in a high-dimensional optimization problem that requires an appropriate meta-heuristic algorithm to effectively avoid local optima. In this paper, the EEFO algorithm [Reference Zhao, Wang, Zhang, Fan, Zhang, Mirjalili, Khodadadi and Cao30] has been selected for this optimization. The EEFO algorithm is a nature-inspired swarm intelligence optimization technique based on the collective foraging behavior of electric eels. By clustering and driving prey into a “bait ball” for a high-pressure attack, eels optimize their hunting efficiency. This strategy inspires the development of the EEFO algorithm to enhance optimization performance. In this study, the algorithm will search within this range to find the vector $\boldsymbol{x}$ that minimizes $G$ .

4.2 The machining stage

In machining stage, the workpiece coordinate system needs to be adjusted for each specific machining instance. Therefore, in each machining instance, the measurement robot is employed to measure the four standard spheres around the workpiece and obtain the homogeneous transformation matrix $_{\mathit{work}\mathit{'}}^{base1}{T}{}$ . Using $_{base1}^{base2}{T}{}$ obtained in the calibration stage, $_{\mathit{work}\mathit{'}}^{base2}{T}{}$ can be derived.

Since the kinematic parameters of the measurement robot have already been calibrated during the calibration stage, constructing the objective function $H$ only requires introducing six deviation quantities ( $\Delta R_{x'}$ , $\Delta R_{y'}$ , $\Delta R_{z'}$ , $\Delta X_{\mathit{work}\mathit{'}}$ , $\Delta Y_{\mathit{work}\mathit{'}}$ , and $\Delta Z_{\mathit{work}\mathit{'}}$ ) for the position and pose of the adjusted workpiece coordinate system. The modeling approach for the objective function is consistent with Section 4.1. The objective function $H$ can be expressed as

(26) \begin{equation}H=\frac{1}{n}\sum _{i=1}^{n}f_{i}\end{equation}

The final nonlinear optimization problem can be expressed as

(27) \begin{equation}minH(\boldsymbol{x});\,s.t.\begin{cases} x_{i,\min}\leq x_{i}\leq x_{i,\max}\\ i=1,2,3,4,5,6 \end{cases}\end{equation}

By applying the EEFO algorithm, the six deviation quantities can be resolved, yielding the transformation matrix $_{\mathit{work}\mathit{'}}^{base1}{T}{}$ . Subsequently, $_{\mathit{work}\mathit{'}}^{base2}{T}{}$ is derived using the following equation:

(28) \begin{equation}_{\textit{work}\mathrm{'}}^{base2}{T}{}={}_{base1}^{base2}{T}{}\cdot {}_{\textit{work}\mathrm{'}}^{base1}{T}{}\end{equation}

By using Eq. (28), the surface machining trajectory in work space can be transformed into the machining robot base system to perform high-precision machining. If another surface part needs to be machined, it is only necessary to repeat the steps in this section to obtain the new fixture position.

5.1 Verification of measurement experiments

The constructed dual-robot system for measurement and machining is illustrated in Fig. 8. By using the proposed method, an objective function as shown in Eq. (24) is formulated, optimizing structural parameters of the measurement-machining system.

Figure 8.

The dual-robot system and the calibration process.

By employing the optimization method proposed in this study, the value of the objective function $G$ for the measurement robot is reduced to 1.250, resulting in a minimized sphere radius deviation of 0.039 mm. Similarly, the value of the objective function $G$ for the machining robot is reduced to 4.025, resulting in a minimized sphere radius deviation of 0.069 mm. Tables I and II list the optimization results of kinematic parameters deviation for measurement robot and machining robot. Tables III and IV list the optimization results of laser sensor coordinate system deviation and workpiece coordinate system deviation for each robot.

Table I.

EEFO-optimized results of kinematic parameters deviation for measurement robot.

Table II.

EEFO-optimized results of kinematic parameters deviation for machining robot.

Table III.

EEFO-optimized results of laser sensor coordinate system deviation.

Table IV.

EEFO-optimized results of workpiece coordinate system deviation.

It should be noted that these parameters are specifically designed to optimize accuracy within a localized region. As the robot’s end-effector moves further away from the workpiece coordinate system, the positioning error may increase. However, this characteristic is advantageous, as these parameters are not intended to reflect the true structural configuration of the robot. Instead, they function as compensation parameters tailored to reduce errors within the specific local space where machining occurs. Since the machining trajectory is confined to a local area, focusing on enhancing local accuracy is far more meaningful than striving for global precision. This approach ensures higher accuracy and efficiency in machining applications, where localized accuracy directly impacts the quality of the machining.

5.2 Calibration accuracy verification experiment

To validate the calibration accuracy of the dual-robot system, measurements and analyses of the standard spheres are conducted both within and outside the valid region. Five positions (Position A∼E) are set within the calibration area for placing the standard spheres, with an additional position (Position F) set outside the effective calibration area. The placement of the standard spheres is shown in Fig. 9.

Figure 9.

Settings of test positions.

During the experiment, the measurement robot scans the standard spheres to capture a considerable number of points. Using the calibrated forward kinematics model of the measurement robot, the positions of the measured points in {base1} are obtained, and the position of the sphere center $_{ball}^{base1}{O}{}$ can be calculated using a fitting algorithm. Then, the position of the sphere center in {base2} $_{ball}^{base2}{O_{cnv}}{}=[x,y,z]^{T}$ can be converted from $_{ball}^{base1}{O}{}$ :

(29) \begin{equation}_{ball}^{base2}{O_{cnv}}{}={}_{base1}^{base2}{T}{}\cdot {}_{ball}^{base1}{O}{}\end{equation}

A laser sensor is mounted on the machining robot to measure the same sphere, thereby directly obtaining the position data $_{\textit{touch}}^{base2}{P_{i}}{}$ of the standard sphere in {base2}. Theoretically, the distance $\| _{\textit{touch}}^{base2}{P_{i}}{}- _{ball}^{base2}{O_{cnv}}{}\|$ between the measured point and the sphere center should equal the actual radius $R_{real}$ . However, due to the deviations in the kinematic model, a discrepancy exists between this distance and $R_{real}$ , which reflects the positioning accuracy error. The error at each point can be calculated by the following formula:

(30) \begin{equation}\delta _{i}=\left| \left\| _{\textit{touch}}^{base2}{P_{i}}{}- _{ball}^{base2}{O_{cnv}}{}\right\| -R_{real}\right| =\left| \left\| _{\textit{touch}}^{base2}{P_{i}}{}- _{base1}^{base2}{T}{}\cdot {}_{ball}^{base1}{O}{}\right\| -R_{real}\right|\end{equation}

$\delta _{i}$ represents the localization error of the $i$ th sampling point. In this experiment, each sphere is sampled at 210 points, thus $i\in [1,210], i\in Z$ . $_{\textit{touch}}^{base2}{P_{i}}{}$ represents the coordinates of each sampling point in {base2}. Figure 10 shows the procedure of the calibration accuracy verification experiment.

Figure 10.

The procedure of the calibration accuracy verification experiment.

Since the calculation of $\delta _{i}$ requires the kinematic model of both robots, using different kinematic models will yield different error values. In this study, both the theoretical and calibrated kinematic models are used in Eq. 30 for comparison. The average error values for the measurements at six different positions (Position A∼F) are shown in Fig. 11. Table V lists the average positioning error of the six positions, and Table VI lists the maximal positioning error of the six positions.

Figure 11.

Comparison of positioning errors of six test positions.

Table V.

The average positioning error of the six positions.

Table VI.

The maximal positioning error of the six positions.

From Tables V, VI, and Fig. 11, it can be summarized that the deviation values obtained using the theoretical kinematic model are significantly higher than those using the calibrated kinematic model, both for average and maximal deviations, whether inside or outside the region. This result demonstrates the effectiveness of the calibration algorithm in improving accuracy. At the same time, for the standard spheres inside the region (Position A∼E), the average positioning error after calibration is less than 0.7 mm, and the maximal error is less than 1 mm. For the standard spheres outside the region (Position F), the average positioning error after calibration is 1.7689 mm, and the maximal positioning error is 3.5178 mm. This indicates that the calibration algorithm has a far greater improvement effect within the local region than outside.

The tables also present the accuracy improvement ratio for both inside and outside the region, $\textit{Ratio}$ in Tables V and VI is defined as

(31) \begin{equation}\textit{Ratio}=\left| 1-\frac{E_{ca}}{E_{th}}\right| \times 100\mathrm{\% }\end{equation}

where $E_{th}$ represents the positioning error calculated with the theoretical kinematic model, and $E_{ca}$ represents the positioning error calculated with the calibrated kinematic model. As shown in the tables, the accuracy improvement ratio for positions inside the region is higher than that for positions outside the region. This further confirms that the calibration algorithm effect on accuracy improvement is more significant within the local region than outside it.

5.3 Algorithm comparison experiment

Due to the nonlinear, high-dimensional, and constrained nature of the dual-robot system calibration problem, traditional gradient-based methods are highly sensitive to initial values, where small deviations may lead to overfitting and cause the algorithm to become trapped in a local optimum. Given that meta-heuristic optimization algorithms possess strong global search capabilities and can better handle such problems, this paper adopts the EEFO algorithm to avoid overfitting and enhance the calibration accuracy. To validate its effectiveness and necessity, this section employs a commonly used gradient-based method, the sequential quadratic programming algorithm (SQP), for calibrating the dual-robot system, and compares the results with those obtained using the EEFO algorithm in Sections 5.1 and 5.2. The relevant results are listed in Tables VII–X.

Table VII.

SQP-optimized results of kinematic parameters deviation for measurement robot.

Table VIII.

SQP-optimized results of kinematic parameters deviation for machining robot.

Table IX.

SQP-optimized results of laser sensor coordinate system deviation.

Table X.

SQP-optimized results of workpiece coordinate system deviation.

By employing the SQP algorithm, the value of the objective function $G$ for the measurement robot is reduced to 58691.815 (EEFO: 1.250), which means the minimized sphere radius deviation is 8.359 mm (EEFO: 0.039 mm). Similarly, the value of the objective function $G$ for the machining robot is reduced to 51258.488 (EEFO: 4.025), and the minimized sphere radius deviation is 7.211 mm (EEFO: 0.069 mm). It can be observed from the objective function values that the value obtained by the SQP algorithm is significantly greater than that obtained by the EEFO algorithm. To further demonstrate the robustness of the EEFO algorithm, 10 comparative experiments were conducted with different initial values, and the results of the two algorithms are compared in Table XI. Since the models of both robots are identical, it is sufficient to validate the robustness of the algorithm on one robot (the machining side).

Table XI.

Mean error and error variance of two algorithms.

To further illustrate the error caused by the SQP algorithm, measurements are conducted on the same standard sphere using both robots, following the same experimental procedure as described in Section 5.2. The point cloud is reconstructed based on the SQP-calibrated model. The experimental results are shown in Fig. 12. In the figure, the red point cloud corresponds to the data collected by the machining robot, while the blue point cloud is acquired by the measurement robot. It can be observed that the reconstructed point cloud exhibits a significant deviation from the actual position of the standard sphere. Moreover, the reconstructed point cloud has lost the spherical characteristics that the actual point cloud should exhibit, making it impossible to fit a sphere as the actual point cloud could.

Figure 12.

Measured points under SQP-calibration and fitted spheres under EEFO-calibration.

5.4 Verification of the spiral line machining experiment

To validate the trajectory accuracy of the dual-robot calibration, this study machines a portion of the Archimedean spiral. The fixture is rotated to form an angle with the $oxy$ plane, and the workpiece is placed in the center of the fixture.

The measurement robot measures four standard spheres around the fixture, from which $_{work}^{base1}{T}{}$ is derived using the previously mentioned method, and $_{work}^{base2}{T}{}$ is subsequently obtained using $_{base2}^{base1}{T}{}$ . The machining process is then carried out using the machining robot, with the results displayed in Fig. 13.

Figure 13.

The machining process and results of the spiral line.

To assess the precision of the machined curve, a high-precision 3D scanner, the ATOS Q 12 M model (as shown in Fig. 14), is employed to capture point clouds and evaluate the accuracy of the curve. Measurements indicate that the average error of the spiral line is 0.073 mm. Figure 15 shows the theoretical curve of the Archimedean spiral processed in this experiment, along with the numerical values and distribution of the trajectory errors.

Figure 14.

The process of precision measurement.

Figure 15.

Theoretical curve diagram and error distribution diagram of the spiral line.

5.5 Verification of the sine surface machining experiment

To prove the ability of the dual-robot system to accommodate small adjustments in the workspace pose, this study further adjusts the position and pose of the fixture and conducts a surface machining experiment. The fixture is rotated to the position shown in Fig. 16, and the workpiece is placed on the fixture.

Figure 16.

The machining process and results of the sine surface.

The calibration and processing are carried out, resulting in the surface shown in Fig. 16. The accuracy of the processed surface is measured using the same equipment as shown in Section 5.3, and the results are presented in Fig. 17. The experimental results indicate that the average accuracy of the surface machining trajectory is 0.109 mm.

Figure 17.

Theoretical surface diagram and error distribution diagram of the sine surface.