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Abstract

This paper proposes a hand-eye LRF-based (laser range finder) welding plane-detection method for autonomous robotic welding in the field of shipbuilding. The hand-eye LRF system consists of a 6 DOF manipulator and an LRF attached to the wrist of the manipulator. The welding plane is detected by the LRF with only the wrist's rotation to minimize a mechanical error caused by the manipulator's motion. A position on the plane is determined as an average position of the detected points on the plane, and a normal vector to the plane is determined by applying PCA (principal component analysis) to the detected points. In this case, the accuracy of the detected plane is analysed by simulations with respect to the wrist's angle interval and the plane angle. As a result of the analysis, an iterative plane-detection method with the manipulator's alignment motion is proposed to improve the performance of plane detection. For verifying the feasibility and effectiveness of the proposed plane-detection method, experiments are carried out with a prototype of the hand-eye LRF-based system, which consists of a 1 DOF wrist's joint, an LRF system and a rotatable plane. In addition, the experimental results of the PCA-based plane detection method are compared with those of the two representative plane-detection methods, based on RANSAC (RANdom SAmple Consensus) and the 3D Hough transform in both accuracy and computation time's points of view.

Keywords

Welding Robot Hand-eye LRF PCA (Principal Component Analysis)Iterative Plane Detection

1. Introduction

Welding operations have been required in various manufacturing industries, such as car and shipbuilding, among others [1]. Since productivity and the quality of the products can be remarkably improved if welding operations are automated, extensive research about autonomous robotic welding has been carried out. As a result, most welding operations in automotive assembly lines have already been automated. On the other hand, welding operations in the shipbuilding industry have been far less automated because of its large-scale and unstructured production environment. Thus, more studies about autonomous robotic welding are still required in the field of shipbuilding.

General robotic welding consists of two steps, such as welding-target recognition and autonomous welding along the welding path, which is predefined by the CAD (Computer Aided Design) model of the welding target. In this case, welding-target recognition is a very important factor that determines the success or failure of the autonomous robotic welding. Thus, much research about welding-target recognition has been conducted by using various kinds of sensors, such as a vision, a structured-light vision and an LRF (laser range finder), among others. In the case of using a vision [2], the computational complexity for target recognition increases remarkably, since a lot of image-processing steps, such as binary thresholding, edge detection, labelling and pattern matching, are needed for robust target recognition. In the case of using a structured-light vision [3 –6], the computational complexity is relatively less complex than that of a vision, but it is still quite high. In addition, the size of the sensor is larger than that of a vision, since a structured-light source should be added to a vision. In our previous work [7], a structured-light vision is also proposed for welding-target recognition. In this case, as many as four laser sources are used with a monocular camera. Since it is sometimes possible to constrain the motion of the welding robot in a case where the sensor is not compact, the sensor must be designed to be as compact as possible. Conversely, in the case of using an LRF [8, 9], it is barely possible to limit the motion of the robot, since the sensing system is already compact enough. Although the LRF can quickly and accurately detect distances in 2D space, the 2D-LRF data are not enough to recognize the welding target. Thus, in most cases, an additional motion is needed to obtain more data. In this case, for improving feasibility, the number of motions should be reduced as far as possible. In [10], a depth sensor, Microsoft Kinect, which is popularly used for various recent applications, as in [11, 12], etc., is used for coarse detection of the welding target. In this case, a laser scanner is then used for the precise detection of the target.

In the field of shipbuilding, most welding operations consist of welding huge steel plates together, and attaching steel plates to parts such as lugs; a lug is a kind of handle used to move a steel plate. As mentioned above, a welding robot should detect welding targets precisely for successful autonomous welding. In this paper, we also focus on welding-target recognition, especially steel-plate recognition. In this case, the steel plate is modeled as a plane which can be defined by its position and normal vectors. In various fields of research, a considerable number of studies have been conducted on plane detection. Most plane-detection methods that have been proposed are based on three representative methods, such as the 3D Hough transform, RANSAC (RAndom SAmple Consensus)[13] and PCA (principal component analysis). In [14], in order to obtain planes, the 3D Hough transform is applied to a continuous point cloud stream generated by a depth sensor over time. The obtained planes are iteratively refined for robust plane detection. The 3D Hough transform originally requires high memory for the parameter space and computational complexity of point accumulations. For reducing the memory requirement and computational complexity, the hierarchical structure is used and tune up experimentally, and a cache space is used, respectively. In [15], the 3D Hough transform and RANSAC are used for automatic detection of 3D-building roof planes from LIDAR data. In this case, RANSAC is more efficient than the 3D Hough transform regarding computation time and quality. In [16], the 3D plane-based map building method is proposed for mobile robot navigation. For plane detection, the 3D Hough transform and RANSAC are used; the plane acquisition times for them are 50ms and 170ms, respectively. In this case, RANSAC is also better than the 3D Hough transform. The generated planer map of the environment is stored in an Octree-based data structure. In [17], the surface elements are first grouped by the 3D Hough transform, and then the best plane is fitted by RANSAC. For efficient use of the available data, a multiresolution approach based on a coarse-to-fine strategy is used. In [18], the plane is detected by PCA after outliers are removed by using the robust scatter matrix, since PCA is very sensitive to outliers. From the experimental results, the proposed method is significantly better than RANSAC in terms of time, accuracy and robustness. According to the results addressed above, in most cases, use of PCA for plane detection in 3D space is better than both the 3D Hough transform and RANSAC. Thus, we used PCA for plane parameter acquisition.

In this paper, a scanning LRF is attached to the wrist of a 6 DOF welding manipulator for steel-plate recognition. Since the LRF data are not sufficient for recognizing the target plate, more data are obtained via the wrist's rotational motions. The plane parameters, such as position and normal vectors, are calculated by using the obtained LRF data. The position vector is defined as an average position of the detected points, and the normal vector is calculated by applying PCA to the detected points. For improving the feasibility and effectiveness of plane detection, the error of the normal vector is first analysed by simulations with respect to the change of the wrist's angle interval and the detected plane angle, considering the intrinsic characteristics of the LRF [19, 20]. According to the results of the error analysis, the number of the wrist's rotations can be reduced as far as possible for rapid detection. In addition, for improving accuracy, an iterative plane-detection method with alignment motions of the manipulator is proposed[21]. After alignment motions, the laser beams of the LRF are nearly perpendicular to the plane, and the number of outliers affected by noise can be remarkably reduced. Thus, PCA can work well, although it is very sensitive to the LRF data's outliers. Finally, the results of PCA are compared with two representative plane-detection methods, such as the 3D Hough transform and RANSAC, with respect to accuracy and computation time.

The organization of this paper is as follows. In Section 2, the hand-eye LRF-based robotic welding system is proposed, and the frames for the manipulator and the sensor are defined. In Section 3, the PCA-based plane detection method is proposed, and its accuracy is analysed with respect to the change of the wrist's angle interval and the plane angle. According to the results of the analysis, the iterative plane-detection method for improving accuracy and reducing acquisition time is proposed by using the manipulator's alignment motions. Section 4 introduces the experimental set-up with the prototype of the hand-eye LRF system, consisting of a 1 DOF wrist joint, an LRF sensor and the rotatable plane. Next, experimental results are shown for verifying the feasibility and effectiveness of the proposed plane-detection method based on PCA. In addition, the comparison results with two representative plane-detection methods based on RANSAC and the 3D Hough transform are given. Finally, Section 5 presents concluding remarks.

2. Hand-eye LRF-based Robotic Welding System

In this section, we propose a hand-eye LRF-based robotic welding system as shown in Fig.1. The welding robot has a welding torch on its end-effector and a scanning LRF on its wrist for precise steel-plate detection. The scanning LRF used in this paper is UBG-04LX-F01 by Hokuyo, Inc., which can detect distances using the rotational motion of a laser beam. The physical specifications of UBG-04LX-F01 are shown in Table 1.

Table 1.

Physical specification of the scanning LRF, UBG-04LX-F01 by Hokuyo, Inc

Parameters	Descriptions
Light source	Semiconductor laser diode (λ = 785nm)
	Laser safety Class 1 (FDA)
	Laser Power: less than 0.67mW
Power source	12V DC ±10%
Detection Distance	Accuracy Range: 60∼4,095mm
Accuracy	Nominal Range 0.06∼1m: ±10mm, 1∼4m: 1% of Distance
Resolution	1mm
Scan Angle	240°
Angular Resolution	0.35° (360°/1024 steps)
Scan Time	28ms/scan
Interface	RS232C
Weight	185 g
Dimensions (W×D×H)	60×60×75mm

Figure 1.

Hand-eye LRF-based 6 DOF welding robot which has an LRF on its wrist

For representing LRF data, coordinate frames for the manipulator and the LRF sensor are first defined with respect to the universal frame {U}, which is a reference coordinate frame as shown in Fig.1. The fixed-base frame and the wrist frame of the manipulator are described as {B} and {W}, respectively. The LRF sensor frame is described as {S}. The homogeneous transformation matrix ^B_ST from {B} to {S} can be obtained via Eq.(1).

_{S}^{B} T =_{W}^{B} T_{S}^{W} T

(1)

where ^B_WT can be obtained by the forward kinematics of the manipulator and ^W_ST is known from design parameters of the hand-eye LRF sensor. Although the scanning angle is 240° with the angular resolution of 0.35°, a scanning angle of only θ_s from $- \frac{θ_{s}}{2}$ to $\frac{θ_{s}}{2}$ with respect to the central line of the LRF is used to detect the plane in front of the sensor. Thus, the total number of the distances obtained using the LRF is calculated as k = θ_s/0.35°, and each distance d_i for i =0,1, …,k-1 can be obtained with respect to {S} by using the homogeneous transformation matrix of rotation about the y_s -axis in Eq.(2).

^{S} P_{i} = [\begin{matrix} \cos (\frac{θ_{s}}{2} - 0.35 i) & 0 & \sin (\frac{θ_{s}}{2} - 0.35 i) & 0 \\ 0 & 1 & 0 & 0 \\ - \sin (\frac{θ_{s}}{2} - 0.35 i) & 0 & \cos (\frac{θ_{s}}{2} - 0.35 i) & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} 0 \\ 0 \\ d_{i} \\ 1 \end{matrix}]

(2)

where [0 0 d_i1] ^T is the ith distance vector in the direction of the laser beam. By ^B_ST in Eq.(1), ^SP_i can be represented with respect to {B} in Eq.(3).

^{B} P_{i} =_{S}^{B} T^{S} P_{i}

(3)

In this way, the distances to the points on the plane can be represented with respect to {B}. However, the plane cannot be determined by only one set of the distance data since all detected points of the scanning LRF are collinear on the plane. Thus, we need more than one set of the distance data to the points, not all of which are collinear. In this paper, we obtain more than one set of the distance data through the rotational motion of the wrist, where the wrist is repeatedly controlled and rotated by the interval angle Δθ_w between two consecutive data sets. Therefore, the number of the sets is determined as m = 360° / Δθ6_w. Consequently, m sets, each of which consists of k distance data, are used for plane detection. From here, the total number of the distance data detected by the hand-eye LRF is defined as n = m × k.

3. Precise Plane Detection

3.1 PCA-based Plane Detection

The huge steel plates used for shipbuilding are modeled as a plane, which is determined by a normal vector to the plane and a point on the plane. The normal vector is obtained by applying PCA to the points P_i(x_i,y_i,z_i) for i=1,…,n detected by the hand-eye LRF, and the point is determined as the mean of the detected points. First, a n × 3 matrix X is defined with n detected points in three-dimensional space in Eq.(4).

X = [\begin{matrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ ⋮ \\ x_{n} & y_{n} & z_{n} \end{matrix}]

(4)

where the means, X̅, Y̅ and Z̅, for three dimensions are defined from Eq.(5) to Eq.(7).

\bar{X} = E (X) = \frac{1}{n} \sum_{i = 1}^{n} x_{i}

(5)

\bar{Y} = E (Y) = \frac{1}{n} \sum_{i = 1}^{n} y_{i}

(6)

\bar{Z} = E (Z) = \frac{1}{n} \sum_{i = 1}^{n} z_{i}

(7)

From Eqs.(5), (6) and (7), the position vector P̂_m of the point on the plane is first determined as (X̅, Y̅, Z̅). Next, the covariance between two dimensions for X and Y can be obtained based on Bessel's correction in Eq.(8).

σ_{x y} = c o v (X, Y) = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - \bar{X}) (y_{i} - \bar{Y})

(8)

where σ_xy is equal to σ_yx, since the covariance satisfies the commutative law. In the same way, the covariances σ_yz and σ_zx can be defined in Eqs.(9) and (10), respectively.

σ_{y z} = c o v (Y, Z) = \frac{1}{n - 1} \sum_{i = 1}^{n} (y_{i} - \bar{Y}) (z_{i} - \bar{Z})

(9)

σ_{z x} = c o v (Z, X) = \frac{1}{n - 1} \sum_{i = 1}^{n} (z_{i} - \bar{Z}) (x_{i} - \bar{X})

(10)

In this case, the covariance between one dimension and itself is the variance. For each dimension, the variance is obtained from Eq.(11) to Eq.(13).

σ_{x x} = c o v (X, X) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{X})}^{2}

(11)

σ_{y y} = c o v (Y, Y) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(y_{i} - \bar{Y})}^{2}

(12)

σ_{z z} = c o v (Z, Z) = \frac{1}{n - 1} \sum_{i = 1}^{n} {(z_{i} - \bar{Z})}^{2}

(13)

From Eq.(8) to Eq.(13), the covariance matrix C can be defined as shown in Eq.(14).

C = [\begin{matrix} σ_{x x} & σ_{x y} & σ_{x z} \\ σ_{y x} & σ_{y y} & σ_{y z} \\ σ_{z x} & σ_{z y} & σ_{z z} \end{matrix}]

(14)

where C is a 3×3 square matrix. The eigenvalue λ and the eigenvector V for the covariance matrix C can be obtained from Eq.(15).

C V = λ V

(15)

where three eigenvalues are obtained, and all eigenvectors for them are mutually perpendicular (or orthogonal). The eigenvector with the largest eigenvalue indicates the direction with the highest variance of the detected points on the plane, and each succeeding eigenvector in turn indicates the direction with the next highest variance, under the constraint that it is orthogonal to the preceding vectors. Let V_i for i=1,2,3 be the eigenvector corresponding to the eigenvalue λ_i, where λ₃ ≥ λ₂≥λ₁≥0. In this case, two eigenvectors, V₃ and V₂, corresponding to the two largest eigenvalues, λ₃ and λ₂, lie on the plane and are orthogonal to each other, since the detected points are scattered only on the plane. Thus, we can find the right vector of the plane as the eigenvector V₁ corresponding to the third eigenvalue λ₁, since V₁ is orthogonal to both V₃ and V₂, which are the basis of the plane. In this way, the normal vector to the plane and the point on the plane can be obtained.

In Fig. 2, the simulation results for the PCA-based plane detection method at a distance of 800mm are shown. Since the distance to the plane is 800mm, we added random noise ranging from −10mm to +10mm to the detected distances of the LRF, taking the accuracy of the LRF in Table 1 into consideration. In this case, the scanning angle θ_S is set as 40°, and the distance d between the wrist's rotational axis and the central axis of the LRF is set as 115mm. Thus, the number of points, k, for one scanning data set is 114 (= 40° /0.35°), where the angular resolution of the LRF is 0.35°. The wrist's roteitional angle Δθ_w is set as 10°, and thus the number of the scanning data sets, m, is 36 (= 360° / 10°). Therefore, the total number of the detected points, n, is 4004 (= 36 × 114). By applying PCA to the points, n, is 4104 (= 36 × 114). By applying PCA to the detected points, the normal vector to the plane can be obtained as the eigenvector V₁ corresponding to the smallest eigenvalue λ₁ among three eigenvalues, where the eigenvectors V₃ and V₂ corresponding to the two largest eigenvalues λ₃ and λ₂, respectively, lie on the plane and are orthogonal to V₁. In this case, the point on the plane is determined as the mean vector P̂_m of the whole of the detected points.

Figure 2.

Simulation results for plane detection according to the plane angle of θ_p

In Fig. 2, for ease of understanding, the coordinate frame {W} is depicted as the wrist frame where there is no rotation of the wrist, although the actual frame {W} is changed depending on the wrist's motion. From Fig. 2(a) to Fig. 2(d), the results of plane detection are shown according to the change of the plane angle θ_p from 0° to 30°. In the figures, three eigenvectors, V₃, V₂ and V₁, corresponding to the eigenvalues λ₃, λ₂ and λ₁, respectively, are depicted, where λ₃ ≥ λ₂ ≥ λ₁. The first two eigenvectors, V₃ and V₂, lie on the plane although their directions change according to the distribution of the detected points. Thus, we can obtain the normal vector to the plane as V₁, since V₁ is orthogonal to both V₃ and V₂. Using the normal vector V₁, the plane angle is measured as the angle θ_n between V₁ and the z_w -axis. In Table 2, the angular error E_θ of the measured angle θ_n is less than 0.1167° for the plane angle θ_p from 0° to 30°. Thus, the normal vector to the plane can be obtained by PCA with a reasonably small error for plane detection. However, the error E_θ tends to increase as the plane angle θ_p increases.

Table 2.

Angular error E_θ of the measured angle θ_n between the normal vector V₁ and the z_w -axis for the plane angle θ_p

Plane angle θ_p	Measured angle θ_n	Angular error E_θ
0°	0°	0°
10°	10.0467°	0.0467°
20°	20.0931°	0.0931°
30°	30.1167°	0.1167°

3.2 Iterative Plane Detection with Alignment Motion of the Manipulator

In Sec.3.1, the angular error E_θ is reasonably small for plane detection, but it tends to increase as the plane angle θ_p increases. In this section, the relationship between the plane angle θ_p and the angular error Eθ is analysed, and an iterative plane-detection method is suggested for precise plane detection. In Fig.3, the simulation results for plane detection with the wrist's rotational angle Δθ_w of 30° are shown. The relationship between the plane angle θ_p and its maximum (or upper bound) error (| E_θ | + σ_θ) is shown in Fig. 3(a), where E_θ is the average angular error and σ_θ is its standard deviation. As shown in Fig. 3(a), (| E_θ | + σ_θ) increases as θ_p increases. If the tolerable error E_T for successful autonomous welding is set as 0.04°, the normal vectors for θ_p = 10°, 20°, 30° are not suitable for autonomous welding since their errors are greater than E_T. From the plane angle's point of view, if the measured angle θ_n for the normal vector is greater than θ_ref, corresponding to the tolerable error E_T, the robotic welding is quite likely to fail, where θ_ref is determined as 7.782°, as shown in Fig. 3(b), which is a magnification of part A in Fig. 3(a). In the case that θ_n is greater than θ_ref, the additional alignment motion of the manipulator is required to reduce the plane angle θ_p close to zero, and then the normal vector to the aligned plane is again detected by the hand-eye LRF, where the alignment motion is to align the z_m -axis parallel with the detected normal vector. In this case, we have to consider the mechanical error E_m caused by the manipulator's alignment motion. If we assume the manipulator has a relatively large error E_m= 3°, then the aligned plane angles for given θ_p = 10°,20°,30° after alignment motion are measured as ①, ② and ③, respectively, in Fig. 3(b). Although the plane angles of ①, ② and ③ about the θ_p axis are slightly greater than 3 because of E_m, they are sufficiently smaller than θ_ref. Thus, the aligned plane can be detected with an error less than E_T by conducting one more measurement of the hand-eye LRF. In the simulations, the plane can be detected precisely by only two measurements of the hand-eye LRF. However, the second measurement may in reality have an error greater than E_T. In this case, two steps, such as manipulator alignment and LRF measurement, are carried out iteratively until the measured angle θ_n is less than θ_ref, as shown in Fig.4. In this way, we can find the normal vector to the plane with an error less than E_T.

Figure 3.

Relationship between the maximum (or upper bound) angular error | E_θ | + σ_θ of the normal vector V₁ and the plane angle θ_p, where E_θ is the average error and σ_θ is its standard deviation

Figure 4.

Flowchart of iterative plane detection

3.3 Error Analysis for Rotational Angle of the Wrist

In Sec.3.1 and Sec.3.2, the wrist's rotational angle Δθ_w is set as 10° and 30°, respectively, thus obtaining 36 and 12 distance data sets. In this case, the amount of distance data is large enough to detect the plane but it is still very inefficient, because as many as 35 or 11 rotational motions of the wrist are additionally required. In this section, we want to reduce the number of the wrist's rotations for improving the efficiency of the proposed method. The number of rotations is determined as m-1, where m is the number of data sets and is determined by the wrist's rotational angle interval Δθ_w as (360° / Δθ_w). In other words, we want to increase Δθ_w as much as possible. In Table 3, the simulation results of plane detection for θ_p = 0°, 10°, 20°, 30° are shown with respect to the change of Δθ_w from 30° to 180°. While the average angular error E_θ for each wrist's angle interval Δθ_w tends to increase as the plane angle θ_p increases, E_θ for each θ_p has no significant change as Δθ_w increases. That is because the variance of points about the second-largest principal axis decreases as θ_p increases, while that about the largest principal axis increases. In this case, the accuracy of the calculated direction of the second-largest principal axis decreases, and E_θ increases consequentially. On the other hand, as Δθ_w increases, the ratio of the variances about the first and second principal axes is not changed significantly, although the number of points decreases. Thus, there is no significant change of E_θ although Δθ_w increases. Next, the larger Δθ_w is more sensitive to θ_p than the standard deviation σ_θ is. That is because the total number of points decreases as Δθ_w increases. If the number of points is small, the efficiency of noise cancellation by using PCA is much reduced. As a result, σ_θ increases as Δθ_w increases. For consideration of both E_θ and σ_θ, the maximum (or upper bound) error is defined as (| E_θ | + σ_θ), and tends to increase as θ_p or Δθ_w increases. If the tolerable error E_T is 0.04°, as in Sec.3.2, the measurement results are not reliable, except for θ_p= 0°, since their errors are greater than E_T. In this case, the alignment motion of the manipulator is needed to reduce θ_p close to zero. If the mechanical error E_m of the manipulator is set as 3°, as in Sec.3.2, θ_p is slightly larger than 3° but is smaller than 3.2637° in the worst-case scenario for θ_p = 30° and Δθ_w = 180° after alignment motion. Thus, the error (| E_θ | + σ_θ) can be remarkably reduced in the second measurement of the LRF.

Table 3.

Angular error analysis from simulation results according to the rotational angle interval Δθ_w of the wrist, where the number m of the wrist's rotations is determined as (360°/Δθ_w)

Plane angle	Wrist angle interval	Angular error	Standard deviation	Maximum error
θ_p	Δθ_w	E_θ	σ _θ	\|E_θ\|+σ_θ
0°	30°	0°	0°	0°
	60°	0°	0°	0°
	90°	0°	0°	0°
	120°	0°	0°	0°
	180°	0°	0°	0°
10°	30°	−0.0468°	0.0046°	0.0514°
	60°	−0.0468°	0.0089°	0.0557°
	90°	−0.0468°	0.0088°	0.0556°
	120°	−0.0466°	0.0125°	0.0591°
	180°	−0.0356°	0.0650°	0.1006°
20°	30°	−0.0931°	0.0031°	0.0962°
	60°	−0.0931°	0.0051°	0.0982°
	90°	−0.0932°	0.0084°	0.1016°
	120°	−0.0935°	0.0069°	0.1004°
	180°	−0.0746°	0.1237°	0.1983°
30°	30°	−0.1167°	0.0086°	0.1253°
	60°	−0.1167°	0.0237°	0.1404°
	90°	−0.1169°	0.0215°	0.1384°
	120°	−0.1162°	0.0320°	0.1482°
	180°	−0.0974°	0.1663°	0.2637°

For example, the maximum error (| E_θ | + σ_θ) for the plane angle θ_p is shown in Fig.5, where the wrist's angle interval Δθ_w is set as 180°. As shown in Fig.5(a), (| E_θ | + σ_θ) monotonically increases as θ_p increases. In Fig.5(b) which is the magnification of part A in Fig. 5(a), the reference plane angle θ_ref for the tolerable angle E_T is determined as 3.976°. For θ_p = 10°,20°,30°, the aligned plane angles after the manipulator's alignment motion are smaller than ①, ② and ③ about the θ_p axis, respectively. In this case, only the second measurements of the LRF are reliable, since ①, ② and ③ are all smaller than θ_ref. Of course, ①, ② and ③ about the (| E_θ | + σ_θ) axis are smaller than the tolerable error E_T. Consequently, the wrist's angle interval Δθ_w of 180° is large enough to measure the plane precisely using the iterative plane-detection method under the constraints of E_T =0.04° and E_m = 3°. Since Δθ_w is 180°, the number of the wrist's rotations can be reduced to one. Although the results of plane detection depend on E_T and E_m, reliable results can be obtained by the iterative plane-detection method with just one rotation of the wrist in the simulation results.

Figure 5.

Simulation results for Δθ_w = 180

4. Experiments

4.1 Experimental Set-up

For verifying the feasibility and effectiveness of the proposed plane-detection method, an experimental set-up consisting of a prototype of the hand-eye LRF and a rotatable plane was devised, as shown in Fig.6. In Fig. 6(a), the prototype of the hand-eye LRF consists of two parts, such as a 1 DOF wrist joint and an LRF sensor system. In this case, the wrist joint consists of a motor, a motor controller and a motor battery. The motor controller is connected to the control computer via the RS-232 serial communication. Also, the LRF sensor system consists of an LRF sensor and a power supply. The LRF sensor is also connected to the control computer via the RS-232 serial communication. Therefore, the control computer can not only control the motor but also receive the LRF sensor data. Using the received LRF data, the control computer detects a plane with 3.3 Ghz quad core CPUs and 8 GB of RAM. In the experimental set-up, the vertical and horizontal positions of the hand-eye LRF can be changed manually. In this case, the motor can be controlled only from 0° to 300°, unlike the simulation from 0° to 360°, because of the configuration of the prototype. Thus, the number of the LRF data sets, m, is determined as ((300° / Δθ_w) + 1) in the experiments. In the bottom part of the experimental set-up, there is a plane whose angle can be manually changed. Thanks to the plane with the changeable angle, the experiment for plane detection can be carried out, taking the aligned motion of the manipulator into consideration. The sensor frame {S} and the wrist frame {W} are defined as shown in Fig. 6(b), where the distance d between the origins of {S} and {W} is set as 68.7mm. The x_w -axis and the x_s -axis are initially aligned parallel to the rotational axis of the plane. The distance between the origin of {W} and the rotational axis of the plane about the z_w -axis is set as 580mm.

Figure 6.

Experimental set-up with the prototype of the hand-eye LRF

4.2 Experimental Results for the PCA-based Plane Detection

Using the experimental set-up in Sec.4.1, the experiments on plane detection for θ_p = 0°,10°,20°,30° are carried out with respect to the change of Δθ_w from 30° to 180°, as shown in Table 4. Both the average angular error E_θ and the standard deviation σ_θ for the Δθ_w of the wrist increase monotonically as the plane angle θ_p increases, as in the simulation results. However, both E_θ and σ_θ for each θ_p do not increase monotonically, but tend to increase as Δθ_w increases. Similarly, the maximum error (| E_θ | + σ_θ) for each Δθ_w increases monotonically as θ_p increases, while (|E_θ | + σ_θ) for each θ_p does not increase monotonically, but tends to increase as Δθ_w increases. For verifying the feasibility of the iterative plane-detection method, experimental results for Δθ_w = 30° and 180° are analysed in Fig.7 and Fig.8, respectively, where the tolerable error E_T and the mechanical error E_m of the manipulator are set as 1.5° and 3°, respectively. Although the mechanical error of the actual manipulator for autonomous welding is much less than that of 3°, as in [22], the error E_m is defined as 3° in consideration of the worst-case scenario.

Table 4.

Angular error analysis from experimental results, according to the rotational angle interval Δθ_w of the wrist where the number m of the wrist's rotations is determined as ((300°/Δθ_w)+1)

Plane angle	Wrist angle interval	Angular error	Standard deviation	Maximum error	Computation time
θ_p	Δθ_w	E_θ	σ_θ	\|E_θ\|+σ_θ	T
0°	30°	−0.1979°	0.0496°	0.2475°	0.0590s
	60°	−0.2788°	0.1657°	0.4445°	0.0552s
	90°	−0.5017°	0.0266°	0.5283°	0.0591s
	120°	−0.5962°	0.2454°	0.8416°	0.0606s
	180°	−0.7898°	0.3977°	1.1875°	0.0458s
10°	30°	1.6228°	0.3618°	1.9846°	0.0596s
	60°	1.6645°	0.4966°	2.1611°	0.0554s
	90°	1.1348°	0.1309°	1.2657°	0.0608s
	120°	1.1809°	0.3282°	1.5091°	0.0606s
	180°	0.6017°	1.3106°	1.9123°	0.0470s
20°	30°	2.2205°	0.3839°	2.6044°	0.0611s
	60°	2.2512°	0.5123°	2.7635°	0.0574s
	90°	1.6980°	0.1613°	1.8593°	0.0638s
	120°	1.8620°	0.4125°	2.2745°	0.0631s
	180°	0.9293°	2.3023°	3.2316°	0.0470s
30°	30°	2.3910°	0.2249°	2.6159°	0.0605s
	60°	2.4255°	0.4371°	2.8626°	0.0564s
	90°	1.8216°	0.1054°	1.9270°	0.0598s
	120°	2.1015°	0.2626°	2.3641°	0.0608s
	180°	1.3826°	2.9204°	4.3030°	0.0471s

Figure 7.

Experimental results for Δθ_w =30°

Figure 8.

Experimental results for Δθ_w = 180 °

For the wrist's angle interval Δθ_w = 30°, the experimental results about iterative plane detection are shown with respect to the change of the plane angle θ_p in Fig.7. As shown in Fig. 7(a), the maximum error (|E_θ | + σ_θ) monotonically increases as the plane angle θ_p increases. When the laser beam of the LRF is parallel to the normal vector to the plane, the distance error is the smallest probabilistically. The number of laser beams which are parallel or subparallel to the plane's normal vector is larger in the case of the level plane than in the case of the inclined plane. As a result, the smaller θ_p is, the smaller (|E_θ | + σ_θ) is. Since the maximum errors except for θ_p = 0° are greater than the tolerable error E_T, the manipulator's alignment motion is needed to reduce the plane angle close to zero. Since the reference angle θ_ref corresponding to E_T is determined as 7.21° as shown in Fig. 7(b) which is the magnification of the part A in Fig. 7(a), the plane detection and the alignment motion should be conducted iteratively until the aligned plane angle θ_p is less than or equal to θ_ref.. The aligned plane angles for the initial plane angle θ_p = 10°,20°,30° are denoted as ①, ② and ③ about the θ_p axis respectively. In this case, ①, ② and ③ are less than θ_ref and their maximum errors are also less than E_T. Therefore, the second measurement of the LRF after just one alignment motion is enough to precisely detect the plane with an error less than the tolerable error E_T.

Next, for Δθ _w = 180°, the experimental results for iterative plane detection are shown with respect to the change of θ_p in Fig. 8. As shown in Fig. 8(a), the maximum error (|E_θ | + σ_θ) monotonically increases as θ_p increases. Similar to the case of Δθ _w = 30°, the smaller that θ_p is, the smaller E_θ is. Since the maximum errors, except for θ_p= 0°, are greater than E_T, similar to the results for Δθ_w = 30°, the manipulator's alignment motion is needed to reduce the plane an angle close to zero. Since the reference angle θ_ref with respect to E_T is determined as 4.316°, as shown in Fig. 8(b), which is the magnification of the part A in Fig. 8(a), the plane detection and the alignment motion should be conducted iteratively until the aligned plane angle θ_p is less than or equal to θ_ref. The aligned plane angles for the initial plane angle θ_p = 10°,20°,30° are denoted as ①, ② and ③ about the θ_p axis, respectively. After the first alignment motion, the plane angles ①, ② and ③, are 4.9123°, 6.2316° and 7.3030°, respectively, and their maximum errors are 1.5435°, 1.6391° and 1.7168°, respectively. Since the aligned plane angles after the first alignment motion are greater than θ_ref, one more alignment motion is needed. After the second alignment motion, the aligned plane angles for ①, ② and ③ will be 4.5435°, 4.6391° and 4.7168°, respectively, and they are still greater than θ_ref. Thus, we can consider that one more alignment motion is needed. However, in this case, additional alignment motions are useless for reducing the plane angle close to zero. Since the mechanical error E_m is 3°, the maximum error of the aligned plane angle should be less than 1.316° in order to obtain the plane angle with an error less than E_T after one more alignment motion. In this case, the maximum error less than 1.316° is achieved for the plane angle less than θ_ref, but the aligned plane angle is greater than θ_ref after additional alignment motions. Thus, we cannot detect the plane precisely when under the constraints of Δθ _w = 180°, E_T =1.5° and E_m=3°. To resolve this problem, we have to reduce the wrist's angle interval Δθ _w , even though the number of the wrist's rotations increases or replaces the manipulator with the mechanical error E_m of 3° with another that has a smaller mechanical error. In this way, we can determine the wrist's angle interval and the specification of the manipulator in order to satisfy the requirements of successful autonomous robotic welding.

4.3 Comparison with RANSAC and 3D Hough Transform-based Plane-Detection Methods

Using the experimental set-up as in Sec.4.1, a comparison between the PCA-based plane-detection method in Sec.4.2 and two other representative methods were conducted. One is the RANSAC-based (RANdom SAmple Consensus) plane detection method, and the other is the 3D Hough transform-based plane-detection method. RANSAC is a randomized procedure that fits an accurate model iteratively to observed data without trying all possibilities[13]. For plane detection, three points are extracted randomly from the detected points on the plane, and used to compute the plane model's parameters. The obtained model's suitability for the remaining points is assessed via a score function. The score is determined by reference to the number of points that are close enough to the obtained plane. After the above process from three random point-extraction is carried out iteratively, the plane model with the best score is taken to be the winner. In this case, the threshold value of the distances to the closest points and the number of iterations should be appropriately determined for precise and fast detection. The threshold value is determined as 2σ, where σ is the standard deviation of the points. The points within the threshold value are defined as inliers and their percentage is about 95.5% in a normal distribution. The points outside the threshold value are defined as outliers. Since the points with random noise are probably included in outliers, the effects of the random noise can be reduced. For determining the number of iterations, N, the probability of a successful detection is first considered as P = 1 - (1 - α^m)^N, where α is the percentage of inliers and m is the number of extracted random points [23]. Thus, N can be obtained as $\frac{\ln (1 - P)}{\ln (1 - α^{m})}$ . If the probability of successful detection P is assumed to be 0.999, N is calculated as $\frac{\ln (1 - 0.999)}{\ln (1 - {0.955}^{3})} = 3.3732$ . Since N is an integer, N is set as Mean Square (LMS) algorithm to inliers for the RANSAC plane model obtained above.

Next, the Hough transform is one of the popular parameter-estimation methods based on a voting mechanism. For plane detection, the original Hough transform proposed for line detection should be extended to the 3D Hough transform. The parameter space of the 3D Hough transform is defined as $(α, β, d) = {[- \frac{π}{2}, \frac{π}{2}]}^{1} \times {[- \frac{π}{2}, \frac{π}{2}]}^{1} \times {[- d_{m a x}, d_{m a x}]}^{1}$ , where α and β are the two Euler angles of the unit plane normal vector v_n, and d is the distance to the plane from the origin. In this case, d_max is set as the maximum distance to the detectable planes. Using the z - y - x Euler angles, the unit plane normal vector v_n = (a,b,c) can be obtained as $R_{z} (α) R_{y} (β) \hat{x} = {[\cos α \cos β \sin α \cos β - \sin β]}^{T}$ , where R_y(α) and R_y(β) are rotation matrices by α about the z -axis and by β about the y -axis, respectively, and where x̂ is a unit vector [100]^T in the direction of the x -axis. If the sampling intervals of the parameter space (α, β, d) are defined as Δα, Δβ and Δd, respectively, then the dimensions of the sampled parameter space are $(\frac{π}{Δ α} + 1) \times (\frac{π}{Δ β} + 1) \times (\frac{2 d_{m a x}}{Δ d} + 1)$ . Each cell in the sampled parameter space is an accumulator that represents discrete parameter values, and is initialized to zero. For each detected point ^sP_i = (x_i,y_i,z_i), each of the accumulators satisfying the plane equation: ax_i + by_i + cz_i + d = 0 are incremented by 1, where (a,b,c) = [cosαcosβ sinαcosβ - sinβ]^T. The resultant plane parameters are determined as peaks of the accumulators. In this case, the sampling intervals should be determined appropriately for successful detection. In the case of small intervals, the computation time increases and the probability of successful detection decreases, while the accuracy of the detected plane can increase. In this paper, the sampling intervals Δα, αβ and Δd are defined as 1°, 1° and 10mm, respectively, taking the trade-off between computation time and accuracy into consideration. The resultant dimensions of the sampled parameter space are 181 × 181 × 201, where d_max is set as 1000mm.

Similar to the experimental results about the PCA-based plane-detection method in Sec.4.2, the experimental results about both plane-detection methods based on RANSAC and the 3D Hough transform are shown in Table 5 and Table 6, respectively. In the case of RANSAC, both the average angular error E_θ and the standard deviation σ_θ for each angle interval Δθ_w of the wrist increase monotonically as the plane angle θ_p increases, similar to PCA. On the other hand, in the case of the 3D Hough transform, neither E_θ nor σ_θ for each Δθ_w seem to be closely related to θ_p. However, in both experimental results, the maximum error (| E_θ | + σ_θ) for each Δθ_w increases monotonically as θ_p increases, while (| E_θ | + σ_θ) for each θ_p does not increase monotonically but tends to increase as Δθ_w increases, similar to PCA. The comparison results for Δθ_w = 30°, 180° are shown in Figs. 9(a) and 9(b). In the case of Δθ_w = 30°, the maximum error for each method increases as the plane angle θ_p increases, as shown in shown in Fig. 9(a). Thus, the accuracy of the plane parameters can be improved by iterative plane detection. In this case, the tolerable error E_T and the mechanical error E_m of the manipulator are set as 1.5° and 3°, like in Sec.4.2. First, the error of the 3D Hough transform is larger than those of the others regardless of θ_p, and the reference angle θ_ref with respect to E_T is 1.47°. If the detected plane angle is greater than θ_ref, the plane detection and the alignment motion should be conducted iteratively until the aligned plane angle is less than or equal to θ_ref. However, the plane with an error of less than E_T cannot be obtained successfully since the plane angle is greater than at least 4.47°, after the manipulator's alignment motion under the constraint of the mechanical error E_m = 3°. On the other hand, the error of RANSAC is approximately equal to that of PCA. In this case, the reference angles θ_ref of RANSAC and PCA are 5.98° and 7.21°, respectively. If the plane angle detected by RANSAC is greater than 5.98°, the alignment motion is needed for precise plane detection. In this case, just one alignment motion is sufficient for detecting the plane within the tolerable error E_T, since the first aligned plane-angle is less than at least 5.6159° in the worst-case scenario for θ_p = 30° under the constraint of E_m = 3°. Nevertheless, PCA is more precise than RANSAC since the error of PCA is less than that of RANSAC in the vicinity of θ_p = 0°. Next, in the case of Δθ_w = 180°, the maximum error of each method is shown in Fig. 9(b). First, the error of the 3D Hough transform does not seem to be closely related to θ_p because the results are increasingly affected by distance measurement noise as the number of detected points decreases. In addition, the error for all θ_p is greater than E_T, and it is not possible to detect the plane successfully. Although the error of RANSAC is less than that of the 3D Hough transform, the error for all θ_p, is also greater than E_T, and thus successful plane detection is not possible. Of course, in the case of PCA, successful plane detection is also very difficult, as described in Sec.4.2. However, by replacing the manipulator with a more accurate one, the probability of successful detection can be increased. On the other hand, in the case of the 3D Hough transform and RANSAC, successful detection is not possible under the constraint of E_T = 1.5°, however small the mechanical error E_m of the manipulator. From the computation time's point of view, the average computation times for the 3D Hough transform, RANSAC and PCA are 95.5318s, 0.4129s and 0.0570s, respectively. In this case, the 3D Hough transform is more than 230 times slower than RANSAC, and 1670 times slower than PCA. Thus, the 3D Hough transform is not appropriate for real-time robot control. RANSAC is much faster, but is itself more than seven times slower than PCA. Consequently, the PCA-based plane detection method is more appropriate for real-time welding robot application with respect to both accuracy and computation time. Of course, through additional post processing or appropriate parameter tuning the accuracy of the 3D Hough transform and RANSAC methods can be improved, but computation time for the additional processing increases, or it takes additional effort and time to find appropriate parameters.

Table 5.

Angular error analysis from experimental results for RANSAC-based plane detection with α=0.955, P=0.999 and N=4 where α, P and N are the ratio of inliers, the probability of successful detection and the number of iterations, respectively

Plane angle	Wrist angle interval	Angular error	Standard deviation	Maximum error	Computation time
θ_p	Δθ_w	E_θ	σ_θ	\|E_θ\|+σ_θ	T
0°	30°	0.5688°	0.2054°	0.7742°	0.7625s
	60°	0.6833°	0.2563°	0.9396°	0.4750s
	90°	0.8680°	0.4000°	1.2679°	0.4166s
	120°	0.9483°.	0.3946°	1.3429°	0.3600s
	180°	−0.9273°	0.6998°	1.6271°	0.4093s
10°	30°	−1.6241°	0.3636°	1.9877°	0.5867s
	60°	−1.7335°	0.5289°	2.2624°	0.4004s
	90°	−1.2177°	0.2331°	1.4508°	0.3333s
	120°	−1.1049°	0.3502°	1.4551°	0.3298s
	180°	0.5502°	1.2832°	1.833°	0.3802s
20°	30°	−2.2241°	0.3861°	2.6102°	0.5008s
	60°	−2.2548°	0.5150°	2.7698°	0.3919s
	90°	−1.6931°	0.1873°	1.8804°	0.3348s
	120°	−1.8687°	0.4235°	2.2922°	0.3363s
	180°	0.9629°	2.3331°	3.2960°	0.3706s
30°	30°	−2.3930°	0.2285°	2.6216°	0.4724s
	60°	−2.4237°	0.4420°	2.8657°	0.3925s
	90°	−1.8216°	0.1070°	1.9286°	0.3133s
	120°	−2.1003°	0.2654°	2.3657°	0.3164s
	180°	1.3781°	2.9254°	4.3035°	0.3755s

Table 6.

Angular error analysis from experimental results for 3D Hough transform-based plane detection where 3D Hough space (α, β, d) is ${[- \frac{π}{2}, \frac{π}{2}]}^{1} \times {[- \frac{π}{2}, \frac{π}{2}]}^{1} \times {[- d_{m a x}, d_{m a x}]}^{1}$ with Δα=1°, Δβ=1° and Δd=10mm

Plane angle	Wrist angle interval	Angular error	Standard deviation	Maximum error	Computation time
θ_p	Δθ_w	E_θ	σ_θ	\|E_θ\|+σ_θ	T
0°	30°	0.6648°	0.5316°	1.1965°	200.4644s
	60°	1.0511°	0.3323°	1.3834°	102.9116s
	90°	2.1858°	1.4942°	3.6800°	79.6204s
	120°	2.5110°	0.7277°	3.2387°	58.4948s
	180°	2.4311°	1.7817°	4.2128°	39.2949s
10°	30°	0.3903°	2.8835°	3.2738°	199.8429s
	60°	−0.8337°	2.7665°	3.6002°	101.0264s
	90°	1.6285°	2.5642°	4.1927°	82.3411s
	120°	2.1343°	1.4709°	3.6052°	58.6103s
	180°	0.2273°	2.5700°	2.7973°	39.1985s
20°	30°	−2.7858°	0.5631°	3.3488°	200.2916s
	60°	−1.9953°	1.6519°	3.6473°	101.2316s
	90°	−0.3719°	2.0777°	2.4496°	82.5586s
	120°	−0.3659°	2.6417°	3.0076°	58.8060s
	180°	1.2000°	2.4506°	3.6506°	41.3662s
30°	30°	−2.7800°	1.1614°	3.9415°	199.8700s
	60°	−1.5469°	1.5138°	3.0607°	103.2374s
	90°	1.1439°	2.2206°	3.3644°	81.0867s
	120°	0.3000°	2.6135°	2.9136°	59.5532s
	180°	−0.1988°	3.0830°	3.2818°	40.8290s

Figure 9.

Comparison results for PCA, RANSAC and 3D Hough transform-based plane detection methods

5. Conclusions

The hand-eye LRF-based precise plane-detection method was proposed for autonomous robotic welding in the field of shipbuilding. The robotic welding system consists of a 6 DOF manipulator with an LRF on its wrist. The plane is detected by PCA (principal component analysis) using the LRF data, which are obtained only by the rotational motion of the wrist in order to reduce the mechanical error caused by additional motion of the manipulator. In the simulations, the accuracy of the plane detection was analyzed with respect to the change of the plane angle, and the iterative plane detection method with the manipulator's alignment motions is proposed to reduce the plane angle close to zero, since the error increases as the plane angle increases. In addition, accuracy was analysed with respect to the change of the wrist's angle interval that determines the number of the wrist's rotations. As a result of the analysis, the number of the wrist's rotations could be reduced, taking the requirements for successful autonomous robotic welding into consideration. Besides this, the results of the analysis can be used to determine an acceptable specification for the manipulator. For verifying the feasibility and effectiveness of the proposed method, experiments were carried out with the prototype of the hand-eye LRF consisting of the 1 DOF wrist joint, the LRF sensor system and the rotatable plane. Although the error is relatively larger than that of the simulations, the tendency of the error with respect to the change of the wrist's angle interval and plane angle is similar to those of the simulations. Using the proposed method, the welding plane can be detected precisely for successful, autonomous robotic welding. Also, the analysis results of plane-detection experiments can be used to determine the specifications of the autonomous robotic welding-system in the field of shipbuilding. In addition, the experimental results for PCA were compared with those for the other representative plane-detection methods, based on RANSAC and the 3D Hough transform, where the parameters of the methods were appropriately defined for the LRF-based plane detection. According to the comparison results, PCA is remarkably better than the others regarding both accuracy and computation time. In other words, PCA is more appropriate than the others for the proposed iterative plane-detection method, using LRF data.

Footnotes

6. Acknowledgements

This work was supported by Brain Korea 21 PLUS Project and Basic Science Research Program (2013R1A1A2012578), through the National Research Foundation of Korea (NRF) funded by the Ministry of Education.

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Hand-Eye LRF-Based Iterative Plane Detection Method for Autonomous Robotic Welding

Abstract

Keywords

1. Introduction

2. Hand-eye LRF-based Robotic Welding System

3. Precise Plane Detection

3.1 PCA-based Plane Detection

3.2 Iterative Plane Detection with Alignment Motion of the Manipulator

3.3 Error Analysis for Rotational Angle of the Wrist

4. Experiments

4.1 Experimental Set-up

4.2 Experimental Results for the PCA-based Plane Detection

4.3 Comparison with RANSAC and 3D Hough Transform-based Plane-Detection Methods

5. Conclusions

Footnotes

6. Acknowledgements

References