Hand-eye calibration method of line laser vision sensor on mechanical arm

Abstract

The invention relates to the technical field of visual calibration, in particular to a hand-eye calibration method of a line laser visual sensor on a mechanical arm, which comprises the following steps: step 1, designing a calibration plate and mounting the calibration plate on a platform of a machine module; and 2, generating a robot scanning point position by controlling a starting position where the laser line coincides with the marking line of the calibration plate. According to the calibration method, only manual operation is needed, the calibration plate is in the visual field of the on-line laser vision sensor, the laser line coincides with the lineation line, other manual intervention is not needed, operation is very simple, the calibration deviation risk caused by manual operation is reduced, other auxiliary equipment is not needed, a large number of calibration point positions can be generated through simple scanning, and the calibration accuracy is improved. The calibration plate is easy to manufacture, low in cost, convenient to use on site, simple in solving mode, free of complex point cloud computing and convenient to integrate into a robot controller.

Description

Technical Field

[0001] The present invention relates to the technical field of vision calibration, and in particular to a hand-eye calibration method of a line laser vision sensor on a robotic arm. Background Art

[0002] Hand-eye calibration generally determines the relationship between the visual sensor coordinate system and the robot arm flange coordinate system through calibration. In order to obtain this conversion relationship, in the prior art, one or more fixed positions are often marked on the calibration block, and the robot base coordinate system coordinates of these positions are obtained by manual alignment through the end of the robot arm, and then the line laser vision sensor is aligned to these positions, and the current position is read from the robot, and the contour obtained by the line laser vision sensor is read and analyzed. The coordinates of the camera coordinate system of these fixed positions are identified through a specific recognition algorithm, thereby solving the conversion relationship between the visual sensor and the robot arm flange coordinate system, that is, the hand-eye calibration matrix. The whole process is too cumbersome and requires a lot of manual intervention.

[0003] In order to make the calibration operation simpler, the prior art also uses a standard sphere as a calibration block, and uses a line laser profile sensor to image different positions of the calibration sphere, obtain a number of data points on the surface of the sphere irradiated by the laser line, and perform arc fitting on the several data points to obtain the arc radius and the center of the circle; the coordinates of the center of the sphere are calculated, and additional auxiliary instruments are used to determine the coordinates of the center of the sphere in the robot arm base coordinates, thereby solving the conversion relationship between the visual sensor and the robot arm flange coordinate system.

[0004] However, this calibration method has high requirements for the processing accuracy of the standard sphere, and requires additional auxiliary instruments to obtain the robot base coordinate system coordinates of the center of the sphere. It also relies on the human eye to judge the positive and negative positions of the center of the calibration ball, which is prone to manual errors and is not suitable for on-site hand-eye calibration. At the same time, when the robot changes position and scans, the surface features obtained based on point cloud processing are not obvious enough, resulting in inaccurate calibration results. Therefore, a hand-eye calibration method for line laser vision sensors on a robotic arm is needed to solve the above problems. Summary of the invention

[0005] The purpose of the present invention is to solve the problems raised in the above background technology.

[0006] In order to achieve the above object, the present invention adopts the following technical solutions:

[0007] A hand-eye calibration method for a line laser vision sensor on a robotic arm comprises the following steps:

[0008] Step 1: Design the calibration board and install it on the platform of the machine module;

[0009] Step 2: Generate the robot scanning point by controlling the starting position where the laser line coincides with the marking line on the calibration plate;

[0010] Step 3: The coordinates Psi of the camera coordinate system and the coordinates Pi of the base coordinate system of the calibration point can be easily obtained by using the a value at each scanning point;

[0011] Step 4: Use the robotic arm to measure the flange position Ti at each scanning point;

[0012] Step 5: The coordinates and positions of all scanning points are obtained by solving the overdetermined equations by the least squares method to obtain the hand-eye calibration matrix.

[0013] Preferably, the machine module in the step one is composed of a robotic arm, a robot welding gun, a linear laser vision sensor, a support seat and a platform seat. A platform seat is installed at the bottom end of the robotic arm, a support seat is connected to the right side of the platform seat, and a calibration plate is arranged on the right side of the top end of the support seat shaft. A robot welding gun is installed at the top end of the robotic arm, and a linear laser vision sensor is installed at the end of the robotic arm close to the robot welding gun.

[0014] Preferably, the calibration plate in step one is installed on a support base, and the position of the calibration plate in the robot base coordinate system can be determined by the designed installation position, or by aligning the robot end with three of the four corners of the calibration plate. This process only needs to be performed once when the robot leaves the factory, and is stored inside the robot as known information.

[0015] Preferably, the step 2 is specifically as follows:

[0016] Before calibration, operate the robotic arm so that the calibration plate is in the center of the field of view of the line laser vision sensor, and the line laser coincides with the marking line. This position is considered to be the starting position of the calibration. Based on this starting position, a series of scanning points of the robotic arm are generated. These points need to keep the line laser shining on the calibration plate at all times, always parallel to the marking line, and try to make the calibration plate be in each corner of the field of view of the line laser vision sensor.

[0017] Preferably, the step three is specifically:

[0018] Establish the calibration plate workpiece coordinate system OXYZ, set the base coordinates of the surface centers of hole 1, hole 2, and hole 3 to P1 (x1, y1, z1), P2 (x2, y2, z3), and P3 (x3, y3, z3), respectively, then:

[0019] The origin of the OXYZ coordinate system is PO(px, py, pz) = ((x1+x2) / 2, (y1+y2) / 2, (z1+z2) / 2), and the x-axis direction vector (nx, ny, nz) = (P2-P1) / ||P2-P1|.

[0020] Y-axis direction vector (ox, oy, oz) = (P3-P1) / / ||P3-P1||.

[0021] Preferably, the step 4 is specifically as follows:

[0022] Through z=xXy, we can get the z-axis direction vector (ax, ay, az), and the relative position of OXYZ relative to the base coordinate system can be obtained:

[0023]

[0024] Preferably, the line laser vision sensor can obtain the distance a between Pl and Pr where the current laser line hits the calibration plate. Since the calibration plate triangle is designed in advance with known width and height of W and H, respectively, then:

[0025] The coordinates of Pl and Pr in the workpiece coordinate system are Pl(x, y, z) = (-a / 2, a*H / W), Pr(x, y) = (a / 2, a*H / W), and the coordinates of Pl and Pr in the robot base coordinates are obtained by Toxyz*Pl, Toxyz*Pr. The coordinates of these two points in the camera coordinates Pls, Prs can be directly read by the line laser vision sensor, and the current robot arm flange position Trob can be directly read by the robot arm.

[0026] Preferably, the point position of the line laser vision sensor on the calibration plate is known at a distance a, that is, the coordinate Ps of the corner point (Pl or Pr) irradiated on the calibration plate in the camera coordinate system and the coordinate P0 in the base coordinate system can be obtained by calculation, the position Ti of the robot flange is set, and the transformation matrix of the camera coordinate system of the line laser vision sensor (3) relative to the robot flange coordinate system is set to Tsensor, then:

[0027] Since there are multiple scanning points, a series of Pi, Ti, Psi are formed.

[0028]

[0029] RiRsRsi+RiTs=Pi-Ti, (i=0, 1, 2,...,n),

[0030] (RiRsi-R0Rs0)Rs+(Ri-R0)Ts=(Pi-Ti)-(P0-T0), (i=0,1,2,…,n)

[0031] Among them Ri=(Nx, Ny, Nz), Psi= (Xsi, Ysi, Zsi), there are (RiXsi-R0Xs0)Nx+(RiYsi-R0Ys0)Ny+(RiZsi-R0Zs0)Nz+(Ri-R0)Ts=(Pi-Ti)-(P0-T0), (i=0,1,2,...,n),

[0032] [(RiXsi-R0Xs0)(RiYsi-R0Ys0)(RiZsi-R0Zs0)(Ri-R0)][Nx Ny Nz Ts] T =(Pi-Ti)-(P0-T0), (i=0,1,2,…,n),

[0033]

[0034] x=[Nx Ny Nz Ts] T ,

[0035]

[0036] Preferably, the step five is specifically:

[0037] Solve x=(A T A) -1 A T b, that is, the hand-eye calibration matrix between the visual sensor and the robot arm flange coordinate system is obtained.

[0038] The present invention has at least the following beneficial effects:

[0039] 1. By setting the support seat with a calibration plate, a quick calibration operation is achieved. Compared with the traditional structure, the calibration method of the present invention only requires manual operation to make the calibration plate within the field of view of the online laser vision sensor and the laser line coincide with the marking line. No other human intervention is required, the operation is very simple, and the risk of calibration deviation caused by manual operation is reduced. No other auxiliary equipment is required, and a large number of calibration points can be generated by simple scanning, which is beneficial to improving the accuracy of solving overdetermined equations by the least squares method. The calibration plate is easy to make, low cost, and convenient for on-site use. The solution method is simple, without the need for complex point cloud computing, and it is convenient to integrate into the robot controller.

[0040] 2. By setting a support base and a calibration plate, the position of the calibration plate can be locked. In the present invention, the calibration plate is installed on a support base fixed to the robot. There is no need to sense the specific position of the calibration plate every time calibration is performed. The calibration plate is not limited to a single form. The design element of the calibration plate is that there is no need to manually align the calibration points, reducing human intervention. The coordinates of the calibration points in the camera coordinate system and the coordinates of the base coordinate system can be directly obtained through the parameters obtainable by the line laser vision sensor, effectively reducing the time required for calibration and improving the calibration efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings required for use in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other accompanying drawings can be obtained based on these accompanying drawings without paying any creative work.

[0042] Figure 1 This is a schematic diagram of the overall external structure of a hand-eye calibration device for a line laser vision sensor on a robotic arm proposed by the present invention;

[0043] Figure 2 A schematic diagram of the hand-eye calibration process of a line laser vision sensor on a robotic arm proposed by the present invention;

[0044] Figure 3 A schematic diagram of the three-dimensional structure of a calibration plate in a hand-eye calibration device for a line laser vision sensor on a robotic arm proposed by the present invention;

[0045] Figure 4 This is a schematic diagram of the top view of the structure of a calibration plate in a hand-eye calibration device of a line laser vision sensor on a robotic arm proposed by the present invention.

[0046] In the figure: 1. Robotic arm; 2. Robot welding gun; 3. Line laser vision sensor; 4. Support base; 5. Calibration plate; 6. Platform base. DETAILED DESCRIPTION

[0047] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0048] Reference Figure 1-4 , a hand-eye calibration method of a line laser vision sensor on a robotic arm, comprising the following steps:

[0049] Step 1: Design the calibration plate 5 and install it on the platform of the machine module;

[0050] Step 2: Generate the robot scanning point by controlling the starting position where the laser line coincides with the marking line of the calibration plate 5;

[0051] Step 3: The coordinates Psi of the camera coordinate system and the coordinates Pi of the base coordinate system of the calibration point can be easily obtained by using the a value at each scanning point;

[0052] Step 4: Use the robotic arm to measure the flange position Ti at each scanning point;

[0053] Step 5: The coordinates and positions of all scanning points are obtained by solving the overdetermined equations by the least squares method to obtain the hand-eye calibration matrix.

[0054] The machine module in step one is composed of a robot arm 1, a robot welding gun 2, a line laser vision sensor 3, a support seat 4 and a platform seat 6. A platform seat 6 is installed at the bottom end of the robot arm 1, and a support seat 4 is connected to the right side of the platform seat 6. A calibration plate 5 is arranged on the right side of the top end of the support seat 4 axis. A robot welding gun 2 is installed at the top end of the robot arm 1, and a line laser vision sensor 3 is installed at the end of the robot arm 1 close to the robot welding gun 2. The line laser vision sensor 3 is fixed to the outer side of the end seat of the platform seat 6 through the support seat 4, so that the support seat 4 supports the calibration plate 5. Since the length and height of the support seat 4 are fixed, that is, the position of the calibration plate 5 installed at the top of the support seat 4 is fixed, it is convenient to input the initial position coordinates and store them in the database. When positioning, the robot welding gun 2 can be quickly moved to the initial position.

[0055] The calibration plate 5 in step one is installed on the support base 4. The position of the calibration plate 5 in the robot base coordinate system can be determined by the designed installation position, or by aligning the robot end with three of the four corners of the calibration plate 5. This process only needs to be performed once when the robot leaves the factory and is stored inside the robot as known information.

[0056] Step 2 is as follows:

[0057] Before calibration, operate the robotic arm so that the calibration plate 5 is at the center of the field of view of the online laser vision sensor, and the line laser coincides with the marking line. This position is considered to be the starting position of the calibration, and a series of scanning points of the robotic arm are generated based on the starting position. These points need to always keep the line laser shining on the calibration plate 5, always parallel to the marking line, and try to make the calibration plate 5 in each corner of the field of view of the line laser vision sensor. Use the line laser vision sensor to align these positions, read the current position from the robot, and read and analyze the contour obtained by the line laser vision sensor. The coordinates of the camera coordinate system of these fixed positions are identified through a specific recognition algorithm, thereby solving the conversion relationship between the vision sensor and the robotic arm flange coordinate system.

[0058] Step three is as follows:

[0059] Establish the workpiece coordinate system OXYZ of calibration plate 5, set the base coordinates of the surface centers of hole 1, hole 2, and hole 3 to P1 (x1, y1, z1), P2 (x2, y2, z3), and P3 (x3, y3, z3), respectively, then:

[0060] The origin of the OXYZ coordinate system is PO(px, py, pz) = ((x1+x2) / 2, (y1+y2) / 2, (z1+z2) / 2),

[0061] x-axis direction vector (nx, ny, nz) = (P2-P1) / ||P2-P1|,

[0062] Y-axis direction vector (ox, oy, oz) = (P3-P1) / / ||P3-P1||.

[0063] Step 4 is as follows:

[0064] Through z=xXy, we can get the z-axis direction vector (ax, ay, az), and the relative position of OXYZ relative to the base coordinate system can be obtained:

[0065]

[0066] The line laser vision sensor 3 can obtain the distance a between the current laser line Pl and Pr on the calibration plate 5. Since the calibration plate 5 triangle is designed in advance with known width and height W and H respectively, then:

[0067] The coordinates of Pl and Pr in the workpiece coordinate system are Pl(x, y, z) = (-a / 2, a*H / W), Pr(x, y) = (a / 2, a*H / W), and the coordinates of Pl and Pr in the robot base coordinates are obtained by Toxyz*Pl, Toxyz*Pr. The coordinates of these two points in the camera coordinates Pls, Prs can be directly read by the line laser vision sensor 3, and the current robot arm flange position Trob can be directly read by the robot arm.

[0068] The point position of the line laser vision sensor 3 on the calibration plate 5, the distance a is known, that is, the coordinates Ps of the corner point (Pl or Pr) irradiated on the calibration plate 5 in the camera coordinate system and the coordinates P0 in the base coordinate system can be calculated, the flange position Ti of the manipulator 1 is set, and the transformation matrix of the camera coordinate system of the line laser vision sensor 3 relative to the flange coordinate system of the manipulator 1 is set to Tsensor, then:

[0069] Since there are multiple scanning points, a series of Pi, Ti, Psi are formed.

[0070]

[0071] RiRsRsi+RiTs=Pi-Ti, (i=0, 1, 2,...,n),

[0072] (RiRsi-R0Rs0)Rs+(Ri-R0)Ts=(Pi-Ti)-(P0-T0), (i=0,1,2,…,n)

[0073] Among them Ri=(Nx, Ny, Nz), Psi= (Xsi, Ysi, Zsi), there are (RiXsi-R0Xs0)Nx+(RiYsi-R0Ys0)Ny+(RiZsi-R0Zs0)Nz+(Ri-R0)Ts=(Pi-Ti)-(P0-T0), (i=0,1,2,...,n),

[0074] [(RiXsi-R0Xs0)(RiYsi-R0Ys0)(RiZsi-R0Zs0)(Ri-R0)][Nx Ny Nz Ts] T =(Pi-Ti)-(P0-T0), (i=0,1,2,…,n),

[0075]

[0076] x=[Nx Ny Nz Ts] T ,

[0077]

[0078] The step five is specifically as follows:

[0079] Solve x=(A T A) -1 A T b, that is, the hand-eye calibration matrix between the visual sensor and the robot arm flange coordinate system is obtained.

[0080] According to the attached Figure 1 As shown, during positioning, the initial position coordinates of the calibration plate 5 are entered into the robot information storage device, and the robot scanning point is generated by controlling the starting position where the laser line of the line laser vision sensor 3 coincides with the marking line of the calibration plate 5. During scanning, the robot arm is operated before calibration to make the calibration plate 5 at the center of the field of view of the line laser vision sensor, and the line laser coincides with the marking line. This position is considered to be the starting position of calibration, and a series of scanning points of the robot arm are generated based on the starting position. These points need to always keep the line laser shining on the calibration plate 5, always parallel to the marking line, and make the calibration plate 5 as far as possible in each corner of the field of view of the line laser vision sensor.

[0081] According to the attached Figure 3 With attached Figure 4 As shown, hole 1, hole 2, and hole 3 are set at the corners of the calibration plate 5, and the coordinates of the base coordinates of the centers of the three hole surfaces are P1, P2, and P3 respectively. At this time, the coordinate origin of the calibration plate 5 is P0, and the z-axis direction vector (ax, ay, az) can be obtained through z=xXy, and the relative position of OXYZ relative to the base coordinate system can be obtained. The distance a between Pl and Pr where the current laser line hits the calibration plate 5 can be obtained through the line laser vision sensor 3, and the coordinates P of Pl and Pr in the robot base coordinates can be obtained through Toxyz*Pl and Toxyz*Pr. l0, Pr0, and the coordinates Pls and Prs of the two points in the camera coordinates can be directly read by the line laser vision sensor 3. The current robot flange position Trob can be directly read by the robot arm. The coordinates Ps of the corner point (Pl or Pr) irradiated on the calibration plate 5 in the camera coordinate system and the coordinates P0 in the base coordinate system are obtained by calculation. The flange position T of the robot arm 1 is set, and the transformation matrix of the camera coordinate system of the line laser vision sensor 3 relative to the flange coordinate system of the robot arm 1 is set. Finally, the least square method of the overdetermined equations is used to solve x=(A T A) -1 A q b, that is, the hand-eye calibration matrix between the line laser vision sensor 3 and the flange coordinate system of the robot 1 is obtained.

[0082] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions only describe the principles of the present invention. The present invention may be subject to various changes and improvements without departing from the spirit and scope of the present invention. These changes and improvements fall within the scope of the present invention. The scope of protection claimed by the present invention is defined by the attached claims and their equivalents.

Claims

1. A hand-eye calibration method for a line laser vision sensor on a robotic arm, characterized in that: The following steps are involved: Step 1: Design the calibration plate (5) and install it on the platform of the machine module; Step 2: Generate the robot scanning point by controlling the laser line to overlap the starting position of the marking line on the calibration plate (5); Step 3: The coordinates Psi of the camera coordinate system and the coordinates Pi of the base coordinate system of the calibration point can be easily obtained by using the a value at each scanning point; Step 4: Use the robotic arm to measure the flange position Ti at each scanning point; Step 5: The coordinates and positions of all scanning points are obtained by solving the overdetermined equations by the least squares method to obtain the hand-eye calibration matrix.

2. According to claim 1, a hand-eye calibration method for a line laser vision sensor on a robotic arm is characterized in that: The machine module in step 1 is composed of a robot arm (1), a robot welding gun (2), a line laser vision sensor (3), a support seat (4) and a platform seat (6); the platform seat (6) is installed at the bottom end of the robot arm (1); the support seat (4) is connected to the right side of the platform seat (6); and a calibration plate (5) is arranged on the right side of the top end of the support seat (4); the robot welding gun (2) is installed at the top end of the robot arm (1); and the line laser vision sensor (3) is installed at the position of the end of the robot arm (1) close to the robot welding gun (2).

3. The hand-eye calibration method of a line laser vision sensor on a robotic arm according to claim 1, characterized in that: The calibration plate (5) in step 1 is installed on the support seat (4). The position of the calibration plate (5) in the robot base coordinate system can be determined by the designed installation position, or by aligning the robot end with three of the four corners of the calibration plate (5). This process only needs to be performed once when the robot leaves the factory, and is stored inside the robot as known information.

4. The hand-eye calibration method of a line laser vision sensor on a robotic arm according to claim 1, characterized in that: The step 2 is specifically as follows: Before calibration, the robot arm is operated so that the calibration plate (5) is at the center of the field of view of the line laser vision sensor, and the line laser coincides with the marking line. This position is considered to be the starting position of the calibration. Based on the starting position, a series of scanning points of the robot arm are generated. These points need to always keep the line laser shining on the calibration plate (5) and always be parallel to the marking line, so that the calibration plate (5) is respectively located at each corner of the field of view of the line laser vision sensor.

5. The hand-eye calibration method of a line laser vision sensor on a robotic arm according to claim 4, characterized in that: The step three is specifically as follows: Establish the workpiece coordinate system OXYZ of the calibration plate (5), set the base coordinates of the surface centers of hole 1, hole 2, and hole 3 to P1 (x1, y1, z1), P2 (x2, y2, z3), and P3 (x3, y3, z3), respectively, then: The origin of the OXYZ coordinate system is PO(px, py, pz) = ((x1+x2) / 2, (y1+y2) / 2, (z1+z2) / 2), x-axis direction vector (nx, ny, nz) = (P2-P1) / ||P2-P1|, Y-axis direction vector (ox, oy, oz) = (P3-P1) / / ||P3-P1||.

6. The hand-eye calibration method of a line laser vision sensor on a robotic arm according to claim 1, characterized in that: The step 4 is specifically as follows: Through z=xXy, we can get the z-axis direction vector (ax, ay, az), and the relative position of OXYZ relative to the base coordinate system can be obtained:

7. The hand-eye calibration method of a line laser vision sensor on a robotic arm according to claim 2, characterized in that: The line laser vision sensor (3) can obtain the distance a between the current laser line Pl and Pr on the calibration plate (5). Since the calibration plate (5) is designed in advance with a triangle with known width and height of W and H, respectively, then: The coordinates of Pl and Pr in the workpiece coordinate system are Pl(x, y, z) = (-a / 2, a*H / W), Pr(x, y) = (a / 2, a*H / W), and the coordinates of Pl and Pr in the robot base coordinate system are obtained by Toxyz*Pl, Toxyz*Pr. The coordinates of these two points in the camera coordinate system, Pls, Prs, can be directly read by the line laser vision sensor (3), and the current robot flange position Trob can be directly read by the robot arm.

8. The hand-eye calibration method of a line laser vision sensor on a robotic arm according to claim 1, characterized in that: The point position of the line laser vision sensor (3) on the calibration plate (5) is known by the distance a, that is, the coordinate Ps of the corner point (Pl or Pr) irradiated on the calibration plate (5) in the camera coordinate system and the coordinate P0 in the base coordinate system can be calculated, the flange position Ti of the manipulator (1) is set, and the transformation matrix of the camera coordinate system of the line laser vision sensor (3) relative to the flange coordinate system of the manipulator (1) is set to Tsensor, then: Since there are multiple scanning points, a series of Pi, Ti, Psi are formed. RiRsRsi+RiTs=Pi-Ti, (i=0, 1, 2,..., n), (RiRsi-R0Rs0)Rs+(Ri-R0)Ts=(Pi-Ti)-(P0-T0), (i=0,1,2,…,n) Where Ri=(Nx,Ny,Nz), Psi=(Xsi,Ysi,Zsi), (RiXsi-R0Xs0)Nx+(RiYsi-R0Ys0)Ny+(RiZsi-R0Zs0)Nz+(Ri-R0)Ts=(Pi-Ti)-(P0-T0), (i=0,1,2,…,n), [(RiXsi-R0Xs0)(RiYsi-R0Ys0)(RiZsi-R0Zs0)(Ri-R0)][Nx Ny Nz Ts] T =(Pi-Ti)-(P0-T0)\,1,2,…,n)\, x=[Nx NZ Ts] T , 9. The hand-eye calibration method of a line laser vision sensor on a robotic arm according to claim 1, characterized in that: The step five is specifically as follows: Solve x=(A T A) -1 A T b, that is, the hand-eye calibration matrix between the visual sensor and the robot arm flange coordinate system is obtained.

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