A Robotic Compliant Tool Calibration Method Based on an Articulated Measuring Arm

Abstract

This invention provides a robot compliant tool calibration method based on an articulated measuring arm, comprising the following steps: S1, controlling the robotic arm to move along the coordinate axes and recording the relevant coordinates of the measuring arm and the robot; S2, controlling the robotic arm to rotate around a fixed point and recording the relevant coordinates of the measuring arm and the robot; S3, establishing a system of equations based on the coordinates from the first two steps, and solving for the transformation matrix using the least squares method; S4, calculating the calibration plane equation using the measuring arm, controlling the robotic arm to contact the plane, and solving for the tool center point position based on the robot's own pose and the plane equation. The robot compliant tool calibration method based on an articulated measuring arm described in this invention introduces a high-precision measuring instrument, achieving high calibration accuracy, not limited by robot accuracy, and with low operational difficulty.

Description

Technical Field

[0001] This invention belongs to the field of robot calibration technology in the automation industry, and in particular relates to a robot compliant tool calibration method based on an articulated measuring arm. Background Technology

[0002] Currently, before a robotic arm is put into operation, the origin (TCP point) of the end-effector coordinate system needs to be manually calibrated. Generally, the four-point method is used for calibration, that is, the worker marks the pre-set TCP point on the end-effector of the robotic arm, and then controls the robotic arm to move four times through the robotic arm teach pendant, ensuring that the robotic arm moves to the same fixed point in a different posture each time. The robotic arm controller collects the posture data of the robotic arm four times, thereby calculating the coordinates of the TCP point of the end-effector of the robotic arm.

[0003] The calibration process described above requires skilled workers. To ensure high accuracy, calibrating a single TCP point typically takes more than five minutes, which is time-consuming, labor-intensive, and cannot guarantee consistent accuracy. For compliant tools, the tool will shift under force, resulting in even greater errors in point-to-point calibration. Furthermore, calibration accuracy is limited by the robot's absolute positioning accuracy. Therefore, in applications requiring a high-precision tool coordinate system, high-precision measuring tools are necessary. Summary of the Invention

[0004] In view of this, the present invention aims to propose a robot compliant tool calibration method based on an articulated measuring arm, in order to solve the problems of the existing "four-point method" which has high requirements for operators, is difficult, and whose accuracy depends entirely on the absolute positioning accuracy of the robot itself.

[0005] To achieve the above objectives, the technical solution of the present invention is implemented as follows:

[0006] A method for calibrating compliant cutting tools for robots based on an articulated measuring arm includes the following steps:

[0007] S1. Control the robotic arm to move along the coordinate axis and record the relevant coordinates of the measuring arm and the robot;

[0008] S2. Control the robotic arm to rotate around a fixed point and record the relevant coordinates of the measuring arm and the robot;

[0009] S3. Establish a system of equations based on the coordinates from the first two steps, and solve for the transformation matrix using the least squares method;

[0010] S4. Use the measuring arm to calculate the calibration plane equation, control the robotic arm to contact the plane, and solve for the tool center point position based on the robot's own pose and the plane equation.

[0011] Furthermore, the specific method of step S1 is as follows:

[0012] S11. Select a fixed point on the robot's end effector as the measurement point P. a ,

[0013] S12. Keeping the robotic arm's posture unchanged, control the robotic arm to move along the X, Y, and Z axes, and use the measuring arm to measure P. a The coordinates in the measuring arm coordinate system are measured five times for each axis, and the corresponding robot end-effector pose is recorded; for five points on a certain axis, the coordinates in the measuring arm coordinate system are recorded as { M P1, M P2, M P3, M P4, M P5}; Record the robot end-effector coordinates in the robot coordinate system as { R P1, R P2, R P3, R P4, R P5}

[0014] Furthermore, the specific method of step S2 is as follows: keeping the position of the robot's end-effector coordinate system unchanged, only changing the robot's posture, and measuring P with a measuring arm. a Position, recording the posture of the robotic arm's end effector in the robot coordinate system. and position { R P e1 , R P e2 , R P e3 , R P e4 , R P e5}, and the measuring arm reading { M P a1 , M P a2 , M P a3 , M P a4 , M P a5}

[0015] Furthermore, the specific method for step S3 is as follows:

[0016] First, calculate the rotation matrix.

[0017] For any five points on one axis in the same coordinate system, the resulting coordinate matrix is ​​as follows:

[0018]

[0019] The covariance matrix of the above coordinate matrix is

[0020]

[0021] The mathematical expectation of coordinate X in equation (1)

[0022]

[0023] Mathematical expectation of coordinate Y

[0024]

[0025] Mathematical expectation of coordinate Z

[0026]

[0027] For two variables M and N with expected values ​​E(M) and E(N), their covariance Cov(M,N) is:

[0028] Cov(M,N)=E(MN)-E(M)E(N) (6)

[0029] Substituting equations (3), (4), (5), and (6) into equation (2), the covariance matrix can be calculated. Then, singular value decomposition is used to calculate the largest eigenvector V of the covariance matrix, which is the direction of the current coordinate axis. The representations of the robot's three axes X, Y, and Z in the robot coordinate system are established sequentially. R X, R Y, R Z} and its representation in the coordinate system of the measuring arm { M X, M Y, M Z}. The two sets of coordinate vectors have the following relationship:

[0030]

[0031] Solving equation (7) yields the rotation matrix:

[0032]

[0033] Then, the offset vector from the measuring arm coordinate system to the robot coordinate system is calculated;

[0034] In step S2, the measurement point P a With the coordinate P of the robotic arm end effector e The following relationship exists:

[0035]

[0036] In the formula R P ai Let P be the measurement point of the robot in the i-th pose. a Representation in robot coordinate systemR P ei Let be the representation of the robot's end effector in the robot coordinate system for the robot's i-th pose. E t ae For the end effector P of the robotic arm e to measurement point P a Offset in the robot's end-effector coordinate system This indicates the robot's current pose.

[0037] Measurement point P a The relationship between the coordinates in the robot coordinate system and the coordinates in the measuring arm coordinate system is as follows:

[0038]

[0039] In the formula To measure the offset vector from the arm coordinate system to the robot arm coordinate system in the robot arm coordinate system, substitute equation (10) into equation (9) to replace... R P ai We can obtain:

[0040]

[0041] Subtracting the data from pose 1 from the data of pose 2 yields the following relationship:

[0042]

[0043] And so on.

[0044]

[0045] beg E t ae Best Least Squares Solution

[0046]

[0047] Will E t ae Bring it back and calculate it using the same method. The best least squares solution.

[0048] Thus, the transformation matrix from the measuring arm coordinate system to the robot coordinate system is obtained.

[0049]

[0050] Furthermore, the specific method for step S4 is as follows:

[0051] Select a fixed, precision-machined plane as the calibration plane. Use the measuring arm to randomly select five points on the plane. The coordinates of these five points are ( M x1,M y1, M z1), ( M x2, M y2, M z2), ( M x3, M y3, M z3), ( M x4, M y4, M z4), ( M x5, M y5, M z5), the transformation matrix Multiplying these five coordinates by the given coordinates yields the five points' representations in the robot coordinate system: (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), (x4, y4, z4), and (x5, y5, z5). The planar regression equation for these five points in the robot coordinate system is then calculated using the following method:

[0052] The general expression for the equation of a plane is:

[0053] Ax + By + Cz + D = 0 (C ≠ 0)

[0054] Transform it into the following form:

[0055]

[0056] make but

[0057] z = a0x + a1y + a2

[0058] At this point, the matrix form of the plane equation can be expressed as:

[0059]

[0060] The least squares solution for coefficients a0, a1, and a2 can be obtained.

[0061]

[0062] Therefore, the plane regression equation Z = a0X + a1Y + a2 can be obtained;

[0063] Since the distance from the probe center point to the plane is equal to the probe radius, the regression equation needs to be shifted in the opposite direction along the plane normal by the length of the probe radius. Let the probe radius be r, then the coefficient a3 of the constant term in the translated plane equation is...

[0064]

[0065] The equation for the calibration plane, Z = a0X + a1Y + a3, can be rearranged as Z = aX + bY + C.

[0066] Control the robot to make the tool's center point contact any five points on the plane in different postures, and record the coordinates of the robot's end effector. R P t1 , R P t2 , R P t3 , R P t4 , R P t5} and posture The tool center point P of the robot in different poses tcp It can be represented as

[0067]

[0068] In the formula R P tcpi Let P be the tool center point of the robot in the i-th pose. tcp Representation in robot coordinate system Let i be the robot's pose at the i-th time. E P tcpi Tool center point P tcp Representation in the robot's end-effector coordinate system R P ei Let be the representation of the robot end effector in the robot coordinate system under the i-th pose;

[0069] Tool center point coordinates under different poses R P tcpi Satisfy the following equation

[0070]

[0071] Substituting the five different poses and equation (15) into the system of equations yields the following set of equations.

[0072]

[0073] beg E P tcp Best Least Squares Solution

[0074]

[0075] Compared with existing technologies, the robot compliant tool calibration method based on an articulated measuring arm described in this invention has the following advantages:

[0076] (1) The robot compliant tool calibration method based on articulated measuring arm described in this invention addresses the problems of the traditional four-point method being too difficult to operate, the calibration accuracy depending on the robot's own accuracy, and the compliant tool being easily deformed under force. By measuring the coordinates of the measuring arm and the robot's pose at the same point, the transformation matrix between the measuring arm and the robot is calculated. This ensures that the coordinates of subsequent points are all measured by the measuring arm, greatly improving the calibration accuracy. To solve the problems of the high difficulty of controlling the robot point-to-point and the easy deformation of the compliant tool during the calibration process, a method is proposed to make the robot tool center point contact the surface. The measuring arm measures five points on the plane, calculates the plane equation, transforms it into the robot coordinate system, controls the robot to make the tool center point contact the plane in different poses, and solves the equation to obtain the coordinates of the tool center point. This reduces the alignment work of three degrees of freedom to the alignment of one degree of freedom, greatly reducing the operation difficulty and human error. Attached Figure Description

[0077] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0078] Figure 1 This is a flowchart illustrating the implementation of the robot compliant tool calibration method based on an articulated measuring arm according to an embodiment of the present invention.

[0079] Figure 2 This is a schematic diagram of the calibration system described in an embodiment of the present invention;

[0080] Figure 3 This is a schematic diagram illustrating the operation of measuring fixed points on the measuring arm probe tool according to an embodiment of the present invention;

[0081] Figure 4 This is a schematic diagram illustrating the operation of contacting the calibration plane with the center point of the tool in different postures. Detailed Implementation

[0082] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0083] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0084] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art will understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0085] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0086] A method for calibrating compliant robotic tools based on an articulated measuring arm, such as... Figures 1 to 4 As shown, by establishing a transformation matrix from the measuring arm coordinate system to the robot coordinate system, high-precision positioning of the robot TCP is achieved. The fixed point in the "four-point method" is changed to a fixed plane. The TCP position coordinates are obtained by inversely calculating the plane equation and the robot end position, reducing the calibration difficulty.

[0087] like Figure 1 As shown, the specific steps include the following:

[0088] The first step is to select a fixed point on the robot's end effector as the measurement point P. a ,like Figure 3 As shown, a hexagonal screw is selected so that the probe can be fixed inside the screw and will not wobble.

[0089] The second step is to keep the robotic arm in the same posture and control its movement along the X, Y, and Z axes, then use the measuring arm to measure P. aThe coordinates in the measuring arm coordinate system are measured five times for each axis, and the corresponding robot end-effector pose is recorded. For five points on a certain axis, the coordinates in the measuring arm coordinate system are recorded as { M P1, M P2, M P3, M P4, M P5}; Record the robot end-effector coordinates in the robot coordinate system as { R P1, R P2, R P3, R P4, R P5}

[0090] Third, keeping the robot's end effector coordinate system position unchanged, only changing the robot's posture, and using the measuring arm to measure P. a Position, recording the posture of the robotic arm's end effector in the robot coordinate system. and position { R P e1 , R P e2 , R P e3 , R P e4 , R P e5}, and the measuring arm reading { M P a1 , M P a2 , M P a3 , M P a4 , M P a5}

[0091] The fourth step is to establish a system of equations based on the coordinates from the first two steps and calculate the transformation matrix from the measuring arm coordinate system to the robot coordinate system.

[0092] First, calculate the rotation matrix.

[0093] For any five points on one axis in the same coordinate system, the resulting coordinate matrix is ​​as follows:

[0094]

[0095] The covariance matrix of the above coordinate matrix is

[0096]

[0097] The mathematical expectation of coordinate X in equation (1)

[0098]

[0099] Mathematical expectation of coordinate Y

[0100]

[0101] Mathematical expectation of coordinate Z

[0102]

[0103] For two variables M and N with expected values ​​E(M) and E(N), their covariance Cov(M,N) is:

[0104] Cov(M,N)=E(MN)-E(M)E(N) (6)

[0105] Substituting equations (3), (4), (5), and (6) into equation (2), the covariance matrix can be calculated. Then, singular value decomposition is used to calculate the largest eigenvector V of the covariance matrix, which is the direction of the current coordinate axis. The representations of the robot's three axes X, Y, and Z in the robot coordinate system are established sequentially. R X, R Y, R Z} and its representation in the coordinate system of the measuring arm { M X, M Y, M Z}. The two sets of coordinate vectors have the following relationship:

[0106]

[0107] Solving equation (7) yields the rotation matrix:

[0108]

[0109] The offset vector from the measuring arm coordinate system to the robot coordinate system is then calculated.

[0110] In the third step, the measurement point P is... a With the coordinate P of the robotic arm end effector e The following relationship exists:

[0111]

[0112] In the formula R P ai Let P be the measurement point of the robot in the i-th pose. a Representation in robot coordinate system R P ei Let be the representation of the robot's end effector in the robot coordinate system for the robot's i-th pose. E t ae For the end effector P of the robotic arm e to measurement point P aOffset in the robot's end-effector coordinate system This indicates the robot's current posture.

[0113] Measurement point P a The relationship between the coordinates in the robot coordinate system and the coordinates in the measuring arm coordinate system is as follows:

[0114]

[0115] In the formula To represent the offset vector from the arm coordinate system to the robot arm coordinate system in the robot arm coordinate system, substitute equation (10) into equation (9) and replace... R P ai We can obtain:

[0116]

[0117] Subtracting the data from pose 1 from the data of pose 2 yields the following relationship:

[0118]

[0119] And so on.

[0120]

[0121] beg E t ae Best Least Squares Solution

[0122]

[0123] Will E t ae Bring it back and calculate it using the same method. The best least squares solution.

[0124] Thus, the transformation matrix from the measuring arm coordinate system to the robot coordinate system is obtained.

[0125]

[0126] Fifth step, select a fixed precision-machined plane as the calibration plane, and use the measuring arm to randomly select five points on the plane. The coordinates of these five points are ( M x1, M y1, M z1), ( M x2, M y2, M z2), ( M x3, M y3, M z3), ( M x4, M y4,M z4), ( M x5, M y5, M z5), the transformation matrix Multiplying these five coordinates by the coordinates of the five points yields the representations of the five points in the robot coordinate system: (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), (x4, y4, z4), (x5, y5, z5). Calculate the planar regression equations of these five points in the robot coordinate system.

[0127] The general expression for the equation of a plane is:

[0128] Ax + By + Cz + D = 0 (C ≠ 0)

[0129] Transform it into the following form:

[0130]

[0131] make but

[0132] z = a0x + a1y + a2

[0133] At this point, the matrix form of the plane equation can be expressed as:

[0134]

[0135] The least squares solution for coefficients a0, a1, and a2 can be obtained.

[0136]

[0137] Therefore, the plane regression equation Z = a0X + a1Y + a2 can be obtained.

[0138] Since the distance from the probe center point to the plane is equal to the probe radius, the regression equation needs to be shifted in the opposite direction along the plane normal by the length of the probe radius. Let the probe radius be r, then the coefficient a3 of the constant term in the translated plane equation is...

[0139]

[0140] The equation for the calibration plane, Z = a0X + a1Y + a3, can be rearranged to Z = aX + bY + C.

[0141] Step 6, as follows Figure 4 As shown, control the robot to make the tool's center point contact any five points on the plane in different postures, and record the coordinates of the robot's end effector. R P t1 , R P t2 , R P t3, R P t4 , R P t5} and posture The tool center point P of the robot in different poses tcp It can be represented as

[0142]

[0143] In the formula R P tcpi Let P be the tool center point of the robot in the i-th pose. tcp Representation in robot coordinate system Let i be the robot's pose at the i-th time. E P tcpi Tool center point P tcp Representation in the robot's end-effector coordinate system R P ei Let represent the robot end effector in the robot coordinate system under the i-th pose.

[0144] Tool center point coordinates under different poses R P tcpi Satisfy the following equation

[0145]

[0146] Substituting the five different poses and equation (15) into the system of equations yields the following set of equations.

[0147]

[0148] beg E P tcp Best Least Squares Solution

[0149]

[0150] Finally, it should be noted that the above embodiments are only used to illustrate the method solutions of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications to the method solutions described in the foregoing embodiments, or equivalent substitutions for some or all of the method features, do not cause the essence of the corresponding method solutions to deviate from the scope of the method solutions of the embodiments of the present invention. The above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method of calibrating a robot compliant cutter based on an articulated measuring arm, characterized by: Comprise the following steps: S1, control the mechanical arm moves along the coordinate axis direction, record the measurement arm and robot related coordinates; S11. Select a fixed point on the robot end tool as the measurement point P a , S12, keep the mechanical arm pose unchanged, control the mechanical arm to move along the X, Y, Z three axes, measure P with the measuring arm a The coordinates in the measuring arm coordinate system are measured five times for each axis, and the corresponding robot end pose is recorded; for five points of an axis, the coordinates in the measuring arm coordinate system are recorded as { M P1, M P2, M P3, M P4, M P5};The robot end coordinates in the robot coordinate system are recorded as { R P1, R P2, R P3, R P4, R P5}; S2, control the mechanical arm rotates around the fixed point, record the measurement arm and robot related coordinates; The specific method of step S2 is: keeping the position of the robot end coordinate system unchanged, only changing the robot pose, measuring P a Position with the measuring arm, recording the pose of the robot end in the robot coordinate system and position R P e1 , R P e2 , R P e3 , R P e4 , R P e5} and the measuring arm reading M P a1 , M P a2 , M P a3 , M P a4 , M P a5}; S3, according to the coordinates of the first two steps to establish equation group, solve the conversion matrix by least square method; S4, use the measurement arm to calculate the calibration plane equation, control the mechanical arm to contact the plane, and solve the tool center point position according to the robot itself pose and plane equation; The specific method of step S4 is: Select a fixed, precision-machined plane as the calibration plane. Use the measuring arm to randomly select five points on the plane. The coordinates of these five points are ( M x1, M y1, M z1), ( M x2, M y2, M z2), ( M x3, M y3, M z3), ( M x4, M y4, M z4), ( M x5, M y5, M z5), the transformation matrix Multiplying these five coordinates by the coordinates of the five points yields the representations of the five points in the robot coordinate system: (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), (x4, y4, z4), and (x5, y5, z5). Calculate the planar regression equations of these five points in the robot coordinate system.

2. The robot compliant cutter calibration method based on articulated measuring arm according to claim 1, characterized in that: The specific method of step S3 is: First, the rotation matrix is calculated For any five points of an axis in the same coordinate system, the coordinate matrix formed is as follows: The covariance matrix of the above coordinate matrix is The mathematical expectation of coordinate X in formula (1) The mathematical expectation of coordinate Y The mathematical expectation of coordinate Z For two variables M and N with expectation E(M) and E(N), the covariance Cov(M, N) is Cov(M, N) = E(MN) - E(M)E(N) (6) The formula (3) (4) (5) (6) is brought into the formula (2), the covariance matrix can be calculated; then the maximum eigenvector V of the covariance matrix is calculated by using the singular value decomposition method, that is, the direction of the current coordinate axis; the representations of the three axes X, Y and Z of the robot in the robot coordinate system and the representations of the three axes X, Y and Z in the measuring arm coordinate system are established in turn R X, R Y, R Z} and in the measuring arm coordinate system{ M X, M Y, M Z}; the two groups of coordinate vectors have the following relationship: Solve formula (7) to obtain the rotation matrix: Then calculate the offset vector from the measurement arm coordinate system to the robot coordinate system; In step S2, the measurement point P a with the mechanical arm end coordinate P e There is a relationship: wherein R P ai is the representation of the measurement point P in the robot coordinate system at the i-th pose of the robot a is the representation of the measurement point P in the robot coordinate system at the i-th pose of the robot R P ei is the representation of the robot end-effector in the robot coordinate system at the i-th pose of the robot E t ae is the offset from the robot end-effector P e to the measurement point P a in the robot end-effector coordinate system is the current pose of the robot measurement point P a The relationship between the coordinates in the robot coordinate system and the coordinates in the measurement arm coordinate system is wherein To measure the representation of the offset vector of the arm coordinate system to the robot coordinate system in the robot coordinate system, equation (10) is brought into equation (9) to replace R P ai It can be obtained that Subtracting the data of pose 1 from the data of pose 2 can obtain the following relationship: Similarly, we can get Find the best least-squares solution for E t ae ​ The E t ae same procedure, the optimal least-squares solution for is calculated. The transformation matrix from the measuring arm coordinate system to the robot coordinate system is thus found 3. The robot compliant tool calibration method based on articulated measurement arm according to claim 1, characterized in that: The specific method of step S4 is: The general expression of the plane equation is: Ax+By+Cz+D=0(C≠0) Change it to the following form: Let then z=a0x+a1y+a2 At this time, the matrix form of the plane equation can be expressed as Solve the least square solution of the coefficients a0, a1, a2 to obtain From this, the plane regression equation Z=a0X+a1Y+a2 can be obtained; Since the center point of the measuring head is a length of measuring head radius away from the plane, it is necessary to translate the regression equation in the normal direction of the plane to the opposite direction by a length of measuring head radius; Let the measuring head radius be r, then the constant term coefficient a3 of the plane equation after translation is Get the equation of the calibration plane Z=a0X+a1Y+a3, which can be arranged as Z=aX+bY+C; Control the robot to make the tool center point contact any five points on the plane with different postures, record the coordinates of the robot end R P t1 , R P t2 , R P t3 , R P t4 , R P t5} and postures The tool center point P of the robot in different postures tcp Can be represented as wherein R P tcpi is the tool center point P in the i-th pose of the robot tcp in the robot coordinate system, is the i-th pose of the robot, E P tcpi tool center point P tcp in the robot end coordinate system, R P ei is the representation of the robot end in the robot coordinate system in the i-th pose. Tool center point coordinates in different poses R P tcpi Satisfy the following equation Bring five different poses and formula (15) into the equation group Find E P tcp Best least squares solution for

Images