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HOMOGENOUS TRANSFORMATION MATRICES

HOMOGENOUS TRANSFORMATION MATRICES. T. Bajd and M. Mihelj. Homogenous matrix. • • • □ • • • □ • • • □ ■ ■ ■ 1.

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HOMOGENOUS TRANSFORMATION MATRICES

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  1. HOMOGENOUSTRANSFORMATION MATRICES T. Bajd and M. Mihelj

  2. Homogenous matrix • • • □ • • • □ • • • □ ■ ■ ■ 1 The homogenous matrix describes either pose (orientation and position) or displacement (rotation and translation) of an object. It consists of a rotation matrix (•), translation column (□), and perspective transformation row (■).

  3. Homogenous matrix Rotation matrix The elements of rotation matrix are direction cosines of the angles between individual axes of the coordinate frames and .

  4. Homogenous matrix Rotation about axis The first three rows correspond to the and axis of the reference frame, while first three columns refer to and axis of the rotated frame. The element of the matrix is cosine of the angle between the axes given by the corresponding column and row.

  5. Homogenous matrix Rotation about andaxes x’ y’ z’ x’ y’ z’ x y z x y z x’ y’ z’ x y z

  6. Homogenous matrix Translation along andaxes x y z x y z x y z

  7. Pose From the homogenous matrix we „read“, that the axis has the same direction as axis of the reference frame, axis the same direction as axis, while axis is directed in the same way as axis. Pose of frame with respect to reference frame

  8. Displacement The homogenous matrix H can be explained by three successive displacements of the reference frame. Displacement of a frame with respect to a relative coordinate frame

  9. Homogenous matrix Position and orientation of the first block O1 with respect to the base block O0.

  10. Homogenous matrix Position and orientation of the second block O2 with respect to the first block O1.

  11. Homogenous matrix Position and orientation of the third block O3 with respect to the second block O2.

  12. Homogenous matrix Position and orientation of the third block O3 with respect to the base block O0 is obtained by successive multiplications of the three matrices.

  13. Homogenous matrix The correctness of the calculated orientation and position of the third block O3 with respect to the base block O0 can be easily verified from the figure.

  14. Geometric robot model When second block rotates around axis 1, and the third block around axis 2, while the last block is elongated along axis 3, the so called SCARA robot is obtained.

  15. Geometric robot model Because of displacements about axis 1 and 2 and along axis 3, the homogenous matrices consist of products of first matrix describing the pose of the object and second matrix describing its displacement.

  16. Geometric robot model Pose and rotationin the first joint

  17. Geometric robot model Pose and rotationin the second joint

  18. Geometric robot model Pose and rotationin the third joint

  19. Geometric robot model The geometric model of a robot describes the pose of the frame attached to the end-effector with respect to the reference frame on the robot base. It is obtained by successive multiplications of homogenous matrices.

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