T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 HOMOGENOUS TRANSFORMATION MATRICES T. Bajd and M. Mihelj

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 □ ■ ■ ■ 1 Homogenous matrix

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Homogenous matrix Rotation matrix

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Homogenous matrix

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 x’ y’ z’ xyzxyz xyzxyz xyzxyz Homogenous matrix

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 xyzxyz xyzxyz xyzxyz Homogenous matrix

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Pose

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Displacement of a frame with respect to a relative coordinate frame The homogenous matrix H can be explained by three successive displacements of the reference frame. Displacement

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Position and orientation of the first block O 1 with respect to the base block O 0. Homogenous matrix

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Homogenous matrix Position and orientation of the second block O 2 with respect to the first block O 1.

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Homogenous matrix Position and orientation of the third block O 3 with respect to the second block O 2.

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Position and orientation of the third block O 3 with respect to the base block O 0 is obtained by successive multiplications of the three matrices. Homogenous matrix

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 The correctness of the calculated orientation and position of the third block O 3 with respect to the base block O 0 can be easily verified from the figure. Homogenous matrix

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 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. Geometric robot model

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 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. Geometric robot model

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Geometric robot model Pose and rotation in the first joint

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Pose and rotation in the second joint Geometric robot model

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Geometric robot model Pose and rotation in the third joint

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 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. Geometric robot model