homogenous transformation matrices
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HOMOGENOUS TRANSFORMATION MATRICES. T. Bajd and M. Mihelj. Homogenous matrix. • • • □ • • • □ • • • □ ■ ■ ■ 1. - PowerPoint PPT PresentationTRANSCRIPT
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
HOMOGENOUSTRANSFORMATION
MATRICES
T. Bajd and M. Mihelj
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
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 (■).
• • • □ • • • □ • • • □ ■ ■ ■ 1
Homogenous matrix
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
The elements of rotation matrix are direction cosines of the angles between individual axes of the coordinate frames and .
Homogenous matrix
Rotation matrix
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
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.
Rotation about axis
Homogenous matrix
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
1000
0cos0sin
0010
0sin0cos
),(
yRot
x’ y’ z’
xyz
1000
0cossin0
0sincos0
0001
),(
xRot
x’ y’ z’
xyz
x’ y’ z’
xyz
Homogenous matrix
Rotation about and axes
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
1000
100
0010
0001
),(c
czTrans
xyz
1000
0100
010
0001
),(b
byTrans
xyz
1000
0100
0010
001
),(
a
axTrans
xyz
Homogenous matrix
Translation along and axes
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Pose of frame with respect to reference frame
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
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 O1 with respect to the base block O0.
Homogenous matrix
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Homogenous matrix
Position and orientation of the second block O2 with respect to the first block O1.
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Homogenous matrix
Position and orientation of the third block O3 with respect to the second block O2.
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Position and orientation of the third block O3 with respect to the base block O0 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 O3 with respect to the base block O0 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 rotationin the first joint
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Pose and rotationin the second joint
Geometric robot model
T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010
Geometric robot model
Pose and rotationin 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