Transformation matrix from global to localframe


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Hello friends
Iam naveen and new to this forum.I have a question regarding the
transformation matrix from world frame to the base frame of a Spatial
serial manipulator. The frames are attached like this. The Z-axes are
along the link lengths. The Xi axis is oriented such that Xi = Zi X Zi+1
and Y-axis such that it completes a right-hand system. For the base frame
of manipulator the Z-axis is along the First link but the X and Y axes are
arbitrary. My world frame is at (0,0,0) at some distance and different
orientation to that of robot base frame. World frame orientation is along
unit vectors xo = [1 0 0],yo = [0 1 0] and zo = [0 0 1]. I want to know
how to calculate the Transformation matrix from world frame to manipulator
base frame.To make it clear further the First link is located at some
distance and oriented differently with respect to world frame.Iam
sucessfully able to calculate the end-effector position withrespect to
base frame.But iam not able to transform it withrespect to world frame. I
tried in many ways but could not succeed.I will describe the context of
the problem below.
Actually iam modelling the backbone atoms of a RNA molecule as a serial
chain hyper-redundant manipulator. The atoms are modelled as joints and
bond-lengths as links. The bond-lengths and bond-angles are constants and
torsion angles are the variable parameters. The frames are attached as
described in the first paragraph. The Z-axes are along the bond-lengths.
The Xi axis is oriented such that Xi = Zi X Zi+1 and Y-axis such that it
completes a right-hand system.For the first bond the Z axis is along the
bond but the X and Y axes can be taken arbitrary.
The DH parameters are defined as follows. a =0 always for this problem as
the bonds intersect(hence the Z-axis), d- distance between X-axis is here
is taken as the bond-length, theta is the torsion angle. All DH parameters
are calculated from atomic positons of atoms.
Initially iam trying to calculate the end-atom position(end-effector)
withrespect to first-atom(base-frame),for some known values of theta. This
iam sucessfully able to do. But Iam not able to calculate end-atom
position withrespect to world frame. So i want to know the transformation
matrix which relates frame attached to first atom(note that the Z-axis of
frame attached to first-atom is oriented along the bond from first-atom to
second-atom but X and Y axis are arbitrary) to the world frame.
Iam assigning frames in a similar manner as given in the lower half of
this page:
formatting link
but note that my world
frame is different from that described there.
Thank you for your patience reading of my long describtion of my question.
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