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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.

Thank you for your patience reading of my long describtion of my question.

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# Transformation matrix from global to loca frame

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