How can I convert homogeneous transformation matrix into rigid3d object?

I got a homogeneous transformation matrix from 'getTransform' function in Robotics System Toolbox in order to use it to convert point clouds in end-effector frame to base frame by 'pctransform' function.(R2022a)

I tried to convert the homogeneous transform matrix into rigid3d object as below and pass it to 'pctransform', but it didn't return the expected result.

 

transform = getTransform( robot, confg, 'sourcebody', 'targetbody');
rot=tform2rotm(transform);
trvec=tform2trvec(transform);
tform=rigid3d(rot,trvec);
pcOut=pctransform(pcIn,tform);

What's wrong in the above process?

Answer 1

[MathWorks Support Team]

In this case, you should transpose rotational matrix 'rot' as

>>tform=rigid3d(rot', trvec);

because 'getTransform' function and 'rigid3d' object implement homogeneous transformation matrix differently.

In 'getTransform' (and many engineering domain), homogeneous coordinates are expressed in column-vector form: [x; y; z; 1], and homogeneous transformation matrix G is defined as G = [Rot Tran; 0 0 0 1], being used as q = G*p (multiplied from left).

https://www.mathworks.com/help/robotics/ref/rigidbodytree.gettransform.html

On the opposite, 'rigid3d' (and some domains in CV) seems to use row-vector form coordinates: [x y z 1], and transformation matrix T means T = [Rot [0;0;0]; Tran 1] and s = r*T (multiplied from right).

https://jp.mathworks.com/help/images/ref/rigid3d.html

 

Obviously, the latter definition will be the same form as the former if transposing the entire equation, namely, s' = T' * r'.

If you create 'rigid3d' object by

>>tform=rigid3d(rot', trvec);

, transposed tform will be T' = [Rot' [0;0;0]; Tran 1]' = [Rot Tran; 0 0 0 1].