Convention for the labelling of parameters of the jacobian of rigid body motion

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Assuming we have a matrix representing rigid body motion i.e. SE3 matrix as
$$
beginbmatrix
r11 & r12 & r13 & r14 \
r21 & r22 & r23 & r34 \
r31 & r32 & r33 & r44 \
0 & 0 & 0 & 1
endbmatrix
$$



Where the matrix is in (mathematical) column major (i.e. the translation is the last column). The jacobian of the matrix w.r.t a point [x,y,z] around the identity is:



$$
beginbmatrix
x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 & 0 \
0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 \
0 & 0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 \
endbmatrix
$$



My question is: why are the rows of the jacobian differentiated against r11,r21,r31,r12,..... and not r11,r12,r13,r14,....?



Does is have to do with the fact that the matrix itself is column major? I.e. it would change in row major, or is it simply arbitrary and one just has to be consistent?



Edit: The context is this paper/thesis https://vision.in.tum.de/_media/spezial/bib/kerl2012msc.pdf p27 of the thesis / p.34 overall




linear-algebra jacobian rigid-transformation




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  • I doubt there’s a universal convention. In this case, the author has chosen a convention that makes his product of matrices work with the fairly natural choices for $J_I$ and $J_pi$.
    – amd
    Jul 19 at 0:20











  • Ok thanks. I was asking because I am writing my thesis on a similar topic and I wanted to know if I could cite something on this "convention", or lack thereof
    – Marc HPunkt
    Jul 20 at 12:09










  • Both conventions are used AFAIK. See this for a brief discussion. As long as you’re consistent, you should be fine.
    – amd
    Jul 20 at 18:23














up vote
0
down vote

favorite












Assuming we have a matrix representing rigid body motion i.e. SE3 matrix as
$$
beginbmatrix
r11 & r12 & r13 & r14 \
r21 & r22 & r23 & r34 \
r31 & r32 & r33 & r44 \
0 & 0 & 0 & 1
endbmatrix
$$



Where the matrix is in (mathematical) column major (i.e. the translation is the last column). The jacobian of the matrix w.r.t a point [x,y,z] around the identity is:



$$
beginbmatrix
x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 & 0 \
0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 \
0 & 0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 \
endbmatrix
$$



My question is: why are the rows of the jacobian differentiated against r11,r21,r31,r12,..... and not r11,r12,r13,r14,....?



Does is have to do with the fact that the matrix itself is column major? I.e. it would change in row major, or is it simply arbitrary and one just has to be consistent?



Edit: The context is this paper/thesis https://vision.in.tum.de/_media/spezial/bib/kerl2012msc.pdf p27 of the thesis / p.34 overall




linear-algebra jacobian rigid-transformation




share|cite|improve this question



















  • I doubt there’s a universal convention. In this case, the author has chosen a convention that makes his product of matrices work with the fairly natural choices for $J_I$ and $J_pi$.
    – amd
    Jul 19 at 0:20











  • Ok thanks. I was asking because I am writing my thesis on a similar topic and I wanted to know if I could cite something on this "convention", or lack thereof
    – Marc HPunkt
    Jul 20 at 12:09










  • Both conventions are used AFAIK. See this for a brief discussion. As long as you’re consistent, you should be fine.
    – amd
    Jul 20 at 18:23












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Assuming we have a matrix representing rigid body motion i.e. SE3 matrix as
$$
beginbmatrix
r11 & r12 & r13 & r14 \
r21 & r22 & r23 & r34 \
r31 & r32 & r33 & r44 \
0 & 0 & 0 & 1
endbmatrix
$$



Where the matrix is in (mathematical) column major (i.e. the translation is the last column). The jacobian of the matrix w.r.t a point [x,y,z] around the identity is:



$$
beginbmatrix
x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 & 0 \
0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 \
0 & 0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 \
endbmatrix
$$



My question is: why are the rows of the jacobian differentiated against r11,r21,r31,r12,..... and not r11,r12,r13,r14,....?



Does is have to do with the fact that the matrix itself is column major? I.e. it would change in row major, or is it simply arbitrary and one just has to be consistent?



Edit: The context is this paper/thesis https://vision.in.tum.de/_media/spezial/bib/kerl2012msc.pdf p27 of the thesis / p.34 overall




linear-algebra jacobian rigid-transformation




share|cite|improve this question











Assuming we have a matrix representing rigid body motion i.e. SE3 matrix as
$$
beginbmatrix
r11 & r12 & r13 & r14 \
r21 & r22 & r23 & r34 \
r31 & r32 & r33 & r44 \
0 & 0 & 0 & 1
endbmatrix
$$



Where the matrix is in (mathematical) column major (i.e. the translation is the last column). The jacobian of the matrix w.r.t a point [x,y,z] around the identity is:



$$
beginbmatrix
x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 & 0 \
0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 & 0 \
0 & 0 & x & 0 & 0 & y & 0 & 0 & z & 0 & 0 & 1 \
endbmatrix
$$



My question is: why are the rows of the jacobian differentiated against r11,r21,r31,r12,..... and not r11,r12,r13,r14,....?



Does is have to do with the fact that the matrix itself is column major? I.e. it would change in row major, or is it simply arbitrary and one just has to be consistent?



Edit: The context is this paper/thesis https://vision.in.tum.de/_media/spezial/bib/kerl2012msc.pdf p27 of the thesis / p.34 overall





linear-algebra jacobian rigid-transformation





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asked Jul 18 at 23:34









Marc HPunkt

31




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  • I doubt there’s a universal convention. In this case, the author has chosen a convention that makes his product of matrices work with the fairly natural choices for $J_I$ and $J_pi$.
    – amd
    Jul 19 at 0:20











  • Ok thanks. I was asking because I am writing my thesis on a similar topic and I wanted to know if I could cite something on this "convention", or lack thereof
    – Marc HPunkt
    Jul 20 at 12:09










  • Both conventions are used AFAIK. See this for a brief discussion. As long as you’re consistent, you should be fine.
    – amd
    Jul 20 at 18:23
















  • I doubt there’s a universal convention. In this case, the author has chosen a convention that makes his product of matrices work with the fairly natural choices for $J_I$ and $J_pi$.
    – amd
    Jul 19 at 0:20











  • Ok thanks. I was asking because I am writing my thesis on a similar topic and I wanted to know if I could cite something on this "convention", or lack thereof
    – Marc HPunkt
    Jul 20 at 12:09










  • Both conventions are used AFAIK. See this for a brief discussion. As long as you’re consistent, you should be fine.
    – amd
    Jul 20 at 18:23















I doubt there’s a universal convention. In this case, the author has chosen a convention that makes his product of matrices work with the fairly natural choices for $J_I$ and $J_pi$.
– amd
Jul 19 at 0:20





I doubt there’s a universal convention. In this case, the author has chosen a convention that makes his product of matrices work with the fairly natural choices for $J_I$ and $J_pi$.
– amd
Jul 19 at 0:20













Ok thanks. I was asking because I am writing my thesis on a similar topic and I wanted to know if I could cite something on this "convention", or lack thereof
– Marc HPunkt
Jul 20 at 12:09




Ok thanks. I was asking because I am writing my thesis on a similar topic and I wanted to know if I could cite something on this "convention", or lack thereof
– Marc HPunkt
Jul 20 at 12:09












Both conventions are used AFAIK. See this for a brief discussion. As long as you’re consistent, you should be fine.
– amd
Jul 20 at 18:23




Both conventions are used AFAIK. See this for a brief discussion. As long as you’re consistent, you should be fine.
– amd
Jul 20 at 18:23















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