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Composition of uncertain rotations

0

$begingroup$

Here, in this tutorial: http://ethaneade.com/lie.pdf, the author gives the composition of uncertain rotations for Gaussians in $SO(3)$ (Eqn. 47, Page 7). The author doesn't give the detailed derivation. However, I have some difficulty to derive this result. Here is what I'd tried,

Assume $bf{epsilon}_1$ and $bf{epsilon}_0$ are samples drawn from $left({bf{R}_1},bf{Sigma}_1right)$ and $left({bf{R}_0},bf{Sigma}_0right)$, respectively. Then the composition rotation by first transforming by sample_0 and then by sample_1 is given by

begin{align}
&exp{left({bf{epsilon}_1}_{times}right)}bf{R}_1exp{left({bf{epsilon}_0}_{times}right)}bf{R}_0 \
=& exp{left({bf{epsilon}_1}_{times}right)}exp{left(left({rm{Adj_{bf{R}_1}}bf{epsilon}_0}right)_{times}right)}bf{R}_1bf{R}_0 \
=& exp{left({bf{epsilon}_1}_{times}right)}exp{left(left({bf{R}_1bf{epsilon}_0}right)_{times}right)}bf{R}_1bf{R}_0
end{align}

If the the first two exponential items could be combined, then this looks pretty close to the final result. But I don't think I could combine those two items, since the matrices inside the exponential functions seem are not commutative. Am I wrong or am I missing something? I'd appreciate any help in getting me through this. Thank you very much.

lie-groups lie-algebras rotations

share|cite|improve this question

edited Mar 22 at 13:42

sshen

asked Mar 22 at 13:16

sshensshen

12

$endgroup$

add a comment | 

0

$begingroup$

Here, in this tutorial: http://ethaneade.com/lie.pdf, the author gives the composition of uncertain rotations for Gaussians in $SO(3)$ (Eqn. 47, Page 7). The author doesn't give the detailed derivation. However, I have some difficulty to derive this result. Here is what I'd tried,

Assume $bf{epsilon}_1$ and $bf{epsilon}_0$ are samples drawn from $left({bf{R}_1},bf{Sigma}_1right)$ and $left({bf{R}_0},bf{Sigma}_0right)$, respectively. Then the composition rotation by first transforming by sample_0 and then by sample_1 is given by

begin{align}
&exp{left({bf{epsilon}_1}_{times}right)}bf{R}_1exp{left({bf{epsilon}_0}_{times}right)}bf{R}_0 \
=& exp{left({bf{epsilon}_1}_{times}right)}exp{left(left({rm{Adj_{bf{R}_1}}bf{epsilon}_0}right)_{times}right)}bf{R}_1bf{R}_0 \
=& exp{left({bf{epsilon}_1}_{times}right)}exp{left(left({bf{R}_1bf{epsilon}_0}right)_{times}right)}bf{R}_1bf{R}_0
end{align}

If the the first two exponential items could be combined, then this looks pretty close to the final result. But I don't think I could combine those two items, since the matrices inside the exponential functions seem are not commutative. Am I wrong or am I missing something? I'd appreciate any help in getting me through this. Thank you very much.

lie-groups lie-algebras rotations

share|cite|improve this question

edited Mar 22 at 13:42

sshen

asked Mar 22 at 13:16

sshensshen

12

$endgroup$

add a comment | 

$begingroup$

Here, in this tutorial: http://ethaneade.com/lie.pdf, the author gives the composition of uncertain rotations for Gaussians in $SO(3)$ (Eqn. 47, Page 7). The author doesn't give the detailed derivation. However, I have some difficulty to derive this result. Here is what I'd tried,

Assume $bf{epsilon}_1$ and $bf{epsilon}_0$ are samples drawn from $left({bf{R}_1},bf{Sigma}_1right)$ and $left({bf{R}_0},bf{Sigma}_0right)$, respectively. Then the composition rotation by first transforming by sample_0 and then by sample_1 is given by

begin{align}
&exp{left({bf{epsilon}_1}_{times}right)}bf{R}_1exp{left({bf{epsilon}_0}_{times}right)}bf{R}_0 \
=& exp{left({bf{epsilon}_1}_{times}right)}exp{left(left({rm{Adj_{bf{R}_1}}bf{epsilon}_0}right)_{times}right)}bf{R}_1bf{R}_0 \
=& exp{left({bf{epsilon}_1}_{times}right)}exp{left(left({bf{R}_1bf{epsilon}_0}right)_{times}right)}bf{R}_1bf{R}_0
end{align}

If the the first two exponential items could be combined, then this looks pretty close to the final result. But I don't think I could combine those two items, since the matrices inside the exponential functions seem are not commutative. Am I wrong or am I missing something? I'd appreciate any help in getting me through this. Thank you very much.

lie-groups lie-algebras rotations

share|cite|improve this question

edited Mar 22 at 13:42

sshen

asked Mar 22 at 13:16

sshensshen

12

$endgroup$

share|cite|improve this question

edited Mar 22 at 13:42

sshen

asked Mar 22 at 13:16

sshensshen

12

share|cite|improve this question

edited Mar 22 at 13:42

sshen

asked Mar 22 at 13:16

sshensshen

12

asked Mar 22 at 13:16

sshensshen

12

asked Mar 22 at 13:16

sshensshen

12