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Rodrigues' Rotation Formula

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Rodrigues' rotation formula gives an efficient method for computing the rotation matrix R in SO(3) corresponding to a rotation by an angle theta about a fixed axis specified by the unit vector omega^^=(omega_x,omega_y,omega_z) in R^3. Then R_(omega^^)(theta) is given by

R_(omega^^)(theta)=e^(omega^~theta)
(1)
=I+omega^~sintheta+omega^~^2(1-costheta)
(2)
=[costheta+omega_x^2(1-costheta) omega_xomega_y(1-costheta)-omega_zsintheta omega_ysintheta+omega_xomega_z(1-costheta); omega_zsintheta+omega_xomega_y(1-costheta) costheta+omega_y^2(1-costheta) -omega_xsintheta+omega_yomega_z(1-costheta); -omega_ysintheta+omega_xomega_z(1-costheta) omega_xsintheta+omega_yomega_z(1-costheta) costheta+omega_z^2(1-costheta)],
(3)

where I is the 3×3 identity matrix

and omega^~ denotes the antisymmetric matrix with entries

omega^~=[0 -omega_z omega_y; omega_z 0 -omega_x; -omega_y omega_x 0].
(4)

Note that the entries in this matrix are defined analogously to the differential matrix representation of the curl operator.

Note that

omega^~omega=0,
(5)

so applying the rotation matrix given by Rodrigues' formula to any point on the rotation axis returns the same point.

See also

Rotation Formula, Rotation Matrix

This entry contributed by Serge Belongie

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References

Brockett, R. W. "Robotic Manipulators and the Product of Exponentials Formula." In Mathematical Theory of Networks and Systems. Proceedings of the International Symposium Held at the Ben Gurion University of the Negev, Beer Sheva, June 20-24, 1983 (Ed. P. A. Fuhrmann). Berlin: Springer-Verlag, pp. 120-127, 1984.Murray, R. M.; Li, Z.; and Sastry, S. S. A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press, 1994.

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Rodrigues' Rotation Formula

Cite this as:

Belongie, Serge. "Rodrigues' Rotation Formula." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/RodriguesRotationFormula.html

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