Rodrigues formula is beautiful, but uneven to sine and cosine. (zz Berkeley's Page)

Posted 2020-07-29

Notes originally by Laura Downs and by Alex Berg

CS184: Computing Rotations in 3D

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Using the Rodrigues Formula to Compute Rotations

Suppose we are rotating a point, p, in space by an angle, b, (later also called theta) about an axis through the origin represented by the unit vector, a.

First, we create the matrix A which is the linear transformation that computes the cross product of the vector a with any other vector, v.

a x v =

ayvz - azvy azvx - axvz axvy - ayvx

=

0 -az ay az 0 -ax -ay ax 0

vx vy vz

= Av, with A =

0 -az ay az 0 -ax -ay ax 0

Now, the rotation matrix can be written in terms of A as

Q = e^Ab = I + A sin(b) + A² [1 - cos(b)]

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A Geometrical Explanation
Rotation as Vector Components in a 2D Subspace

Suppose we are rotating a point, p, in space by an angle, b, about an axis through the origin, represented by the unit vector, a. The component of p parallel to a, p_{par a}, will not change during the transformation. The component of p perpendicular to a, p_{per a} will rotate about the axis in the plane perpendicular to the axis the same as in 2D The vectors p_{per a} and p_biper ar of the correct length and orientation to act as the x and y vectors in this 2D rotation.
p‘ = p_{par a} + cos(b) p_{per a} + sin(b) p_biper
p‘ = (p + (a x (a x p))) + cos(b) (-(a x (a x p))) + sin(b) (a x p)
p‘ = p + sin(b) (a x p) + [1 - cos(b)] (a x (a x p))
p‘ = (I + sin(b) A + [1 - cos(b)] A²) p

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An Algebraic Explanation
Rotation as a Differential Equation

Suppose we are rotating a point, p, in space by an angle, b (called theta in the formatted equations), about an axis through the origin, represented by the unit vector, a. We will form a differential equation describing the motion of the point from time t=0 to time t=b. Let p(t} be the position of the point at time t. The velocity of the point at any time is

p‘(t) = a x p(t)

Now, if we use the matrix formula for cross products in our differential equation, we have a first order, linear system of differential equations, p‘(t) = A p(t). The solution to that system is known to be p(t) = e^Atp(0) so the location of the rotated point will be p(b) = e^Abp(0).

Taylor Expansions

Now we need to evaluate e^Ab, so we examine its Taylor expansion.

Considering how we constructed A, it is easy to verify that A^3 = -A Every additional application of A turns the plane of p_{par a}A^2 in the appropriate places, we get

Now, we recognize the Taylor expansions for sin(b) and cos(b) in the above expression and find that

eAb = I + A sin(b) + A2 [1 - cos(b)] with A = 0 -az ay az 0 -ax -ay ax 0

gives us the rotation matrix. This formula is known as Rodrigues‘ Formula.

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Useful Note Given an arbitrary rotation matrix, can we find the corresponding rotation axis vecto and the angle of rotation
Consider R=e^Ab then by some algebra based on A =- A^t we have, R-R^t = 2Acos( b ) Using this and solving for a unit axis, and an angle we can recover the axis (up to a factor of +/-1) and angle up to a factor of +/- 2pi.

References

• "A Mathematical Introduction to Robotic Manipulation", Richard M. Murray, Zexiang Li, S. Shankar Sastry, pp. 26-28

 

 

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