Notes originally by Laura Downs and by Alex Berg

CS184: Computing Rotations in 3D

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.

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

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

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

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

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

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